mechanics and dynamics of milling thin walled structures
TRANSCRIPT
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
6.1 Introduction 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
7.1 Introduction 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
8.1 Introduction 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
Chapter 1. Introduction 3
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.
Chapter 1. Introduction 4
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
Chapter 1. Introduction 5
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
Chapter 1. Introduction 6
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
Chapter 2. Literature Survey 9
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
Chapter 2. Literature Survey 10
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
Chapter 2. Literature Survey 11
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.
Chapter 2. Literature Survey 12
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
Chapter 2. Literature Survey 13
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
Chapter 2. Literature Survey 14
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
Chapter 2. Literature Survey 15
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.
Chapter 2. Literature Survey 16
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
Chapter 2. Literature Survey 17
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
Chapter 2. Literature Survey 18
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.
Chapter 2. Literature Survey 19
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.
Chapter 2. Literature Survey 20
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 23
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 24
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 25
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)
Chapter 3. Structural Modeling of the Workpiece and End Mill 26
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
following is obtained:
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 —
Chapter 3. Structural Modeling of the Workpiece and End Mill 27
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 28
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 29
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)
Chapter 3. Structural Modeling of the Workpiece and End Mill 30
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.
Chapter 3. Structural Modeling of the Workpiece and End Mill 31
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 32
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)
Chapter 3. Structural Modeling of the Workpiece and End Mill 33
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)
Chapter 3. Structural Modeling of the Workpiece and End Mill 34
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 35
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 36
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 37
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 39
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 40
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 41
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)
Chapter 3. Structural Modeling of the Workpiece and End Mill 42
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:
Chapter 3. Structural Modeling of the Workpiece and End Mill 43
[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)
Chapter 3. Structural Modeling of the Workpiece and End Mill 44
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.
Chapter 3. Structural Modeling of the Workpiece and End Mill 45
Figure 3.5: Flow chart of the developed Finite Element Program.
Chapter 3. Structural Modeling of the Workpiece and End Mill 46
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 47
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}
Chapter 3. Structural Modeling of the Workpiece and End Mill 48
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):
Chapter 3. Structural Modeling of the Workpiece and End Mill 49
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 50
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 52
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 53
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
Chapter 3. Structural Modeling of the Workpiece and End Mill 54
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).
Chapter 3. Structural Modeling of the Workpiece and End Mill 55
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
Chapter 4. Modeling of Milling Forces 58
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)
Chapter 4. Modeling of Milling Forces 59
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
Chapter 4. Modeling of Milling Forces 60
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,
Chapter 4. Modeling of Milling Forces 61
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 —
Chapter 4. Modeling of Milling Forces 62
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.
Chapter 4. Modeling of Milling Forces 63
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.
Chapter 4. Modeling of Milling Forces 64
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
Chapter 4. Modeling of Milling Forces 65
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
Chapter 4. Modeling of Milling Forces 66
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.
Chapter 4. Modeling of Milling Forces 67
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.
Chapter 4. Modeling of Milling Forces 68
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
Chapter 4. Modeling of Milling Forces 69
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
Chapter 4. Modeling of Milling Forces 70
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
Chapter 4. Modeling of Milling Forces 71
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
Chapter 4. Modeling of Milling Forces 72
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.
Chapter 4. Modeling of Milling Forces 73
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)
Chapter 4. Modeling of Milling Forces 74
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
Chapter 4. Modeling of Milling Forces 75
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)
Chapter 4. Modeling of Milling Forces 76
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
Chapter 4. Modeling of Milling Forces 77
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/’
Chapter 4. Modeling of Milling Forces 78
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)
Chapter 4. Modeling of Milling Forces 79
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
Chapter 4. Modeling of Milling Forces 80
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)
Chapter 4. Modeling of Milling Forces 82
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.
Chapter 4. Modeling of Milling Forces 83
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
Chapter 4. Modeling of Milling Forces 84
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.
Chapter 4. Modeling of Milling Forces 85
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.
Chapter 4. Modeling of Milling Forces 86
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
Chapter 4. Modeling of Milling Forces 87
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)
Chapter 4. Modeling of Milling Forces
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.
Chapter 4. Modeling of Milling Forces
()
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
Chapter 4. Modeling of Milling Forces 90
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
Chapter 4. Modeling of Milling Forces 91
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].
Chapter 4. Modeling of Milling Forces 92
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
Chapter 4. Modeling of Milling Forces 93
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.
Chapter 4. Modeling of Milling Forces 94
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
Chapter 4. Modeling of Milling Forces 95
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.
Chapter 4. Modeling of Milling Forces 97
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
Chapter 4. Modeling of Milling Forces 98
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
Chapter 4. Modeling of Milling Forces
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.
Chapter 4. Modeling of Milling Forces 100
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.
Chapter 4. Modeling of Milling Forces 101
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.
Chapter 4. Modeling of Milling Forces 102
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
Rotation Angle (deg)
270 315 360
zci)0LI
600
400
200
0
-200
-400
270 315 360
Chapter 4. Modeling of Milling Forces 103
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
Rotation Angle (deg)
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
Rotation Angle (deg)
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 106
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 107
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 108
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 109
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 110
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 111
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.
5.4 Simulation and Experimental Results
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 112
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 113
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 114
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.
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 116
1500
1000
500
0
-500
-1000
-1500
-2000
-2500 -
0 45 90 135 180 225
Rotation Angle (deg)
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
Rotation Angle (deg)
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 117
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
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 118
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
(.
