Stra ategies for Reducin ng Vibrat tions dur

Stra ategies for Reducin ng Vibrat tions dur
Straategies for Reducin
ng Vibrattions durring Milliing of
Thin-walled Com
mponentts
J. Bertil
B
Waanner
Licentiate Thhesis
Stock
kholm, Swedeen, 2012
©J. Bertil Wanner, 2012
TRITA IIP- 12- 03
ISSN 1650-1888
ISBN 978-91-7501-322-0
KTH School of Industrial Engineering and Management
Royal Institute of Technology
SE-100 44 Stockholm
Sweden
Stockholm, Sweden, November 2012
Abstract
Factors such as environmental requirements and fuel efficiency have pushed
aerospace industry to develop reduced-weight engine designs and thereby
light-weight and thin-walled components. As component wall thickness gets
thinner and the mechanical structures weaker, the structure becomes more
sensitive for vibrations during milling operations. Demands on cost
efficiency increase and new ways of improving milling operations must
follow.
Historically, there have been two “schools” explaining vibrations in milling.
One states that the entry angle in which the cutting insert hits the work
piece is of greater importance than the exit angle. The other states that the
way the cutter leaves the work piece is of greater importance than the cutter
entry. In an effort to shed some light over this issue, a substantial amount of
experiments were conducted. Evaluations were carried out using different
tools, different tool-to-workpiece offset positions, and varying workpiece
wall overhang. The resultant force, the force components, and system
vibrations have been analyzed.
The first part of this work shows the differences in force behavior for three
tool-to-workpiece geometries while varying the wall overhang of the
workpiece. The second part studies the force behavior during the exit phase
for five different tool-to-workpiece offset positions while the overhang is
held constant. The workpiece alloy throughout this work is Inconel 718.
As a result of the project a spread sheet milling stability prediction model is
developed and presented. It is based on available research in chatter theory
and predicts the stability for a given set of variable input parameters.
Keywords
Milling, vibrations, chatter, stability, prediction, thin-wall, Inconel 718.
i
Acknowledgments
I would like to express deep appreciation and gratitude to Prof. Mihai
Nicolescu who has served as examiner and advisor for this work.
I would like to thank Prof. Lars Pejryd and Assoc. Prof. Tomas Beno who have
made this project possible and provided guidance and advice throughout.
Dr. Mahdi Eynian has made himself available for many vital discussions and
has assisted with mathematical issues and complex drawings and graphs. He
has also performed the simulations presented in the published papers.
Ulf Hulling and Edisa Sidzak have provided their time and expertise at the
Production Technology Center to make the experiments possible during this
project.
Seco Tools is acknowledged for providing specially designed tools and inserts.
This work is based on the research project “Vibrations during Milling of Thinwalled Aerospace Components” and supported by a grant from the Vinnova
NFFP program and Volvo Aero. This support is gratefully acknowledged.
The experiments were carried out at the Production Technology Center (PTC)
at Trollhättan, Sweden.
J. Bertil Wanner
Stockholm, November, 2012
iii
Publications
The following papers have been published as a result of this thesis:
Paper 1
B. Wanner, M. Eynian, T. Beno, L. Pejryd. Process Stability
Strategies in Milling of Inconel 718. 4th Manufacturing
Engineering Society International Conference (MESIC), Cadiz,
Spain, 2011. Published in the American Institute of Physics
Conference Proceedings 1431, 465-472 (2012).
Paper 2
B. Wanner, M. Eynian, T. Beno, L. Pejryd. Milling Strategies for
Thin-walled Components. Journal of Advanced Materials
Research special edition “Advances in Materials Processing
Technologies” Vol. 498, pp. 177-182 (2012).
Paper 3
B. Wanner, M. Eynian, T. Beno, L. Pejryd. Cutter Exit Effects
during Milling of Thin-walled Inconel 718. Journal of Advanced
Materials Research special edition “Mechatronic Systems and
Materials Application” Vol. 590, pp. 297-308 (2012).
v
Nomenclature
alim
Critical depth of cut [mm]
ap
Axial depth of cut [mm]
ar
Radial depth of cut [mm]
c
damping coefficient [Ns/m]
f
Feed rate [mm/revolution]
fz
Feed per tooth [mm/tooth]
Fp
Radial force, passive force (Fx) [N]
F
Resultant cutting force [N]
Ff
Axial force, feed force, (Fz) [N]
l
Length [mm]
H
Uncut chip thickness [mm]
k
Spring constant or stiffness [N/m]
kc
Specific cutting force [N/mm2]
n
Spindle rotational speed [rpm]
r
vc
Nose radius [mm]
vch
Chip speed [m/min]
m
Mass [kg]
Kt
Tangential cutting constant
Kr
Radial cutting constant
Kf
Cutting coefficient in the feed direction
Ω
Angular speed [rad/sec.]
Ф
Immersion angle [degrees]
ψ
Effective exit angle [degrees]
ωn
Natural frequency [Hz]
ωc
Chatter frequency [Hz]
Cutting speed in primary cutting direction [m/min]
vii
Acronyms
CAD
Computer Aided Design
CNC
Computer Numerical Control
FEM
Finite Element Method
FFT
Fast Fourier Transform
MRR
Material Removal Rate
SLD
Stability Lobe Diagram
ix
Table of Contents
Abstract
Acknowledgments
Publications
Nomenclature
Acronyms
Table of Content
i
iii
v
vii
ix
xi
1 Introduction
1.1 Background
1.2 Practical Example –Jet Engine Diffuser Case
1.3 Introduction to Milling Vibration Research
1.4 Fixtures and Tooling
1.5 Damping
1.6 Industrial and Scientific Issues
1.6.1 Industrial Problem
1.6.2 Scientific Approach
1.6.3 Research Questions
1.6.4 Research Scope and Limitations
1
1
2
2
7
8
8
8
9
9
10
2 Mechanics of Milling
2.1 Cutter Geometry Terminology on a Macro Level
2.1.1 Rake Angles
2.1.2 Lead Angle
2.1.3 Clearance Angle
2.1.4 Other Features
11
11
11
13
13
14
3 Milling System Vibrations
3.1 Introduction to Milling Vibrations
3.2 Free, Forced, and Chatter Vibrations
3.3 The Stability Lobe Diagram
3.4 The Dynamics of the Stability Lobes
3.5 Experimental Modal Analysis
17
17
17
20
22
23
xi
4 Modeling Work
4.1 General Simulation Modeling
4.2 Determination of the kc curve
4.3 The Milling Stability Prediction Model
25
25
25
27
5 Experimental Work
5.1 Definitions
5.2 Tools and Procedures
5.3 Dynamics of Stability Lobes Experiments
5.4 Tool-to-Workpiece Position Experiments
5.5 Cutting Geometry Sensitivity
5.5.1 Approach
5.5.2 Down Milling Position
5.5.3 Zero Rake Zero Exit Position
5.5.4 Zero Rake Zero Entry Position
5.5.5 Up Milling Position
5.5.6 Zero Offset Position
5.5.7 Force Profile versus Exit Angle
31
31
32
33
35
38
38
39
40
41
42
43
44
6 Conclusions
47
47
48
49
49
6.1 Research Questions – an Analysis
6.2 Vibration Prediction Modeling
6.3 Recommendations to Operators
6.4 The Diffuser Case – a Recap
7 Future Work
51
8 References
55
Appendix 1: Regenerative Stability Theory
Appended Papers
xii
1. Introduction
1.1 Background
Taking 3 kg across the Atlantic requires about 1 kg of fuel!
A European initiative “Clean Sky”1 has been established to support research in
an effort to reduce aircraft emissions and fuel consumption. Volvo Aero is an
Associate Member of Clean Sky (Sustainable & Green Engines or SAGE), a
Joint Technology Initiative (JTI) that was initiated 2008. The goal of the
project is to reach the environmental requirements for flights within Europe.
One example of these requirements is to reduce carbon dioxide emissions to
50% of current levels by 2020. The project covers various advanced materials
and process technologies for aircraft engine structures.
To further increase pressure on the airline industry, the European Union has
implemented legislatives regarding emissions trading starting 2012. In these,
airlines are requested to purchase emission rights for operating aircraft.
The Lufthansa Group’s2 aircraft consume on average 4.2 liters of fuel per 100
passenger kilometers. A major factor is their consistent modernization of the
fleet. The Airbus A380, for example, consumes on average only 3.41 liters of
fuel per 100 passenger kilometers. By 2016, a total of 160 new aircraft will be
delivered to Lufthansa, thus ensuring further reduction in fuel consumption
for their fleet.
Because of programs like Clean Sky or SAGE, aerospace development has led
to reduced-weight engine designs. Requirements for reduced emissions and fuel
consumption have made aerospace components thinner at the same time as
alloys more difficult to machine have been introduced. In addition, the
demands in manufacturing have forced production speeds to increase in order
to reduce production cost. One of the consequences of increased speeds is an
increased risk for vibrations during the milling operation. This, in turn, may
lead to poor surface finish, reduced dimensional accuracy, excessive tool wear
or complete failure, and noisier work environment. It may also lead to rework
and scrap components.
1
2
www.cleansky.eu
www.lufthansa.com/responsibility
1
11.2 Practical Examp
ple –Jet E
Engine Difffuser Casse
A typical component that exhibitss certain maachining diffficulties is th
he jet
engine difffuser case, Figure 1. It
I is a com
mponent locaated between
n the
compressorr inlet case an
nd the combustion chambber case. It coonsists of an inner
and outer ccase and a nu
umber of hol low struts. FFigure 1 show
ws the fuel injjector
clamping m
mounts whicch are typicaal locations fo
for milling op
perations. In
nconel
718, a nickkel-base high temperature alloy, is usedd for the diffu
user case.
Figure 1: D
Diffuser case.
When macchining the diffuser case flanges at hhigh cutting speeds, vibraations
may becom
me uncontroollable, makiing the macchining process very diffficult.
Vibrations can be so violent thatt the machinne-tool basee develops crracks.
Because off chatter vibrrations (self-eexcited vibraations), noisee may reach levels
harmful foor the operattor. Where ceramic insertts are used at
a high mach
hining
rates, vibraations still preesent a challeenge for a co ntrolled proccess. This can
n lead
to tool failu
ure.
11.3 Introd
duction to
o Milling V
Vibration
n Research
h
Concerns aabout millingg vibrations are nothing neew. As early as
a 1907, Fred
derick
W. Taylor [1] described
d machining vibrations ass the most ob
bscure and deelicate
of all the pproblems faccing the mach
hinist. As shhown in man
ny publication
ns on
machiningg, this obserrvation still holds true today. Therre will alwayys be
vibrations in machinin
ng systems. Whether
W
theyy are forced vibrations orr selfexcited vibbrations, theyy need to be controlled
c
inn order not too cause damaage to
the component or the machining
m
sysstem.
A simple vvibratory sysstem may bee representedd by a mass, a spring, and
a a
damper as depicted in Figure
F
2.
2
Figure 2: Mass-sprin
ng-damper sysstem.
(Hooke’s law
w) and awaay from equ
uilibrium,
(Newton’s second laaw), where m=mass, c=
=damping
coefficiient, and k=spring constaant or stiffneess. It is assu
umed that the damper
and th
he spring haave no masss. Free vibraations with damping (cc≠0), free
vibratioon without damping (c=
=0), and unsstable (chatter) vibration
ns can be
visualizzed in Figuree 3.
At equ
uilibrium,
Figure 3: Free vibrrations with damping
d
(topp), free vibrattions withoutt damping
(middlle), and unstabble vibrationss (bottom).
In a m
milling situattion, the maass in Figuree 2 symbolizzes the inertia of the
workpiiece/fixture system, thee spring syymbolizes th
he stiffness of the
workpiiece/fixture and/or
a
the to
ool/spindle syystem, and th
he damper syymbolizes
the tottal milling syystem’s abilitty to reduce vibrations du
uring and affter cutter
engageement.
3
The dynam
mic state of a milling system can be desscribed by a frequency
f
resp
ponse
function, Φ
. The real and im
maginary parrts of the frrequency resp
ponse
function, F
Figure 4, can be written ass
where
,
Φ
Eq. 1
Φ
Eq. 2
is the stiffness, and
d
is the dam
mping ratio.
crosses over
The real paart
Φ
o from poositive to negaative at the natural
Φ
frequency,
. The imaginary part
is at minim
mum at the natural
frequency,
. If the curve
c
for thee imaginary ppart becomess very large in the
negative ddirection, it indicates th
hat the dam
mping ratio nears
n
zero. In a
machining system, the imaginary part
p should aalways be negative. A po
ositive
o the
imaginary part could suuggest an errror in the m easurement. The shape of
maginary partss is dependen
nt on the stiff
ffness and thee damping ratio of
real and im
the system.
Figure 4: R
Real and imagginary part off transfer funct
ction.
Chatter th
heory may bee used for determining thhe likelihood of chatter during
d
machiningg. Under certaain condition
ns, the amplittude of vibrattion grows an
nd the
cutting sysstem becomees unstable. Self-excited chatter vibrrations in milling
m
develop ddue to dynaamic interacttion betweenn the cuttin
ng tool and
d the
workpiece. This results in regenerattion of wavinness on the cu
utting surface and
modulation
n of the chip thickness.
Tobias et al. [2.3] and
d Tlusty et al.
a [4] presennted between
n 1958 and 1967
basics of cchatter vibrattions on macchine tools bby explainingg the fundam
mental
mechanics of regeneratiion of chip thickness
t
andd calculating the uncondittional
4
stability boundary. Their theory was based on the physics of orthogonal cutting
and the regenerative mechanism. If the values of the system parameters are
such that the system is unstable, the smallest disturbance (such as a hard spot
in the material) is sufficient to induce the system to leave steady state of
motion and burst into oscillation (to chatter). Tobias analysis was concerned
with the problem of stability, i.e., whether or not the machine will chatter
under certain working conditions. His observation was that little use was made
of the results by production engineers and machine tool designers of his time
because “some of the recommendations based on theoretical considerations
appeared to be contrary to practical experience”. He thus developed a
regeneration theory that took into account that in a dynamic system, the chip
thickness may vary independently of the machine tool feed rate.
Shridhar et al. [5] presented in 1968 a detailed mathematical model of the
dynamic milling process. They developed a comprehensive stability theory for
milling that is based on the numerical integration of the milling equations for
one period of the cutter’s revolution. Their computer algorithm permitted the
determination of the stability boundaries in the space of controllable
parameters associated with the cutting operation. It was able to handle six
directions and any number of modes.
Minis and Yanushevsky [6] proposed in 1990 a comprehensive analytical
method and solved the two-dimensional milling problem by introducing the
theory of periodic differential equations. They also improved the early analysis
work of Shridhar et al. [5] by applying the theory of periodic differential
equations on the milling dynamics equations. Although the algorithm
depended on the numerical evaluation of the stability limits, it provided for
comprehensive modeling in determining the stability limits for milling and
described the aspects of milling dynamics. They concluded that if each tooth of
the milling cutter remains in contact along the entire length of the arc being
machined, then the dynamics of the milling system are well described by a set
of linear differential equations with periodic coefficients. In the non-linear case
where the cutting teeth loose contact with the workpiece at some point along
the machined arc, the stability theory yields accurate results for most practical
cases of milling. This is according to Minis and Yanushevsky due to the fact
that prior to the onset of chatter, the nonlinearity occurs mainly at the very
beginning of the machining arc.
Altintas and Budak presented in 1995 an alternative method for analytical
prediction of stability lobes in milling utilizing a transfer function [7]. They
modeled the milling cutter and workpiece as multi degree-of-freedom
5
structures. The dynamic interaction in the cutting zone was modeled by
including the variations in the cutter and workpiece dynamics in the axial
direction. It was demonstrated with numerical examples that also for highly
flexible workpieces, the accuracy of the predictions can increase. Their analysis
resulted in analytical relations for the chatter frequency and chatter stability
limit which were used to generate stability diagrams. Time-varying dynamic
cutting force coefficients were approximated by their Fourier series
components, and the chatter-free axial depth of cuts and spindle speeds were
calculated directly from linear analytical expressions without any numeric
iterations. The model can be used to determine the chatter free axial and radial
depth of cuts without resorting to time domain simulations. Further
development of their method and applications can be found in [8,9].
Quintana and Ciurana [10] recently made an extensive review of publications
on chatter in machining processes. The article reviews the state of research on
the chatter problem and defines the differences between free vibrations, forced
vibrations, and self-excited vibrations. In addition, different forms of chatter
such as frictional chatter, thermo-mechanical chatter, mode-coupling chatter,
and regenerative chatter are explained. Further, the existing methods developed
to ensure stable cutting are classified. Same authors [11] applied sound
mapping methodology in order to determine the stability lobe diagram.
During a milling operation, the three different types of mechanical vibrations
(free vibrations, forced vibrations, and self-excited vibrations) propagate
through air and generate a sound that intrinsically contains information about
the process. The article presents the information in the form of a 3-D stability
lobe diagram. Regarding further developments and improvements in the field,
it suggests that models could become more sophisticated and accurate by a
deeper consideration of process damping, part behavior, and changes in
structure or system dynamics along the tool path. Advances in computers and
sensors would undoubtedly play an important role in this field. A useful idea
would be to identify the Stability Lobe Diagram, SLD, of a given system
composed of a certain machine tool, tool holder, cutting tool and workpiece
material system in the process planning. Finally, the article [10] mentions that
some aspects in milling are still difficult to model, e.g., the spindle dynamic
behavior variation at high rotational speeds, where the centrifugal force on the
bearings, the gyroscopic effect, and thermal effects change the performance of
the spindle.