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 120
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)
Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 122
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,
Chapter 6. Static Structure-Milling Process Interaction 125
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
Chapter 6. Static Structure-Milling Process Interaction 126
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
Chapter 6. Static Structure-Milling Process Interaction 127
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)
Chapter 6. Static Structure-Milling Process Interaction 128
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
Chapter 6. Static Structure-Milling Process Interaction 129
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)
Chapter 6. Static Structure-Milling Process Interaction 130
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
Chapter 6. Static Structure-Milling Process Interaction 131
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)
Chapter 6. Static Structure-Milling Process Interaction 132
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
Chapter 6. Static Structure-Milling Process Interaction 133
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
Chapter 6. Static Structure-Milling Process Interaction 134
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
Chapter 6. Static Structure-Milling Process Interaction 135
0
-100
-200
z-300
xU-
-400
-500
-600
0 90 180 270 360
Rotation Angle (deg)
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
Chapter 6. Static Structure-Milling Process Interaction 136
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)
Chapter 6. Static Structure-Milling Process Interaction 137
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
Chapter 6. Static Structure-Milling Process Interaction 138
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
Chapter 6. Static Structure-Milling Process Interaction 139
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
Chapter 6. Static Structure-Milling Process Interaction 140
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
Chapter 6. Static Structure-Milling Process Interaction 141
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.
Chapter 7. Peripheral Milling of Plates 144
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
Chapter 7. Peripheral Milling of Plates 145
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)
Chapter 7. Peripheral Milling of Plates 146
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
Chapter 7. Peripheral Milling of Plates 148
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)
Chapter 7. Peripheral Milling of Plates 149
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,
Chapter 7. Peripheral Milling of Plates 150
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
Chapter 7. Peripheral Milling of Plates 151
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
Chapter 7. Peripheral Milling of Plates 152
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
Chapter 7. Peripheral Milling of Plates 153
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.
7.4 Simulation and Experimental Results
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
Chapter 7. Peripheral Milling of Plates 154
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
Chapter 7. Peripheral Milling of Plates 155
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
Chapter 7. Peripheral Milling of Plates 156
600
500
400z
300
LL200
100
0
Rotation Angle (deg)
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
Chapter 7. Peripheral Milling of Plates 157
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)
Chapter 7. Peripheral Milling of Plates
L0c
(a),
Db -4-
(C)
c0c
C)
C.
-
Chapter 7. Peripheral Milling of Plates 160
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
Chapter 7. Peripheral Milling of Plates 161
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
Chapter 7. Peripheral Milling of Plates 162
172
170
- 168
166
164C,)
162
< 160
158160 180 200 220 240 260 280 300
Rotation Angle (deg)
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
Chapter 7. Peripheral Milling of Plates 163
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
Chapter 7. Peripheral Milling of Plates 164
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.
Chapter 7. Peripheral Milling of Plates
()
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
Chapter 7. Peripheral Milling of Plates 166
- 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-
Chapter 7. Peripheral Milling of Plates 167
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)
Chapter 7. Peripheral Milling of Plates 168
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
Chapter 7. Peripheral Milling of Plates 169
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 172
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 173
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)
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 174
Workpiece
vibration marksleft by tooth )
/
/
x
vibration marksleft by tooth (i-i)
Figure 8.1: Dynamic milling process.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 175
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 176
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 177
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 178
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))
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 179
• 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+...
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 180
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 181
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 182
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)
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 183
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 184
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)
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 185
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 186
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 187
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 188
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 189
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 190
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)
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 191
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)
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 192
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 193
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 194
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
following is obtained:
{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)
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 195
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 196
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)
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 197
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 198
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)]
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 199
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 200
(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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 201
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 202
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 203
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 204
8
7
6
12
0153045607590
Rotation Angle (deg)
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 205
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 206
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 207
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 208
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)
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 209
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,
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 210
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 211
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 212
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 213
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)
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 214
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 215
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 216
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 217
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 218
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)
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 219
Workpiece
kcx
Figure 8.9: Milling system model with two degree-of-freedom cutter and workpiece.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 220
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 221
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 223
b=R12
b=2R
Another example of a 2 DOF milling system is taken from Smith and Tlusty [70].
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 224
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 225
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 226
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 227
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)]
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 228
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 229
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 230
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,
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 231
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 232
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 233
—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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 234
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 235
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 237
•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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 238
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 239
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.
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 240
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
Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 241
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
Chapter 9. Conclusions 244
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
Chapter 9. Conclusions 245
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.
Chapter 9. Conclusions 246
— 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.
Bibliography
[1] F.W. Taylor. On the Art of Cutting Metals. Transactions of ASME, 28, 1907.
[2] M.E. Merchant. Basic Mechanics of the Metal Cutting Process. Trans. ASMEJournal of Applied Mechanics, pages A—168—175, 1944.
[3] S.A Tobias. Machine Tool Vibration. Blackie and Sons Ltd., 1965.
[4] N.N. Zorev. Metal Cutting Mechanics. Pergamon Press, 1966.
[5] F. Koenigsberger and J. Tlusty. Machine Tool Structures- Vol.1 : Stability AgainstChatter. Pergamon Press, 1967.
[6] E.J.A. Armarego and R.H. Brown. The Machining of Metals. Prentice-Hall, 1969.
[7] G. Boothroyd. Fundamentals of Machining and Machine Tools. McGraw-Hill,1975.
[8] E.M. Trent. Metal Cutting. Butterworths, 1977.
[9] M.C. Shaw. Metal Cutting Principles. Oxford University Press, 1984.