Studies by Pekelharing [12,13,14] show that interrupted cutting such as
milling may yield excessive tool wear due to multiple tool entry and exit.
6
Initiallly, the entry shock
s
was blamed whereaas the exit waas considered
d harmless
or of m
minor influeence [12]. The issue of eentry was coonsidered two-fold: A
mechan
nical impactt shock and
d a thermal shock. Thee latter depeended on
machin
nability, coolling time, an
nd cutting speeed and feed
d. However, according
a
to Pekkelharing, thee exit can cau
use immediatte and progreessive chippin
ng. It can
also caause severe bu
urr build-up where the cuutter exits thee workpiece. Burrs are
formedd because maaterial from th
he exit face iss pushed asid
de by each cuttter. Each
tooth ttakes a little less than th
he proper cutt and adds this to the bu
urr. Over
med but thee teeth take too
certain
n ranges of exxit angles, no
o burr is form
t much
insteadd of too littlee. This “too much” is fouund at the end of the ch
hip and is
sometimes referred to as a “foot”” and is depiccted in Figure 5. This foot forming
starts w
where the furrthermost pro
ogressed partt of the edge starts its exiit. It then
spreadss over the wiidth of the cut. The foot is largest wh
here the last exit takes
place [[13]. The pap
per states thaat the most efffective remedy against cu
utter wear
and faiilure is to prrevent foot fo
ormation as m
much as possible. This iss done by
using ssmall diameteer cutters, sho
ort cuts, and most importantly, to shiftt the path
of the center of thee cutter awayy from the exxit face of thee workpiece whenever
possiblle [14]. So according
a
to Pekelharing, the exit con
nditions are of
o greater
importtance than initially expecteed.
ns of process sttages.
Figure 5: Definition
1.4 Fixxtures and
d Tooling
A com
mmon solutioon to avoid vibration prroblems is too design fixttures that
stabilizze the components. How
wever, it is nnot always possible
p
or feeasible to
design suitable macchining fixtu
ures. Some fiixtures, for example,
e
mak
ke certain
parts oof the comp
ponent inacccessible for m
machining. Because
B
of their
t
size,
weightt, complexityy, and cost, many
m
fixturess are only maade in single quantity.
This leads to delaays while reeloading macchines as weell as an in
ncrease in
interm
mediate parts storage.
s
Wheen such a fixtture fails, it may
m cause prroduction
interru
uption and in
ncur additionaal manufactuuring costs. Fiixtures are deesigned to
7
increase the stiffness of the workpiece. As the workpiece wall thickness becomes
thinner, the workpiece itself becomes more flexible. Finding solutions to these
issues becomes more and more urgent.
1.5 Damping
Damping is a capacity of a vibrating system to transform a fraction of energy of
the vibratory process during each cycle of vibration into another form of
energy, mainly heat. This energy transformation leads to reducing intensity
(amplitude) of a forced vibratory process or to a gradual decay and fading of a
free vibration process. If a vibratory process is of a self-excited type (e.g.,
chatter vibrations in metal cutting operations), then the damping capacity of
the system may partially suppress or completely prevent development of the
self-excited vibrations [15]. Damping can also be described as an irreversible
physical process that dissipates energy through the conversion of work into
heat. In engineering structures it is present in several forms: internal hysteresis,
friction via the rubbing action of surfaces or particles, viscous friction in fluids,
radiation damping, electromagnetic damping etc. As demonstrated by
Nicolescu et al. [16], damping plays a critical role in structural dynamics as it is
the primary means by which resonant amplitudes are controlled thus
enhancing durability, life cycle behavior and cost reduction. Damping is of
great interest in vibration cases in machining operations. Layers of high
damping capacity material may be applied at various places of the machining
structure, such as the tool holder/tool interfaces. Although not covered in this
work, it should not be neglected in importance. Rather, it should be
considered carefully when developing further vibration control systems.
1.6 Industrial and Scientific Issues
1.6.1.
Industrial Problem
As higher material removal rates are required, ceramic inserts become an
attractive solution. Using ceramic inserts, the cutting speed may be up to ten
times faster than for carbide inserts. This would increase production rates with
an equivalent amount. However, milling thin-walled components can induce
vibrations which in turn can cause unfavorable machining stability. While
large, complex, and costly fixtures have been the immediate solution to avoid
vibrations, it is by no means the long-term or final solution. Vibration risks are
still prevalent. The industry needs a robust methodology for assessing vibration
8
risks long before the machining operations take place. Stability effects should
be assessed already during early stages in the process planning phase.
1.6.2.
Scientific Approach
As part of the preparation work a State-of-the-Art study was performed to
investigate what work had already been done in this research area. A large
number of experiments were conducted and the results observed and analyzed.
Regarding insert material, it was decided to use carbide inserts instead of
ceramic inserts in order to minimize tool failure during robust machining. The
research presented in this thesis is divided up into several steps. First, the
analysis will determine whether to approach the vibration issue through an
examination of the cutter entry or the cutter exit. The path chosen depends on
which of these two options that show the least amount of vibrations. Then, the
vibration patterns will be examined using a variety of tool-to-workpiece offset
positions. The plotting of the results will be used to acquire an understanding
of how to better predict vibration risk during milling.
1.6.3.
Research Questions
There have been two “schools” on whether the cutter entry or the cutter exit is
of greater importance. The research questions, therefore, are outlined to
investigate this issue. The strategy is to investigate changes in structure or
system dynamics along the tool path. It includes the dynamics explained by the
stability lobe diagram as the cutter moves through the workpiece. The stability
lobe diagram can be identified for a given system composed of a certain
machine tool, tool holder, cutting tool, and workpiece geometry and material.
In addition, micro features of the inserts can be studied more in depth.
This thesis centers on the following four research questions:
R1. How critical is the choice of offset between tool and workpiece
during milling?
R2. What effects do cutter entry and cutter exit have on system
vibrations?
R3. How does the effective exit angle affect vibrations during and after
cutter exit?
R4. What is the dynamic effect on the stability lobe diagram as the
cutter moves through the workpiece?
9
1.6.4.
Research Scope and Limitations
The scope of this thesis is to investigate and analyze strategies for reducing
vibrations during milling of thin-walled components. As an application of the
industrial problem, an easy to use spread sheet milling stability prediction
model is presented. The vibration studies conducted are limited to thin-walled
Inconel 718 components. Inconel 718 was chosen as workpiece material
because of its widespread usage in aerospace components and its difficult
machining characteristics. The workpiece is assumed to be the most flexible
part of the machining system. The tooling used has been limited to coated
cemented carbide inserts mounted in two different 1-fluted milling tools.
Cutter micro geometry effects are not considered. All cutting is done without
cooling or cutting fluids.
10
2. M
Mechanics of Millling
Millingg mechanicss includes th
he relationshhip between workpiece and tool
surfacees, cutting and
a
insert angles,
a
and cutting forcces. The geo
ometry is
generallly significantly more com
mplicated thaan for other cutting meth
hods such
as turn
ning.
2.1 Cu
utter Geom
metry Term
minologyy
This iss a brief sum
mmary of som
me of the mosst frequently used cutter geometry
termin
nology. A general milling tool
t is depicteed in Figure 6.
6
Figure 6: A generall milling tool serving as exaample for this section. It is a six-flute
face miill with negatiive radial andd positive axiaal rake angles.
2. 1.1.
Rakke Angles
Radial Rake Angle
The raadial rake anggle of a milling cutter is tthe angle bettween the rak
ke face of
the toooth and a rad
dial line passiing through the cutting edge,
e
Figure 7. It may
be posiitive, negativve, or zero and
d controls thhe chip flow accordingly.
a
A positive
radial rrake angle reqquires less force and creattes less heat than a zero orr negative
radial rrake angle. However,
H
a po
ositive rake aangle also putts greater streess on the
cuttingg edge. A neggative radial rake angle ccutter starts the cut away from the
edge off the rake face where the cutter
c
is stronnger.
11
Radial Rake angle
a
(showingg a negative raadial rake anggle).
Figure 7: R
Axial Rakee Angle
For tools w
with inserts, the axial rak
ke angle is foormed between the axis of
o the
tool and th
he cutting edgge, Figure 8. Just like the radial rake an
ngle, the axial rake
angle can bbe positive, negative,
n
or zero and conttrols the chip
p flow accordingly.
It influences the cuttingg forces and the strength of the cuttin
ng edge. A po
ositive
axial rake angle allows for an easierr chip flow aas it lifts thee chip and cu
urls it
away from
m the workpieece surface. A negative axxial rake anglee cutter bend
ds the
chip forwaard and down
nward under pressure, cauusing potential chip evacu
uation
difficulties for soft alloyys.
Figure 8: A
Axial rake anggle (showing a positive axiaal rake angle).