[10] P.L.B. Oxley. The Mechanics of Machining. Ellis Horwood Limited, 1989.
[11] J. Airey and C.J. Oxford. On the Art of Milling. Transactions of the ASME,43:549, 1921.
[12] F. Parsons. Power Required for Cuttillg Metal. Transactions of the ASME, 49:193—227, 1923.
[13] N.N Sawin. Theory of Milling Cutters. Mechanical Engineering, 48:1203—1209,1926.
[14] C. Salomon. Die Frasarheit. Werkststtstechnik, 20:469—474, 1926.
[15] O.W. Boston and C.E. Kraus. Elements of Millillg. Transactions of the ASME,54:74—104, October 1932.
[16] O.W. Boston and C.E. Kraus. Elements of Milling-Part ii. Transactions of theASME, 56:355—371, 1932.
247
Bibliography 248
[17] M.E. Martellotti. An Analysis of the Milling Process. Transactions of the ASME,63:677—700, 1941.
[18] M.E. Martellotti. An Analysis of the Milling Process. Part II: Down Milling. Transactions of the ASME, 67:233—251, 1945.
[19] N. Zlatin. Establisment of Production Machinability Data. Report afml-tr-75-120,Air Force Materials Lab., August 1975.
[20] A.J.P. Sabberwal. Chip Section and Cutting Force during the Milling Operation.Annals of the CIRP, 10:197—203, 1961.
[21] F. Koenigsberger and A.J.P. Sabberwal. An Investigation into the Cutting ForcePulsations During Milling Operations. International Journal of Machine Tool Design and Research, 1:15—33, 1961.
[22] J. Tlusty and P. MacNeil. Dynamics of Cutting Forces in End Milling. Annals ofthe CIRP, 24:21—25, 1975.
[23] W.A. Kline, R.E. DeVor, and W.J. Zdeblick. A Mechanistic Model for the ForceSystem in End Milling with Application to Machining Airframe Structures. InNorth American Manufacturing Research Conference Proceedings, page 297, Dearborn, MI, 1980. Society of Manufacturing Engineers. Vol. XVIII.
[24] J.W. Sutherland and R.E. DeVor. An Improved Method for Cutting Force andSurface Error Prediction in Flexible End Milling Systems. Transactions ASME,Journal of Engineering for Industry, 108:269—279, 1986.
[25] Altintas, Y. and Spence, A. End Milling Force Algorithms for CAD Systems.Annals of the CIRP, 40(1):31—34, 1991.
[26] E.J.A. Armarego and C.J. Epp. An Investigation of Zero Helix Peripheral UpMilling. International Journal of Machine Tool Design and Research, 10:273—291,1970.
[27] E.J.A. Armarego and R.C. Whitfield. Computer Based Modelling of Popular Machining Operations for Force and Power Predictions. Annals of the CIRP, 34:65—69,1985.
• [28] I. Yellowley. Observations of the Mean Values of Forces, Torque and Specific Powerin the Peripheral Milling Process. International Journal of Machine Tool Designand Research, 25(4) :337—346, 1985.
[29] R.H. Brown and E.J.A. Armarego. Oblique Machining With a Single Cutting Edge.International Journal of Machine Tool Design and Research, 4:9—25, 1964.
Bibliography 249
[30] E.J.A. Armarego and N.P. Deshpande. Computerized Predictive Cutting Modelsfor Forces in End-Milling Including Eccentricity Effects. Annals of the CIRP,38(1):45—49, 1989.
[31] E. Budak and Y. Altintas. Prediction of Milling Force Coefficients from OrthogonalCutting Data. In K.F. Ehmann, editor, Manufacturing Science and Engineering,pages 453—459. 1993 ASME Winter Annual Meeting, New Orleans, USA, 1993.PED-Vol. 64.
[32] S. Smith and J. Tlusty. An Overview of Modelling and Simulation of the MillingProcess. Trans. ASME Journal of Engineering for Industry, 113(2):169—175, 1991.
[33] W.P. Wang. Solid Modeling for Optimizing Metal Removal of Three-dimensionalNC End Milling. Journal of Manufacturing Systems, 7(1):57—65, 1988.
[34] W.A. Kline, R.E. DeVor, and R. Lindberg. The Prediction of Cutting Forces inEnd Milling with Application to Cornering Cuts. International Journal of MachineTool Design and Research, 22(1):7—22, 1982.
[35] W.A. Kline, R.E. DeVor, and l.A. Shareef. The Prediction of Surface Accuracyin End Milling. Trans. ASME Journal of Engineering for Industry, 104:272—278,1982.
[36] E.J.A. Armarego and N.P. Deshpande. Computerized End-Milling Force Predictions with Cutting Models Allowing Eccentricity and Cutter Deflections. Annalsof the CIRP, 40(1):25—29, 1991.
[37] D. Montgomery and Altintas Y. Mechanism of Cutting Force and Surface Generation in Dynamic Milling. Trans. ASME Journal of Engineering for Industry,113:160—168, 1991.
[38] F. Ismail, M.A. Elbestawi, R. Du, and K. Urbasik. Generation of Milled SurfacesIncluding Tool Dynamics and Wear. Trans. ASME Journal of Engineering forIndustry, 115:245—252, 1993.
[39] J. Tlusty and M.A.A Elbestawi. Constraints in Adaptive Control with FlexibleEnd Mills. Annals of the CIRP, 28:253—255, 1979.
[40] J. Oh. Model Reference Adaptive Control of the Milling Process. PhD thesis,University of California, Berkeley, 1985.