Helix Anglle and Helicaal Axial Rake Angle
When the cutting edgee is formed along
a
a helix about the toool axis (as in the
case of a ssolid carbide tool), the reesulting rake is called hellical rake anggle or
helix anglee (see Figure 9). It is a fu
unction of thhe tool radiuss and is steep
per or
12
lower ttoward the ceenter of the tool.
t
The hellix angle is th
herefore convverging to
zero att the center of
o the tool and larger towaards the perip
phery. The helix angle
for soliid tools correesponds to thee axial rake anngle for toolss with inserts.
Figure 9: Helix anggle, flute, corner radius, andd tooth.
2. 1.2.
Leaad Angle
The leaad angle is deefined accord
ding to Figurre 10. An increase in the lead
l
angle
results in an increasse in the axiall force and a reduction in the radial forrce. From
point oof view of vibrations,
v
th
he magnitudee of the radiial force is important
when iit is directed in the weakeest direction of the workp
piece wall. A properly
chosen
n lead angle allows
a
the cuttter to enter aand exit the cut
c more smoothly by
reducin
ng the shockk load on thee cutting edgge. This is because
b
the leead angle
providees for a longeer entry and exit
e phase forr cases wheree the axial rak
ke angle is
not eqqual to zero, Figure 10. The lead anngle is also important
i
fo
or process
dampin
ng.
Figure 10: Lead anggle.
2. 1.3.
Clearance Anggle
The cleearance anglee (both radiall and axial) iss the angle th
he tool formss with the
workpiiece and preevents the to
ool from rubbbing againsst the workp
piece (see
Figure 11). It shou
uld be large enough forr the tool to clear the workpiece.
w
Howevver, if it is tooo large, it will
w weaken tthe cutter. Itt may be divvided into
13
primary an
nd secondary clearance angles. The cleaarance angle can have an effect
on the dam
mping properties of the sysstem.
Figure 11: Clearance an
ngles (showingg the radial cle
learance angless).
2.1.4..
Other Features
F
The radial runout or offfset of a millling cutter is a variation of
o the cuttingg edge
relative to the outer diaameter of thee tool. The axxial runout iss a variation of
o the
cutting edgge relative to the tool face, Figure 12. T
The runout could
c
be a pro
oduct
of either ddesign or malfunctioning intolerances.
i
For examplee, the axial ru
unout
could be deesigned to function as a wiper.
w
Figure 12: Radial and axial
a
runout
s
of th
he cutting to oth against which
w
the ch
hip is
The rake fface is that surface
formed in the metal cuttting operatio
on, Figure 133.
14
The laand is the part of the to
ooth adjacennt to the cuttting edge. It
I aids in
avoidin
ng interferencce between the tool itselff and the surfface of the workpiece,
w
Figure 13.
The filllet is the curvved surface att the bottom of the flute. The shape off the fillet
contribbutes to the strength
s
of th
he tooth for solid cutters and defines the space
where the chip flow
ws, Figure 13..
d
of the cylinder passing through the
The outside diameeter is the diameter
peripheeral cutting edges. It deffines the larggest slot that the tool can
n cut in a
single ppass, Figure 13.
1
The rooot diameter is the diameter of the cyllinder passing through th
he bottom
of the fillet. The larger root diameter a ttool has, thee greater its torsional
strengtth, Figure 13.
Figure 13: Tool rakke face, fillet, land,
l
flute, annd outer and root
r diameterss.
The cu
utting edge is
i the interseection of thee rake face of
o the tooth with the
leadingg edge of the land.
The flu
ute is a groovve on the periiphery of a cuutter that alloows for chip flow
f
away
from th
he cut.
The teeeth are the cutting
c
pointss on a cuttingg tool. The number
n
of teeth is the
same aas the numberr of flutes.
The cu
utter pitch orr density is deetermined byy the numberr of teeth in the
t cutter
body aand is depicteed in Figure 14. It rangess from coarsee to fine. Coaarse pitch
allows for large feed
d and chip th
hickness, wheereas fine pittch may be used when
15
high efficieency and goood workpiece finish are reequired. Increeasing the nu
umber
of teeth w
will increase the risk for regenerativee chatter while decreasing the
number off teeth may induce vibrrations if thee system is working clo
ose to
resonance. Therefore, th
he number of teeth on a ccutting tool should
s
be carefully
consideredd. Staggered cutters
c
with unequal
u
spacinng (so-called differential pitch)
p
may be useed to reduce the risk for chatter
c
vibrattions in the syystem. The reason
is that the chip load on
n each cutter is varied, andd thus, the effective tooth
h pass
frequency is reduced. Differential pitch cutteers are thus used for sp
pecific
cutting appplications.
Figure 14: Coarse, mediium, fine, andd differential ppitch.
16
3. Milling System Vibrations
3.1 Introduction to Milling Vibrations
Vibrations may be divided into free vibrations, forced vibrations, and selfexcited vibrations (so-called chatter). Chatter may be divided into primary and
secondary chatter [17]. Primary chatter is caused by friction between the tool
and the workpiece, the thermodynamics of the cutting process, and from mode
coupling. Secondary chatter or regenerative chatter results from a modulated
chip thickness.
3.2 Free, Forced, and Chatter Vibrations
Vibrations in milling arise from a flexible workpiece, a flexible machine tool, or
both. Free vibrations occur when the mechanical system is displaced from its
equilibrium and is allowed to vibrate freely. This could be for example as a
result of a collision between the cutting tool and the workpiece. Forced
vibrations appear due to external harmonic excitations, particularly when the
cutting edge enters and exits the workpiece. They can also stem from
unbalanced bearings or cutting tools. Chatter vibrations extract energy from
the interaction between the cutting tool and the workpiece, and grow during
the machining process to bring the system into instability.
As thin-walled workpieces are flexible, chatter may occur. The reason is that
structural modes of the machining structure are excited by the cutting forces.
The first three modes of the workpiece used in this study are shown in Figure
15.
17
Figure 15: Mode 1, modde 2, and modde 3 of the woorkpiece used in
i study.
The wavy surface from
m a specific cu
utting tooth is removed by
b the subseq
quent
tooth, also this leaving a wavy surfaace. The phasse shift betweeen the two waves
w
implies a ch
hange in the chip thickneess as well as tthe cutting foorce, Figure 16.
No viibrations
No vibratioons
Forced vibrations
0
In phase waves
Constantt chip load
No chatter
Chattter vibration
ns
Out of phase waves
Unstable chatter
Figure 16:: Description of
o no vibratioon, forced vibrrations, and chhatter vibrations as
ness variation during
d
milling
ng.
visualized iin chip thickn
is the ph
hase shift anggle between the current aand previouss surface wavviness.
Regenerativve chatter occcurs due to the differencce in vibratioon phase bettween
the currentt cut and the previous cutt (relating to the chip thicckness). When
n two
vibration w
waves are ou
ut of phase, chatter occuurs. From reegenerative ch
hatter
theory, a reelationship is formed betw
ween spindle speed and criitical depth of
o cut.
Here follow
ws a brief sum
mmary of ho
ow to extract the stability lobe diagram
m. For
details, see Appendix 1,, Regenerativee Stability Theeory.
A wavy surrface finish leeft by one too
oth is removeed by the succceeding oscilllatory
tooth. Thee resulting ch
hip thickness will itself beecome oscillattory and produces
18
oscillattory cutting forces
f
havingg magnitudess proportionaal to the tim
me-varying
chip looad. In order to obtain thee stability lobbe diagram, ceertain parameeters such
as the natural frequ
uency, damping ratio, an d stiffness neeed to be dettermined.
From tthese values, the real and imaginary paarts of the traansfer functio
on may be
calculaated and the dynamic cuttting coefficieents evaluateed. The stabiility lobes
are calcculated as follows (also seee the schemattics in Figuree 17):
Select a chaatter frequenccy ωc aroundd a dominant mode ωn.
i.
ii.
Solve an eiggenvalue equ
uation and callculate the critical depth of
o cut alim.
iii.
Calculate the spindle sp
peed n for eacch stability lobe.
iv.
Repeat thee procedure by
b scanning the chatter frequencies
f
around all
dominant modes
m
of the structure.
Figure 17: Simplifieed schematics of procedure ffor graphing the
t stability loobes.
19
3.3
The Stability Lobe Diagrram
Regenerativve stability theory
t
produ
uces a stabiliity lobe diaggram that maay be
used to predict and control
c
chattter. The stabbility lobe diagram
d
plotts the
boundary bbetween stabble and unstaable regions aas a function
n of spindle speed
and depth of cut. Thee locations an
nd shapes off the stabilityy lobes depen
nd on
many variaables, such as
a material properties
p
annd tool-to-woorkpiece position.
Each setupp and set off parameters gives a uniqque stability lobe diagram
m. By
applying su
uch diagramss in milling or turning proocesses, the maximum
m
dep
pth of
cut may bbe optimized at the highest spindle sspeed used, thereby
t
increeasing
material reemoval rates (MRR) and improving prroductivity. The
T spindle speed
can be direectly related to
t the tooth-p
pass frequenccy. As the nu
umber of teeth
h in a
cutting toool increase, the tooth-p
pass frequenncy increases. The work
kpiece
experiencess a greater number of cuts per unit ttime and thiss correspondss to a
higher spin
ndle speed.