[41] L.K. Lauderbaugh and A.G. Ulsoy. Dynamic Modeling for Control of the MillingProcess. Trans. ASME Journal of Engineering for Industry, 110:367—375, November 1988.
Bibliography 250
[42] B.K. Fussel. Modeling and Adaptive Force Control of End Milling Operations. PhDthesis, The Ohio State University, 1987.
[43] A.D. Spence and Y. Altintas. CAD Assisted Adaptive Control of the Milling Process. In ASME 1989 Winter Annual Meeting—Control Issues in ManufacturingProcesses, pages 33—43, San Francisco, December 1989. ASME. DSC-Vol. 18.
[44] Y. Altintas and J. Peng. Design and Analysis of a Modular CNC System forMachining Control and Monitoring. Journal of Engineering for Industry, 1988.
[45] Y. Altintas and I. Yellowley. In-Process Detection of Tool Failure in Milling UsingCutting Force Models. Trans. ASME Journal of Engineering for Industry, 111:149—157, 1989.
[46] P. Albrecht. Self-Induced Vibrations in Metal Cutting. Trans. ASME Journal ofEngineering for Industry, 84:405, 1962.
[47] R.L. Kegg. Cutting Dynamics in Machine Tool Chatter. Trans. ASME Journal ofEngineering for Industry, pages 464—470, November 1965.
[48] P.W. Wallace and C. Andrew. Machining Forces: Some Effects of Tool Vibration.Journal of Mechanical Engineering Science, 17, 1965.
[49] P.W. Wallace and C. Andrew. Machining Forces: Some Effects of Removing aWavy Surface. Journal of Mechanical Engineering Science, 8(2):129—139, 1966.
[50] M.K. Das and S.A. Tobias. The Relation Between Static and Dynamic Cutting ofMetals. International Journal of Machine Tool Design and Research, 7, 1967.
[51] W.A. Knight. Some Observations on the Vibratory Metal Cutting Process Em—ploying High Speed Photography. International Journal of Machine Tool Designand Research, 10:221—247, 1970.
[52] J. Tlusty and S.B. Rao. Verification and Analysis of Some Dynamic Cutting ForceCoefficients Data. NAMRC, U. of Florida, pages 420—426, 1978.
[53] K. Srinivasan and C.L. Nachtigal. Investigation of the Cutting Process Dynamics inTurning Operations. Trans. ASME Journal of Engineering for Industry, 100:323—331, August 1978.
[54] J. Tlusty. Analysis of the State of Research in Cutting Dynamics. Annals of theCIRP, 27/2:583—589, 1978.
[55) J.F. Sarnicola and G. Boothroyd. Machine Tool Chatter, Factors Which InfluenceCutting Forces During Wave Removing. Proc. 1st NAMR Conference, 1973.
Bibliography 251
[56] J.F. Sarnicola and G. Boothroyd. Machine Tool Chatter, Effect of Work SurfaceSlope on Shear Angle During Wave Generation. Proc. 2nd NAMR Conference,1974.
[57] M.M. Nigm and M.M. Sadek. Experimental Investigation of the Characteristics ofthe Dynamic Cutting Process. Trans. ASME Journal of Engineering for Industry,pages 410—418, May 1977.
[58] T.R. Sisson and R.L. Kegg. An Explanation of Low Speed Chatter Effects. Trans.ASME Journal of Engineering for Industry, 91:951—955, November 1969.
[59] D.W. Wu. A New Approach of Formulating the Transfer Function for DynamicCutting Process. Trans. ASME Journal of Engineering for Industry, 111:37—47,1989.
[60] H.J.J Kals. On the Calculation of Stability Charts on the Basis of the Dampingand Stiffness of the Cutting Process. Annals of the CIRP, 19:297, 1971.
[61] J. Peters, P. Vanherck, and H. Van Brussel. The Measurement of the DynamicCutting Force Coefficient. Annals of the GIRP, 21/2, 1972.
[62] B.S. Goel. Measurement of Dynamic Cutting Force Coefficients. PhD thesis, MciViaster University, September 1967.
[63] S.B. Rao. Analysis of the Dynamic Cutting Force Coefficient. Master’s thesis,McMaster University, 1977.
[64] K. Srinivasan and C.L. Nachtigal. Identification of Machining System Dynamics byEquation Error Minimization. Trans. ASME Journal of Engineering for Industry,100:332—339, 1978.
[65] 0. Heczko. New Method for Testing the Dynamic Cutting Force Coefficients.Master’s thesis, McMaster University, July 1980.
[66] P.E. Gygax. Dynamic of Single-Tooth Milling. Annals of the CIRP, 28/1:65—70,1979.
[67] I. Yellowley. A Note on the Significance of the Quasi-Mean Resultant Force andthe Modeling of Instantaneous Torque and Forces in Peripheral Milling Operations.Trans. ASME Journal of Engineering for Industry, 110:300—303, 1988.
[68] P. Doolan, M.S. Phadke, and S.M. Wu. Computer Design of a Minimum Vibration Face Milling Cutter Using an Improved Cutting Force Model. Trans. ASMEJournal of Engineering for Industry, pages 807—8 10, August 1976.
Bibliography 252
[69] J. Tiusty. Dynamics of High-Speed Milling. Trans. ASME Journal of Engineeringfor Industry, 108(2):59—67, May 1986.
[70] S. Smith and J. Tiusty. Update on High Speed Milling Dynamics. Trans. ASME,Journal of Engineering Industry, 112:142—149, 1990.
[71] R.N Arnold. The Mechanism of Tool Vibration in the Cutting of Steel. Proc.Institution of Mechanical Engineers, 54:261—284, 1946.
[72] R.S. Hahn. Metal Cutting Chatter and its Elimination. Trans. ASME Journal ofEngineering for Industry, 75:1073—1080, 1953.
[73] J. Tiusty and M. Polacek. The Stability of Machine Tools Against Self Excited Vibrations in Machining. International Research in Production Engineering, ASME,pages 465—474, 1963.
[74] S.A. Tobias and W. Fishwick. A Theory of Regenerative Chatter. The Engineer -
London, 1958.
[75] J. Tlusty and F. Ismail. Basic Nonlinearity in Machinig Chatter. Annals of theCIRP, 30:21—25, 1981.
[76] J. Tlusty and F. Ismail. Special Aspects of Chatter in Milling. Trans. ASME,Journal of Vibration, Stress and Reliability in Design, 105:24—32, 1983.
[77] J. Tiusty. Machine Dynamics. Handbook of High Speed Machining Technology,pages 49—153, 1985.
[78] R. Sridhar, R.E. Hohn, and G.W. Long. General Formulation of the Milling ProcessEquation. Trans. ASME Journal of Engineering for Industry, pages 317—324, May1968.
[79] R.E. Hohn, R. Sridhar, and G.W. Long. A Stability Algorithm for a Special Caseof the Milling Process. Trans. ASME Journal of Engineering for Industry, pages325—329, May 1968.
[80] R. Sridhar, R.E. Hohn, and G.W. Long. A Stability Algorithm for the GeneralMilling Process. Trans. ASME Journal of Engineering for Industry, pages 330—334, May 1968.
[81] I. Minis and T. Yanushevsky. A New Theoretical Approach for the Predictionof Machine Tool Chatter in Milling. Trans. ASME Journal of Engineering forIndustry, 115:1—8, 1993.
• Bibliography 253
[82] I. Minis, T. Yanushevsky, R. Tembo, and R. Hocken. Analysis of Linear andNonlinear Chatter in Milling. Annals of the CIRP, 39:459—462, 1990.
[83] J. Slavicek. The Effect of Irregular Tooth Pitch on Stability in Milling. In Proceeding of the 6th MTDR Conference. Pergamon Press, London, 1965.
[84] P. Vanherck. Increasing Milling Machine Productivity by Use of Cutter with Non-Constant Cutting-Edge Pitch. In Proc. Adv. MTDR Conf., volume 8, pages 947—960, 1967.
[85] K. Jemielniak and A. Widota. Suppression of Self-Excited Vibration by the Spindle Speed Variation Method. International Journal of II/Iachine Tool Design andResearch, 23(3):207—214, 1984.
[86] Y. Altintas and P. Chan. In-Process Detection and Supression of Chatter in Milling.International Journal of Machine Tool Design and Research, 32:329—347, 1992.
[87] R. Radulescu, S.G. Kapoor, and R.E. DeVor. An Investigation of Variable SpindleSpeed Face Milling for Tool-Work Structures with Complex Dynamics, part i andii. In Manufacturing Science and Engineering (Editor: K.F. Ehmann), pages 603—629, New Orleans, USA, 1993. ASME 1993 Winter Annual Meeting.
[88] S. Smith and J. Tlusty. Update in High Speed Milling Dynamics. ASME 1987• Winter Annual Meeting, 25:153—165, 1987.
[89] C. Nachtigal. Design of a Force Feedback Chatter Control System. Trans. ASMEJournal of Dynamic Systems, Measurement and Control, pages 5—10, March 1972.
[90] K. Srinivasan and C.L. Nachtigal. Analysis and Desing of Machine Tool ChatterControl Systems Using the Regeneration Specturum. Trans. ASME Journal ofDynamic Systems, Measurement and Control, 100:191—200, 1978.
[91j M. Shiraishi and E. Kume. Suppression of Machining Chatter by Controller. Annals of the CIRP, 37:369—372, 1988.
[92] C.R. Liu and T.M. Liu. Automated Chatter Suppression by Tool Geometry Control. Trans. ASME Journal of Engineering for Industry, pages 95—98, 1985.
[93] K.J. Liu and K.E. Rouch. Optimal Passive Vibration Control of Cutting ProcessStability in Milling. Journal of Materials Processing Technology, 28:285—294, 1991.
[94] H. Opitz and F. Bernardi. Investigation and Calculation of the Chatter Behaviorof Lathes and Milling Machines. Annals of the CIRP, 18:335—343, 1970.
[95] J Tlusty. High-Speed Machining. Annals of the CIRP, 42/2, 1993.
Bibliography 254
[96] N. Wood. 30000 RPM Machine Cuts Precision Radar Wave Guides Fast Accurately. CNC/WEST, pages 101—104, February 1989.
[97] D. Montgomery. Milling of Flexible Structures. Master’s thesis, University ofBritish Columbia, 1990.
[98] Y. Altintas, D. Montgomery, and E. Budak. Dynamic Peripheral Milling of FlexibleStructures. Trans. ASME Journal of Engineering for Industry, 114(2), 1992.
[99] R. Sagherian and M.A. Elbestawi. A simulation System for Improving MachiningAccuracy in Milling. Computers in Industry, 14:293—305, 1990.
[100] M. Anjanappa, D.K. Anand, and J.A. Kirk. Identification and Optimal Control ofThin Rib Machining. ASME Winter Annual Meeting, 6:272—278, 1987.