Stability loobe diagramss are created by intersectting a series of scallop-sh
haped
stability boorderlines [188]. These inteersections deefine the deep
pest stable cu
ut at a
given spinddle speed and
d form the lim
mits for chat ter. Locally, for each lobee, it is
stable below the lobe and
a unstable above the lobbe. Since thee lobes intersect, a
ne lobe coulld be above the neighboring lobe. Su
uch a
point locatted below on
point musst be consid
dered as unstable. Globaally, the relaationship bettween
adjacent loobes needs too be considerred when dettermining staability. The upper
u
portions aabove the pooint of interrsection of two adjacent lobes coulld be
trimmed ooff, connectin
ng all the lob
bes into chattter lines, Figure 18. All points
p
below the lines are stabble, whereas all points aboove the liness are unstablee. For
increasing spindle speeeds, the stability lobes bbecome widerr, the intervening
spaces betw
ween consecu
utive lobes greeater, and thee intersection
n points higheer.
Figure 18:: Stability lobbe diagram before
b
and aft
fter connectingg lobes into ch
hatter
lines
20
Observving the lobees in Figure 19, the lobe to the far riight (lobe #0
0) has the
maxim
mum stable deepth of cut at
a its intersecttion with thee lobe on its left (lobe
#1). O
On the other hand,
h
its righ
ht branch haas no intersecction with oth
her lobes,
theorettically allowin
ng for unlim
mited depth oof cuts at veryy high spindle speeds.
The loobes on the faar left move closer
c
togetheer the further left they are located.
Also tthe intersectiion points move
m
downw
ward eventu
ually approacching the
minim
mum depth off cut, alim.
Figure 19: Stabilityy lobe diagram
m with upper aand lower bou
undary limits
ntire range off the stability lobe diagram
m may be diviided into threee regions
The en
of stabbility: uncond
ditionally staable, conditioonally stable,, and unconditionally
unstable.
nconditionallly stable regio
on: The loweest points on the lobes (all the same
The un
value) represent alimm, the minimu
um depth off cut. A lowerr horizontal borderline
b
may th
hereby be draawn by conn
necting the loowest points on all the lo
obes. The
region below this lower
l
border line is uncoonditionally stable,
s
indepeendent of
spindlee speed or chatter
c
frequ
uency. Manyy process en
ngineers and machine
operatoors prefer thiis region beccause it is chhatter free. However,
H
it allso means
low prooductivity an
nd low MRR.
The un
nconditionallly unstable reegion: An uppper border lin
ne may also be
b formed
by fittiing a curve through
t
all th
he intersectioon points of the lobes. Th
he region
above this upper border
b
line is
i unconditioonally unstab
ble. This imp
plies that
chatterr vibrations will
w always occcur during m
machining in this
t region.
The cconditionally stable regio
on: The in--between reggion, i.e., th
he region
betweeen the lower and the upp
per border linnes, is condittionally stable. In this
21
region, poiints are stablee when they are
a below thee lobes and unstable
u
when
n they
are above tthe lobes. If this
t region iss explored annd certain stab
ble regions fo
ound,
then produ
uctivity can be
b increased without jeoppardizing maachined surfaace or
tool integriity.
When the spindle speeed approachees zero, the upper and lower border lines
converge in
nto a point of minimum
m depth of ccut. The influ
uence of diffferent
parameterss on the stabiility lobe diaggram is descrribed more in
n detail in Seection
4.3, The M
Milling Stabiliity Prediction Model.
33.4 The Dynamics
D
of Stabiliity Lobes
Bravo et all. [19] showeed that the naatural frequenncy of a thin
n-walled work
kpiece
changes for each pass of
o the cutter. An extensionn to this is th
hat the shapees and
locations oof the stabilitty lobes conttinually channge during a single cutterr pass
through th
he workpiecce material. As depictedd in Figure 20, they ch
hange
throughout the milling process as material
m
removval and cutterr contact variiation
influence the natural freequency of th
he workpiece . This is particularly noticceable
during machining of th
hin-walled co
omponents si nce the mateerial removed
d may
constitute a considerablle portion of the starting sstock. The instantaneous cutter
c
direction aalso defines th
he stiffness th
he system exp eriences during the cut.
Figure 20: The natural frequency
f
of the
t workpiece changes as material is remooved.
22
Budak et al. [20] confirm that workpiece dynamics continuously change due to
mass removal and variation of cutter contact. The article proposed an analytical
method for modeling varying workpiece dynamics and its effects on process
stability. The method is based on a finite element mesh used to obtain the
frequency response function of the workpiece. It is updated by using the
removed elements along the tool path as defined by the cutter location. The
stability diagrams are then generated from the updated frequency response
functions. Similarly, Shamoto et al. [21] reaffirms that chatter stability depends
on the tool path relative to the dynamically most compliant direction. The
article proposes a concept to optimize the tool path to avoid chatter vibrations
in machining operations and claims that the optimum tool path can be defined
based on given tool geometry and cutting conditions.
3.5 Experimental Modal Analysis
By means of structural dynamic tests, the transfer function of an elastic
structure may be identified. An impact hammer equipped with a piezoelectric
force transducer is used to excite the machining structure. By means of a short
impact by the hammer, an impulse is generated and a range of frequencies are
excited that contain the natural modes of the system. The resulting vibrations
are measured using an accelerometer. The frequency response function is
measured and a stability lobe diagram may be extracted. In order to analyze the
dynamics of machining systems, the interaction between the structural
dynamics and the process dynamics must be analyzed. There is some
inaccuracy associated with this method since the impulse is generated while the
tool is stationary. The modal analysis is therefore done either on the
workpiece/fixture structure or on the tool/machine structure. The results may
be combined for a more accurate reading of the total system. Alternatively, if
the machining forces and their directions are known, the milling tool may be
statically placed against the workpiece exerting similar force magnitudes and
directions. Then the hammer impulse can be recorded for the complete system.
This may give a better indication of a system in motion than when only doing
an impact test on the tool and the workpiece separately.
23
4. Modeling Work
4.1.
General Simulation Modeling
There are several tools currently available for predicting vibrations, including
FEM and modal analysis. The aim has been to simulate or model vibration
problems in milling of thin-walled components. Weck et al. [22] who
suggested stability models where existing pocketing routines in a CAD/CAM
system were corrected using a stability data bank during NC (Numerical
Control) tool path generation. Bayly et al. [23] used FEM analysis localizing
unstable zones within stable zones. Biermann et al. [24] published a simulation
system consisting of an FE model of the workpiece coupled with a geometric
milling simulation for computing regenerative workpiece vibrations during
five-axis milling.
FEM modeling is a relatively complex way to describe the process and requires
a high level of understanding of vibration theory. An easy to use stability
prediction model would therefore be of great value. A spread sheet with a
simple user interface and scroll buttons for input parameters would be
relatively easy to use. Therefore, building a prediction model based on a spread
sheet would make it accessible to many engineers and operators. This chapter
describes how a spread sheet prediction model could be built and applied to an
elementary milling setup.
4.2.
Determination of the Specific Cutting Force
One way to extract the cutting forces of a certain alloy is by determining the
specific cutting force, , from turning. This parameter is used to develop the
milling prediction stability model in this work. The variation of
depends on
the workpiece material, tool geometry and coating, and cutting parameters.
Also the tool wear influences
because of removal of coating and changes in
tool geometry during cutting. Feed is the parameter that has the greatest
influence on the specific cutting force.
The kc curve was extracted for Inconel 718 using a lathe equipped with a
piezoelectric force sensor. The cutting speed was 50 m/min and the depth of
cut 3.5mm. The cutting forces were measured with the force sensor and
recorded into data files. Three measurements were taken and averaged for each
25
value of u
uncut chip th
hickness and
d all measureements were done in ran
ndom
order. Thee
value was
w then extrracted for eaach averaged cutting forcce. 34
values of uncut chip thickness weere chosen bbetween 0.00
015 and 0.25mm
which equaals the feed per
p revolution. The cuurve ( as a function of uncut
u
chip thickn
ness) was theen plotted acccording to Fiigure 21. Using a curve fitting
f
routine, th
he equation foor the curvve was derivedd to be
k
1139
9.7h
.
Eq. 3
m turning Incconel 718.
Figure 21: The kc curvee extracted from
26
4. 3.
The Milling Stability Prediction Model
Figure 22: The geneeral layout off the Milling SStability Prediiction Model.