[101] S.P. Timoshenko and J.M. Gere. Theory of Elastic Stability. McGraw-Hill BookCompany, Inc., 1961.
[102] S.P. Timoshenko and S. Woinowsky-Krieger. Theory of Plates and Shells. McGraw-Hill Book Company, Inc., 1959.
[103] M. Olson. Finite element method-lecture notes. University of British Columbia.
[104] A.W. Leissa. Vibration of Plates. NASA, 1969.
[105] R.D. Cook, D.S. Malkus, and M.E. Plesha. Concepts and Applications of FiniteElements Analysis, 3rd Edition. John Wiley and Sons, 1989.
[106] E. Budak. The Computer Programs Developed for the Structural Modeling of Cantilever Plates. Internal report-manufacturing automation laboratuary, Departmentof Mechanical Engineering, UBC, 1994.
[107] W.H. Press, Flannery B.P., S.A. Teukolsky, and W.T. Vetterling. Numerical Recipies in C. Cambridge University Press, 1988.
[108] G. Aksu. Effect of transverse shear and rotary inertia on vibrations of steppedthickness plates. Trans. ASME Journal of Vibration and Acoustics, 112:40—44,1990.
[109] T. Irie, G. Yamada, and H. Ikari. Natural frequencies of stepped thickness rectangular plates. International Journal of Mechanical Sciences, 22:767—777, 1980.
[110] I. Chopra. Vibration of stepped thickness plates. International Journal of Mechanical Sciences, 14:337—344, 1974.
Bibliography 255
[111] P. Paramasivam and J.K.S. Rao. Free vibrations of rectangular plates of abruptlyvarying stiffness. International Journal of Mechanical Sciences, 11:885—895, 1969.
[112] P.A.A. Laura and C Filipich. Fundamental frequency of vibration of stepped thickness plates. Journal of Sound and Vibration, 50:157—158, 1977.
[113] S. Timoshenko and G.H. MacCullough. Elements of Strength of Materials. D. VanNostrand Inc., 1940. 2nd Edition.
[114] L. Kops and D.T. Vo. Determination of the Equivalent Diameter of an End MillBased on its Compliance. Annals of the CIRP, 39(1):93—96, 1990.
[115] T. Justine and A. Krishnan. Effect of Support Flexibility on Fundamental Frequency of Beams. Journal of Sound and Vibration, 68(2):310—312, 1980.
[116] H. Abramovich and 0. Hamburger. Vibration of a Uniform Cantilever TimoshenkoBeam with Translational and Rotational Springs and with a Tip Mass. Journal ofSound and Vibration, 154(1):67—80, 1992.
[117] S.I. Alvarez, G.M.F. Iglesias, and P.A.A. Laura. Vibrations of an Elastically Restrained, Non-uniform Beam with Translational and Rotational Springs, and witha Tip Mass. Journal of Sound and Vibration, 120(3):465—471, 1988.
[118] M.J. Maurizi, D.V. Bambill De Rossit, and P.A.A. Laura. Free and Forced Vibrations of Beams Elastically Restrained Against Translation and Rotation at theEnds. Journal of Sound and Vibration, 120(3):626—630, 1988.
[119] B.A.H. Abbas. Vibrations of Timoshenko Beams with Elastically Restrained Ends.Journal of Sound and Vibration, 97(4):541—548, 1984.
[120] B.A.H. Abbas and H. Irretier. Experimental and Theoretical Investigations of theEffect of Root Flexibility on the Vibration Characteristics of Cantilever Beams.Journal of Sound and Vibration, 130(3):353—362, 1989.
[121] D. Afolabi. Natural Frequencies of Cantilever Blades with Resilent Roots. Journalof Sound and Vibration, 110(3):429—441, 1986.
[122] F. Ismail. Identification, Modelling, and IViodification of Mechanical StructuresFrom Modal Analysis Testing. PhD thesis, McMaster University, 1982.
[123] J. Tiusty and T Moriwaki. Experimental and Computational Identification ofDynamic Structural Methods. cirp, 25:497—503, 1976.
[124] J. Tlusty and F. Ismail. Dynamic Structural Identification Tasks and Methods.Annals of the CIRP, 29, 1980.
Bibliography 256
[125] E.J.A. Armarego and N.P. Deshpande. Force Prediction IViodels and cad/cam Software for Helical Tooth Milling Processes. i. Basic Approach and Cutting Analyses.International Journal of Production Research, 31(8):1991—2009, 1993.
[126] S. Smith and J. Tlusty. Modelling and Simulatioll of the Milling Process. 1988ASME Winter Annual Meeting Publication, 33:17—26, 1988.
[127] M. Masuko. Fundamental Researches on the Metal Cutting (l2nd Report)-ForceActing on a Cutting Edge and Its Experimental Discussion. Trans. Japanese Society of Mechanical Engineers, 22:371—377, 1956.
[128] P. Albrecht. The Ploughing Process in Metal Cutting. Trans. ASME Journal ofEngineering for Industry, pages 348—358, November 1960.
[129] E.J.A. Armarego and C.J Epp. An Investigation of Zero Helix Peripheral Up-Milling. International Journal of Machine Tool Design and Research, 10:273—291,1970.
[130] R.C. Whitfield. A Mechanics of Cutting Approach for the Prediction of Forcesand Power in Some Commercial Machining Operations. PhD thesis, University ofMelbourne, 1986.
[131] E.H. Lee and B.W. Shaffer. Theory of Plasticity Applied to the Problem of Machining. Journal of Applied Mechanics, 18:405—413, 1951.
[132] W.B. Palmer and P.L.B. Oxley. Proc. Instn. Mechanical Engineers, 1959.
[133] M.C. Shaw, N.H. Cook, and I. Finnie. The Shear Angle Relationship in MetalCutting. Transactions of the ASME, 175:273, 1953.
[134] P.C. Pandey and H.S. Shan. Analysis of Cutting Forces in Peripheral and FaceMilling Operations. International Journal of Production Research, 10(4):379—391,1972.
[135] C.J. Epp. Mechanics of the Milling Process. PhD thesis, University of Melbourne,1973.
[136] E.J.A. Armarego, D. Pramanik, A.J.R. Smith, and Whitfield R.C. Forces andPower in Drilling-Computer Aided Predictions. Journal of Engineering Production,6:149—174, 1983.
[137] G.V. Stabler. Fundamental Geometry of Cutting Tools. Proceedings of the Institution of Mechanical Engineers, pages 14—26, 1951.
Bibliography 257
[138] M.C. Shaw, N.H. Cook, and P.A. Smith. The Mechanics of Three DimensionalGutting Operations. Trans. ASME Journal of Applied Mechanics, 74:1055—1064,1952.
[139] G.V. Stabler. The Chip Flow Law and Its Consequences. Advances in MachineTool Design and Research, pages 243—251, 1964.
[140] J.K. Russel and R.H. Brown. The Measurement of Chip Flow Direction. International Journal of Machine Tool Design and Research, 6:129—138, 1966.
[141] E. Usui, A. Hirota, and M. Masuko. Analytical Prediction of Three DimensionalCutting Process - part 1. Trans. ASME Journal of Engineering for Industry,100:222—228, 1978.
[142] R.C. Coiwell. Predicting the Angle of Chip Flow for Single Point Gutting Tools.Trans. ASME, 76:199—204, 1954.
[143] A.K. Pal and F. Koenigsberger. Some Aspects of the Oblique Cutting Process.International Journal of Machine Tool Design and Research, 8:45—57, 1968.
[144] G.C.I. Lin and P.L.B. Oxley. Mechanics of Oblique Machining: Predicting ChipGeometry and Gutting Forces from Work Material Properties and Cutting Conditions. Proceedings of the Institution of Mechanical Engineers, 186:813—820, 1972.
[145] M. Yang and H. Park. The Prediction of Cutting Force in Ball-End Milling. International Journal of Machine Tool Design and Research, 31(1):45—54, 1991.
[146] M. Yang and H. Park. Reduction of Machining Errors by Adjustment of Feedratesin the Ball-End Milling Process. International Journal of Production Research,31(3):665—689, 1993.
[147] G. Yucesan and Y. Altintas. Mechanics of Ball End Milling Process. In K.F.Ehmann, editor, Manufacturing Science and Engineering, pages 543—551. ASME1993 Winter Annual Meeting, New Orleans, USA, 1993. PED-Vol.64.
[148] C. Sim and M. Yang. The Prediction of the Cutting Forces in Ball-End Milling witha Flexible Cutter. International Journal of Machine Tool Design and Research,33(2):267—284, 1993.
[149] E.M. Lim, Feng, H.Y., and C.H. Menq. The Prediction of Dimensional Errorsfor Machining Sculptured Surfaces Using Ball-End Milling. In K.F. Ehmann, editor, Manufacturing Science and Engineering, pages 149—156. ASME 1993 WinterAnnual Meeting, New Orleans, USA, 1993. PED-VoL64.
Bibliography 258
[150] A.R. Machado and J. Walibank. Machining of Titanium and Its Alloys-A Review. Proceedings of the Institution of Mechanical Engineers-Part B Journal ofEngineering Manufacture, 204(B 1) :53—60, 1990.
[151] H.E. Chandler. Machining of Reactive Materials. Metals Handbook (ASM International), 9th Edition, 16:844—850, 1989.
[152] M.E. Merchant. Mechanics of the Cutting Process. Journal of Applied Physics,16:267— and 318—, 1945.
[153] B. Mills and A.H. Redford. Machinability of Engineering Materials. Applied Science Publishers, 1983.
[154] R.I. King. Handbook of High Speed Machining Technology. Chapman and Hall,1985.
[155] M. Oguri, II. Fujii, K. Yamaguchi, and S. Kato. On the Transient Cutting Mechanics at the Initial Stage of Peripheral Milling Process. Bulletin of JSME, 19:61—70,1976.
[156] F. Koenigsberger and J. Tlusty. Machine Tool Structures, volume 1. PergamonPress, 1970.
[157] M. Tomizuka, J.H. Oh, and D.A. Dornfield. Model Reference Adaptive Control ofthe Milling Process. In D.E. Hardt and W.J. Book, editors, Control of Manufacturing Processes and Robotics Systems, pages 55—63. ASME, 1983.
[158] F.M. Kolarits and W.R. DeVries. A Mechanistic Dynamic Model of End Milling forProcess Controller Simulation. Trans. ASME Journal of Engineering for Industry,113(2):176—183, 1991.
[159] W.A. Kline, R.E. DeVor, and l.A. Shareef. The Effect of Runout on Cutting Geometry and Forces in End Milling. International Journal of Machine Tool Designand Research, 23:123—140, 1983.
[160] 5. Smith and J. Tlusty. Efficient Simulation Programs for Chatter in Milling.Annals of the CIRP, 42/1:463—466, 1993.