The m
milling stabilitty prediction model shoulld be able to predict the shape
s
and
locations of the sttability lobess for a givenn dynamic system
s
and show
s
the
frequen
ncy responsee function. It should alsoo be a tool that can be used for
visualizzation duringg class room instruction oor workshops. With this in mind,
the miilling stabilityy prediction model was ddeveloped in the form off a spread
sheet ggraphics reprresentation, Figure
F
22. Itt is a tool th
hat allows th
he user to
extractt a stability loobe diagram and
a the frequuency responsse function fo
or a set of
workpiiece and tool parameters. The stabilityy lobe diagram
m, Figure 23,, provides
inform
mation on how
w to choose the
t depth of cut for a givven spindle sp
peed. The
frequen
ncy response function is used
u to characcterize the dyynamics of th
he system.
Variable input parrameters inclu
ude workpieece width, len
ngth, height, density,
Young’s Modulus, damping ratio,
r
numb er of teeth,, and start and exit
immersion angles. The parameters are channged by means of scroll bars.
b
The
stabilitty lobe diagrram and thee real and iimaginary paarts of the frequency
f
response function vary
v accordin
ngly. The wayy they vary ass input param
meters are
changeed is depicted
d in Table 1. One applicattion of the model
m
is analyyzing how
27
to compen
nsate the stabbility when a system channge occurs. For example, if the
overhang increases byy 10%, the model can be used to determine what
parameterss need to be changed and
d by how muuch in order to
t compensate for
the changee.
The modeel predictionss are based on the proccess described
d in Append
dix 1,
Regenerativve Stability Theory, and Reference [17]. Princip
pally, the ch
hatter
frequency iis scanned arround the nattural frequen cy of the worrkpiece. From
m this
the real an
nd imaginaryy parts of the transfer fuunction are calculated toggether
with the crritical depth of
o cut and th
he spindle speeed. The stab
bility lobe diaagram
and transfeer function will
w change dyynamically toogether with all
a parameterrs tied
to the userr-changed parrameter. For example, if tthe workpiecce wall thickn
ness is
changed, th
he graphs ch
hange accordingly and so do affected parameters
p
su
uch as
workpiece stiffness and natural frequ
uency.
The curren
nt model inccludes the ab
bility to geneeralize basic workpiece shapes
such as rou
und and squaare bars. Lateer modeling ddevelopmentts may includ
de the
ability to suggest cuttin
ng parameterss from a know
wn location in
i a given staability
lobe diagraam.
m as shown inn the Milling Stability
S
Pred
diction
Figure 23:: The stabilityy lobe diagram
Model.
28
Table 1: The effectss in the stabillity lobe diagrram due to an
n
input pparameter.
Lo
obe
Workpiece
aliim
Effect
intersection natural
Param
frequency
meter
Wall th
hickness
Compoonent
length
Overalll height
Densitty
N/C
Dampiing ratio
Number of
teeth
Young’s
Modullus
Entry aangle
Exit an
ngle
increase in a variable
Workpiece
Stiffness
+
+
N/C
+
-
N/C
N/C
N/C
N/C
N/C
+
+
N/C
N/C
N/C
N/C
The arrrows represeent the direcction in whicch the pointss in the stab
bility lobe
diagram
m move durring an increease of a varriable input parameter. Regarding
R
workpiiece natural frequency and
a
stiffnesss, “+” mean
ns an increase, “-“ a
decreasse, and “N/C
C” no changee. The positioon and shapee of the stabiility lobes
are parrticularly senssitive to component wall thickness and
d wall height, more so
than too wall length
h. Also when
n the numberr of teeth vaaries, the stab
bility lobe
diagram
m displays significant changes. Reegarding thee alim and the lobe
intersections, they divide the staability lobe ddiagram into the regions described
in Secttion 3.3, Thhe Stability Lobe
L
Diagram
m. The millin
ng stability prediction
p
model is an easy way
w to visuaalize how theese regions behave
b
under varying
millingg conditions.
29
5. Exxperimen
ntal Worrk
5. 1.
Definitions
This seection clarifiees some of th
he definitionns used in this work. Thee effective
exit an
ngle ψ is defi
fined as the angle
a
betweeen the rake face
f
and the exit face,
Figure 24. This angle is depend
dent on the ggeometry of the cutting tooth
t
and
the toool offset from
m the exit face of the workppiece.
Figure 24: Effectivee exit angle iss defined in a and b. The zero rake zero exit tool
a depicted inn (c) and (d), respectively.
and thee zero rake zerro entry tool are
The exxit phase is defined as the
t part of tthe cut where the chip thickness
(cuttin
ng load) contiinually decreaases. In proceesses such as down millingg, the exit
31
phase includes the main part of the cutting process, including the in-process
region depicted in Figure 5. The zero rake zero exit tool position, on the other
hand, has a zero degree effective exit angle and therefore an instantaneous exit
phase.
5.2.
Tools, Experimentals, and Procedures
Two different tools were used; a standard on-the-market tool and a zero rake
custom-made tool.
The standard tool was a single coated cemented carbide cutting insert mounted
on a 25mm diameter three-flute milling cutter. The milling tool had an axial
rake angle of 8.0 degrees, a radial rake angle of -7.6 degrees, and a nose radius
of rε=0.8mm. The feed was set at fZ=0.08mm/tooth.
The zero rake tool was a single coated cemented carbide cutting insert
mounted on a 25mm diameter two-flute milling tool. The insert was
configured for zero degree radial and axial rake angles. It had a nose radius of
rε=0.8mm and the feed was set at fZ=0.08mm/tooth.
A three-axis milling machine was utilized for the experiments. The cutting
speed was fixed at vc=50m/min corresponding to a spindle speed of n=637rpm.
The axial depth of cut was set at ap=0.5mm. There were no cutting or cooling
fluids used during any of the experiments. Using a three-component
piezoelectric force sensor mounted under the workpiece fixture, the cutting
forces were measured and recorded into data files. From these, the force
components and the resultant cutting forces were extracted. The data was
analyzed and plotted using numerical software. The workpiece was Inconel 718
plate with a cross section of 5mm x 40mm and was mounted in a fixture as
shown in Figure 25. The workpiece overhang was varied between 5mm and
40mm. The cutting insert was replaced for each machining run.
By altering the offset position between the tool and the workpiece, down
milling, zero offset milling, and up milling can be obtained. These three offset
positions were chosen for the standard tool. The zero offset position was
defined as head-on face milling. The effective exit angles for these positions
were 82.4 (down milling), 3.9 (zero offset milling), and -44.5 (up milling)
degrees respectively. In addition, two offset positions were chosen for the zero
rake tool; a zero entry angle position corresponding to a 23.5-degree effective
exit angle, and a zero exit angle position corresponding to a 0-degree effective
exit angle. The five offset positions are depicted in Figure 26.
32
n machining ffixture
Figure 25: Workpieece installed in
Figure 26: Top vieew of the offsett positions usedd in this workk
5. 3.
Dyynamics off Stabilityy Lobes - Experimen
E
nts
As is illlustrated in Figure
F
27, th
he likelihood to find stablee regions is dependent
d
on how
w fast the spin
ndle is rotatin
ng.
The daata for the graaphs were derived from onne pass of a milling
m
cutterr during a
face milling operatiion for the zeero offset geoometry. The workpiece
w
waas Inconel
718 wiith a cross section of 5mm
m x 40mm annd a wall oveerhang of 40mm. The
33
axial depth
h of cut was 0.5mm, the spindle speedd 250rpm, an
nd the uncutt chip
thickness 00.08mm.
Three poin
nts from the data
d file weree analyzed: onne at the begginning, one at
a the
data,
middle, an
nd one at the end of the cu
ut. From the FFT of the accelerometer
a
the naturaal frequenciess were found
d to be 19990Hz, 2004H
Hz, and 202
24Hz,
respectivelyy. As noted in Section 3.4, The Dyynamics of Sttability Lobess, the
change in natural freqquency is atttributed to material rem
moval and cutter
c
contact varriation. The stability lobee diagrams w
were derived for
f each value and
then overlaaid in the graaphs shown. This
T indicatess a change in natural frequ
uency
by about 1.7% from staart of cut to end
e of cut.
At lower sppindle speeds, Figure 27 left graph, tthe three curvves show a sp
pread
making it very difficullt to define a region aboove alim that would stay stable
s
throughout the cut. Altthough the stability lobe diagram givees some indiccation
of the critiical depth off cut when the
t system gooes from stab
ble to unstab
ble, it
does not provide mu
uch useful information
i
on how too choose milling
m
parameterss.
At higher spindle speeeds, Figure 27 right graaph, the reggions between
n the
stability lobbes are markkedly more well-defined
w
annd, thereforee, the diagram
m will
give a clearrer indication
n on stable reggions above a lim. The stability lobe diaggram,
therefore, is significanttly more useeful for deterrmining milling parameteers at
high spindle speeds than
n at low spindle speeds.
Figure 27:
7: Stability Loobe Diagram at the beginnning (1990H
Hz), at the middle
m
(2004Hz), and at the end
e (2024Hz)
z) of cut for loow spindle speed (left) and
d high
spindle speeed (right).
34
5.4.