[161] Y.Y. Lee and W.F. Lu. An Improved Method For Dynamic Behaviors in EndMilling Process. In The Physics of Machining Process, Anaheim, USA, 1992. ASMEWinter Annual Meeting.
[162] M. Weck, Y. Altintas, and C. Beer. Cad Assisted Chatter Free Nc Tool PathGeneration in Milling. International Journal of Machine Tool Design and Research.(accepted for publication, 1993).
Bibliography 259
[163] H. Opitz. Chatter Behavior of Heavy Machine Tools. Quarterly technical report no.2 af 61 (052)-916, Research and Technology Division, Wright-PattersonAir Force Base, Ohio, 1968.
[164) A.C. Lee and C.S. Liu. Analysis of Chatter Vibration in the End Milling Process.International Journal of Machine Tool Design and Research, 31(4):471—479, 1991.
[165] A.C. Lee, C.S. Liu, and S.T Chiang. Analysis of Chatter Vibration in a CutterWorkpiece System. International Journal of Machine Tool Design and Research,31(2):221—234, 1991.
[166] L.A. Zadeh and C.A. Desoer. Linear System Theory. McGraw-Hill, N.Y., 1963.
[167] J. Thisty and M. Polacek. Experience with Analyzing Stability of Machine ToolsAgainst Chatter. Proceedings of the 9th International MTDR Conference, pages521—571, 1968.
[168] Y.N. Sankin. The Stability of Milling Machines During Cutting. Soviet EngineeringResearch, 4(4):40—43, 1984.
[169] 5. Smith. Automatic Selection of the Optimum Spindle Speed in High Speed Milling.PhD thesis, University of Florida, 1987.
[170) W. Magnus and S. Winkler. Hill’s Equation. John Wiley and Sons, N.Y., 1966.
[171] L.A. Pipes. Matrix Solution of Equations of the Mathieu-hill Type. Journal ofApplied Physics, 24(7) :902—910, 1953.
[172] G.W. Hill. On the Part of the Motion of the Lunar Perigee Which is a Functionof the Mean Motions of the Sun and Moon. Ada Mathematica, 8:1—36, 1886.
[173] N.W. McLachlan. Theory and Application of Mathieu Functions. Oxford University Press (Reprinted by Dover, N.Y.), 1964.
[174] J.A. Richards. Modeling Parametric Process-A Tutorial Review. Proceedings ofthe IEEE, 65:1549—1557, November 1977.
[175] W.W. Cooley, R.N. Clark, and R.C. Buckner. Stability of Linear Systems Havinga Time Variable Parameter. IEEE Transactions on Automatic Control, AC-9:426—434, 1964.
[176] E.A. Coddington and N. Levinson. Theory of Ordinary Differential Equations.McGraw-Hill, N.Y., 1955.
[177] H. Dangelo. Linear Time Varying Systems: Analysis and Synthesis. Allyn andBacon, Boston, 1970.
Bibliography 260
[178] N.P. Erugin. Linear Systems of Ordinary Differential Equations with Periodic andQuasiperiodic Coefficients. Academic Press, N.Y., 1966.
[179] V.A. Yakubovitch and Starzhinskii. Linear Differential Equations with PeriodicCoefficients. John Wiley and Sons, N.Y., 1975.
[180] S.R. Hall. Generalized Nyquist Stability Criterion for Linear Time Periodic Systems. In Proceedings of the 1990 Automatic Control Conference, San Diego, CA,volume 2, pages 1518—1525, 1990.
[181] L. Meirovitch. Elements of Vibration Analysis. McGraw-Hill Book Company, 1986.
[182] R. Bellman and K.L. Cooke. Differential Difference Equations. Academic Press,N.Y., 1963.
[183] K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Press, Netherlands, 1992.
[184] N. Minorsky. Self Excited Oscillations in Dynamical Systems Possesing RetardedActions. Journal of Applied Mechanics, 64:A65—71, 1942.
[185] S. Sherman. A Note on Stability Calculations and Time Lag. Quarterly AppliedMathematics, 5:92—97, 1947.
[186] H.I. Ansoff and J.A. Krumhansl. A General Stability Criterion for Linear Oscillating Systems With Constant Time Lag. Quarterly Applied Mathematics, 6:337—341,1948.
[187] W. Hahn. On Difference-Differential Equations with Periodic Coefficients. Journalof Mathematical Analysis and Applications, 3:70—101, 1961.
[188] K.L. Cooke and Z. Grossman. Discrete Delay, Distributed Delay and StabilitySwitches. Journal of Mathematical Analysis and Applications, 86:592—627, 1982.
[189] N. Olgac and Y. Wang. On the Stability of Linear Dynamic Systems with Unrelated Time Delays. ASME International Computers in Engineering Conference,Anaheim, USA, July 1989.
[190] L.A. Zadeh. Frequency Analysis of Variable Networks. Proceedings of IRE, 38:291—299, 1950.
[191] E.N. Rozenvasser. Computation and Transformation of of Transfer Functions ofLinear Periodic Systems. Automation and Remote Control, 33(1):221—227, 1972.
[192] W. Kaplan. Ordinary Differential Equations. Addison Wesley Publishing Co.,Mass., 1958.
Bibliography 261
[193] I. Minis, E.B. Magrab, and 1.0 Pandelidis. Improved Methods for the Predictionof Chatter in Turning, part 3:A Generalized Linear Theory. Trans. ASME Journalof Engineering for Industry, 112:28—35, 1990.
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.