Tool-to-Workpiece Position Experiments
This part of the research focused on three common offset positions using the
standard milling insert, namely down milling, zero offset milling, and up
milling. The vibration behavior was investigated during cutter entry, in-process
milling, and cutter exit for a thin-walled Inconel 718 component. These offset
positions were chosen to exemplify the impact the milling geometry has on the
resultant cutting force and on the onset of vibrations. The manner in which
the component overhang affects the overall stability of the system depends
greatly on the offset position.
The resultant forces from the measurements taken during the first three offset
positions are shown in Figure 28. For a component wall overhang of 5mm, the
depth of cut ap=0.5mm does not cause any instability in the system. As the wall
overhang increases, a higher degree of instability is observed, especially for zero
offset and up milling. The highest levels of vibrations are seen for the zero
offset geometry.
Figure 28 also shows that onset of vibrations occurs at different component
wall overhang for the three offset geometries. For down milling, vibrations
starts at a wall overhang of 25mm, for the zero offset case at a wall overhang of
20mm, and for up milling at a wall overhang of 30mm. These are critical
points where the dynamic system changes from chatter-free vibrations to
chatter vibrations. The three geometries show significant variations in resultant
force amplitudes. Greater forces are observed for zero offset than for down
milling or up milling. Up to a component wall overhang of 30mm, down
milling and up milling require more or less the same amount of resultant force
to achieve cutting. For wall overhangs of 35 mm and above, the down milling
plots of the resultant force exhibits minor vibrations, whereas the up milling
plots exhibits more severe vibrations.
35
Figure 28 : Down milllling (left), zeero offset millling (middle)
e), and up milling
m
(from bottom to top) wall height
h
(right) at 55, 10, 15, 20,, 25, 30, 35, and 40mm (f
overhang.
Figure 28 indicates thaat the offset position
p
has ggreater influeence on vibraations
than the am
mount of waall height oveerhang of thee workpiece. This is espeecially
true for sm
mall amounts of workpiecee overhang. FFor large amoounts of overhang,
both the zeero offset and
d the up millling geometriies show sign
nificant vibrations.
The plot su
uggests that the
t down miilling positionn is preferablle in order to keep
vibrations to a minimu
um. Because of the dynaamic circumsstances durin
ng the
cutting prrocess, it is not obvious that chatteer vibrations can be preecisely
predicted using existin
ng methods. As noted bby Surmann et al. [25], even
chatter-freee milling proocesses can produce
p
a higgh surface loocation error since
chatter-freee does not necessarily mean vibrattion-free. Su
uch circumsttances
include fleexible deform
mations of thee workpiece before cuttin
ng takes placee and
burr formaation during cutter exit. The
T results oof this study can be used as an
indicator fo
for how to avooid chatter viibrations.
For down m
milling, maxximum force is
i observed dduring cutter entry as the cutter
c
is travelingg in a relativeely flexible direction
d
of thhe workpiecee. The cutterr exit,
on the oth
her hand, occcurs as the cu
utter is travelling in the sttiffest direction of
36
the woorkpiece. Forr up milling,, we have thhe opposite scenario.
s
For the zero
offset ggeometry, the cutter is traveling in thhe predominaantly flexible direction
througghout the cu
ut, includingg entry and exit. From Figure 28 it
i can be
concluded that dow
wn milling (ccharacterizedd by a smooth cutter exit) exhibits
significcantly less vibbrations than
n up milling (characterizeed by a smoo
oth cutter
entry) for same com
mponent overrhang. A genneralization off this is that it is more
importtant to have a smooth cutter
c
exit thhan a smooth cutter en
ntry. This
supporrts the findin
ngs of Peckelharing [12,113,14] mentioned in Secction 1.3,
Review
w of Milling Vibration
V
Reseaarch History.
In ordeer to determine whether the vibrationns displayed were
w forced vibrations
v
or chattter vibration
ns, the force measurement
m
ts from a ran
ndom cut was overlaid
with th
he force meassurements fro
om the subseequent cut. Iff the two are in phase,
forced vibrations arre inherent in
n the processs, whereas if they are out of phase,
chatterr vibrations prevail.
p
Figuree 29 shows exxamples of th
hese two casess.
Figure 29: Two subsequent forcee measuremennts: at left, in
n phase depictting forced
ight, out of phhase depicting chatter vibrattions.
vibratioons, and at rig
If the ddepth of cut ap is held con
nstant, the crritical depth of cut alim deecreases as
the component ovverhang increeases, Figure 30. For a dynamic system with
constan
nt depth of cut
c and increeasing compoonent overhaang, chatter vibrations
v
will eveentually be in
ntroduced.
Figure 30: The effeects on alim as the wall heighht overhang iss increased froom 35mm
mer impact tessts.
(left) too 40mm (righht). The diagraams are deriveed from hamm
37
5.5.
Cutting Geometry Sensitivity
5.5.1
Approach
In this part of the research, a zero rake milling insert is used in addition to the
standard cutting insert described above. Besides the three standard tool
positions, two offset positions are defined as the zero rake zero entry position
corresponding to an effective exit angle of 23.5 degrees and the zero rake zero
exit position at an effective exit angle of zero degree. The resultant force and its
components are presented for each of the five cases. At the end of the section,
the relationship between the force profile and the exit angle is analyzed.
38
5. 5.2
Dow
wn Millingg Position
Duringg down milliing, Figure 31, the cutterr exit phase iss mainly or altogether
a
in the least flexible direction. The
T exit phas e starts from
m a relatively low force
level aand exhibits a long smoo
oth decline. IIn addition, the chip thickness is
reduced throughou
ut the cut. According tto the definiition of Section 5.1,
Definittions, the exitt phase thus includes thee entire cuttin
ng process affter cutter
entry. Because of a smooth cutter exit, vibraations are con
nsiderably lesss than for
the oth
her offset possitions studieed and vibratiions are in efffect absent. Since the
cutter exits in the least flexiblee direction off the workpiiece, the cuttting force
requireed is also sign
nificantly lesss. There is nno after-ringin
ng visible in the force
profile except from free vibration
ns.
Figure 31: Force prrofile during down
d
millingg at 40mm ovverhang. Resulltant force
and off
ffset position (top)
(t and forcee components (bottom). Thhe effective exxit angle is
82.4 de
degrees.
39
5.5.3
Zero Rake
R
Zero Exit
E Positionn
For the Zeero rake zeroo exit case, the
t exit phas e starts from
m a relatively high
force level and the exiit takes placee abruptly. A hump (a su
udden increaase in
force and vvibrations) iss clearly visib
ble during th e post-exit phase
p
althouggh the
vibrations rapidly decaay at the end
d of the hum
mp. Significaant vibrationss and
forces are ggenerated beccause the cuttter exits in thhe most flexib
ble direction of
o the
workpiece,, Figure 32. It exhibits basically no ex
exit phase and
d has the shortest
exit of the offset positioons considereed in this stuudy. The resu
ultant force profile
p
shows a hu
ump behaviorr which is also clearly visibble in the y and
a z compon
nents.
It is, howevver, completeely absent in the x componnent.
Figure 322: Force profi
file for the zeero rake zeroo exit tool att 40mm overrhang.
ffset position (top) and foorce components (bottom). The
Resultant fforce and off
effective exiit angle is zeroo degree.
40
5. 5.4
Zerro Rake Zerro Entry Poosition
By inccreasing the exit angle byy 23.5 degreees, the vibrrations and the
t forces
decreasse to the leveel shown in Figure 33 (zeero rake zeroo entry positiion). The
exit ph
hase starts froom a low to moderate
m
leveel. Less vibrattion is observved at the
same ttime as less force
f
is required for the ccutting proceess. Also, the post-exit
after-riinging almostt completely disappears.
Figure 33: Force profile
p
for thee zero rake zzero entry toool at 40mm overhang.
Resultaant force andd offset positiion (top) andd force compponents (botttom). The
effectivve exit angle iss 23.5 degrees.
41
5.5.5
Up Mi lling Position
Up millingg, Figure 34 exhibits the opposite
o
behhavior from down
d
milling. The
exit phase starts from a moderately high forcee level and the
t chip thicckness
increases th
hroughout th
he cut. The cutter
c
directiion changes to
t an all the more
flexible diirection of the workpiiece and, ttherefore, viibrations inccrease
throughout the cut. Cu
utting forces are also muchh larger than
n for down milling
m
since the ccutter exits paartly in the most
m flexible direction. It can be seen from
the graphss that the y and z com
mponents sttart to dom
minate over the
t x
component after the cu
utter exit ph
hase. The afteer-ringing ob
bserved, therefore,
mainly stem
ms from the cutting forcee in these twoo directions. A hump is clearly
c
seen in thee resultant foorce graph bu
ut is not as nnoticeable in the graphs fo
or the
force compponents. Altthough up milling
m
causess more vibraations than down
d
milling, it exhibits less vibration
v
thaan the zero offfset and the zero rake zero exit
positions. Up millingg causes sevvere burr foormation du
uring cutter exit.
Characterisstic only for up milling is
i that the viibrations con
ntinue on thrrough
the post exxit phase.
Figure 34:: Force profile during up milling at 40m
mm overhang. Resultant
R
forcce and
offset positioon (top) and force
f
componeents (bottom).. The effectivee exit angle is -44.5
degrees.
42
5. 5.6
Zerro Offset Po
osition
The strrongest vibraations are seen
n for the zeroo offset posittion with thee standard
tool, F
Figure 35. The
T exit phasse starts at hhigh force leevels and theen decays
rapidlyy down to a minimum after which a large-sized
d hump appeears. The
vibratioons continuee through thee hump but disappear almost compleetely after
the hump. It generaates after-ringging similar iin nature to the zero rakee zero exit
positioon. The cutterr exits near th
he most flexibble direction of the workp
piece and,
therefoore, strong viibrations are present duriing and afterr cutter exit. Also the
cuttingg forces are considerablyy larger throoughout the cut compareed to up
millingg and down milling. In this tool possition, the ch
hip thicknesss remains
relativeely constant throughout the
t cut. The after-ringingg is characterrized by a
large-siized hump having
h
approxximately the same span ass the tool engagement
region of the force profile. This hump is barrely noticeablle in the x co
omponent
but cleearly visible in
n the y and z components .
Figure 35: Force prrofile during zero offset miilling at 40m
mm overhang. Resultant
nd force compponents (botto
tom). The effe
fective exit
force aand offset posiition (top) an
angle iss 3.9 degrees.
43
5.5.7
Force Profile
P
versu
us Exit Anggle
The force profile behaavior in the region
r
follow
wing the cuttter exit (the afterringing and the hump)) may be cau
used by severral factors. In
n addition to
o free
vibrations and built-up
p edge, the beeat phenomennon, Figure 36, may influ
uence
the force bbehavior. In the
t beat phen
nomenon, thee response is composed of preexisting ch
hatter vibratioons at frequency ωc and ffree vibration
n of the work
kpiece
at frequenccy ωn. The reesponse could
d be written aas:
sin
sin
2 cos
sin
Eq. 4
Here the ccosine factorr is an envelo
ope for the ssine wave. Iff the two staarting
frequenciess are close to each other, the
t frequencyy of the cosin
ne on the righ
ht side
is too slow
w to be percceived as a pitch.
p
Insteadd it is perceiived as a perriodic
variation oof the sine in the expresssion with a frequency of
o
, i.ee. the
average of the two freqquencies. Thee successive vvalues of maxxima and miinima
form a waave whose freequency equaals the differeence between
n the two staarting
frequenciess.
b producedd (1400Hz vibbration with 58Hz
5
beat)
Figure 36: Example of beat
At an effecctive exit anglle of 82.4 deggrees (down m
milling positiion), the optimum
vibration-ffree region is found, Figu
ure 37. This is no doubt what experieenced
machine ooperators woould expect. However, seeemingly favvorable ampllitude
versus effecctive exit anggle scenario iss seen at 23.55 degrees, corrresponding to
t the
zero rake zero entry position.
p
This merits furtther studies to investigatte the
sensitivity of exit angle choice and to
t what degreee a model coould be developed
to predictt vibration in
i the proccess. In ordeer to predicct vibration risk,
geometricaal factors havve to be tak
ken into acccount. The model shoulld be
capable off handling th
hese phenomeena. To furthher develop this, cutter micro
m
geometry sshould be con
nsidered. Forr the five offs
fset positions studied, exceept in
44
the casse of down milling,
m
a hump is seen inn the force profile
p
after the
t cutter
exit. T
This hump is most noticeaable for the zeero offset possition for thee standard
tool an
nd for the zerro exit positio
on for the zeero rake tool. When usingg the zero
rake toool, it was nooted that a change
c
from zero to 23.5
5 degrees in exit
e angle
significcantly reducees the size off the hump aand the amou
unt of vibratiions. It is
notewoorthy that the
t
hump amplitude
a
foollows the maximum amplitude
a
througghout the eff
ffective exit angle
a
range. The conclu
usion is that the exit
directioon should bee chosen in the least flexibble direction
n possible and
d that the
effectivve exit angle should be as large as posssible (so that the tool-to-w
workpiece
positioon is as close to
t the down milling
m
positiion as possiblle).
Figure 37: Maxim
mum amplitudde during exxit phase andd hump ampplitude vs.
effectivve exit angle
After ccutter exit, th
he workpiece vibrates at a certain freq
quency and am
mplitude.
These vvibrations deepend on facttors such as thhe exit angle and force levvel at start
of exitt phase. As the
t effective exit angle neears zero deggrees, these vibrations
v
become stronger. In
ndependentlyy of whether tthe standard cutter or the zero rake
cutter is used, the most
m favorable cutter exitt is in the leaast flexible dirrection of
the woorkpiece. By choosing
c
an offset
o
positio n such as dow
wn milling, vibrations
v
are avooided almost altogether, evven at relativvely large worrkpiece overh
hang. The
force reequired for th
he cut is also reduced. In addition, burrr formation is kept to
a minimum. It wass determined that exit forrces should be avoided in the most
flexiblee direction off the work piiece and thatt down millin
ng is more ro
obust and
less proone to generaate vibrationss than up millling and zeroo offset millin
ng. In the
case of down millling, it prod
duces a cleann surface with
w
virtually no burr
ormation.
formattion. Up millling, on the other hand,, produces severe burr fo
Thin-w
walled compoonents are paarticularly proone to burr formation
f
at high exit
45
forces and large chip thickness since the material tends to be plastically
deformed rather than cut. Burr formation mechanisms are explained in [26]
and reviews on burr minimization techniques are found in [27].
46
6. Conclusions
6.1.
Research Questions - an Analysis
The research questions raised in the introduction give rise to queries much
greater than the scope for this thesis. The answers are therefore given in regards
the limitations of the scope.
R1. How critical is the choice of offset between tool and workpiece
during milling?
The results from the experiments indicate that by changing the offset location
of the tool in relation to the workpiece, the amount of vibrations in the system
may change significantly. This is particularly noticeable when the offset
location is changed from zero offset to down milling. Generally, for small
component overhangs, the vibration risk is limited. However, as the
component overhang increases, the choice of offset position becomes crucial.
Down milling is more robust and less prone to generate vibrations than up
milling and the zero offset geometry. It also displays the smallest force
amplitude during the exit phase and no hump in the post exit phase. In
addition, it produces the least amount of burr formation.
R2. What effects do the cutter entry and cutter exit have on system
vibrations?
Down milling (characterized by a smooth cutter exit) exhibits significantly less
vibration than up milling (characterized by a smooth cutter entry) for same
component overhang. A generalization of this is that it is more important to
have a smooth cutter exit than a smooth cutter entry in order to avoid
vibrations. Entry and exit forces should be avoided in the most flexible
direction of the workpiece.
R3. How does the effective exit angle affect vibrations during and
after cutter exit?
The tool position should be chosen so that the cutter exits in the least flexible
direction possible for the workpiece. As was observed for the standard cutting
tool, down milling clearly is more favorable than up milling regarding
47
vibrations, force requirements, and burr formation. The effective exit angle,
therefore, should be chosen so that the cutter exits in the least flexible direction
possible at the same time as the tool-to-workpiece position is as close to the
down milling position as possible. The region around the zero degree effective
exit angle is very sensitive. Even small changes in the processing conditions,
such as the effective exit angle, may result in substantial changes in vibrations.
R4. What is the dynamic effect on the stability lobe diagram as the
cutter moves through the workpiece?
The shapes and locations of stability lobes change as the cutter passes through
the workpiece material. They change throughout the milling process as
material removal and cutter contact variation influence the natural frequency
of the workpiece, Figure 19. This is particularly noticeable during machining
of thin-walled components since the material removed may constitute a
considerable portion of the starting stock. The instantaneous cutter direction
also defines the stiffness the system experiences during the cut. The usefulness
of the stability lobe diagram is dependent on how fast the spindle rotates,
Figure 27. Under the same milling situation, it is easier to define stable regions
above alim during milling at high spindle speeds than at low spindle speeds.
6.2.
Vibration Prediction Modeling
In addition to these research questions, this work has discussed some aspects of
modeling. In order to better predict the vibration risk for a component during
early stages in the process planning phase, an easy to use milling stability
prediction model has been developed. It predicts the shape and locations of the
stability lobes and the frequency response functions for a given set of milling
parameters. It also shows dynamic changes in parameters such as cutting forces,
chip thickness, and workpiece natural frequency and stiffness. It is a tool that
can be used for visualization during class room instruction or workshops. It is
in the form of a spread sheet graphics representation. The workpiece geometry
discussed within this thesis has been a rectangular block, but the model is
generalizable for other basic workpiece shapes such as round and square bars.
48
6.3.
Recommendations to Operators
Process engineers and operators may consider that vibration risks are reduced
in the following cases:




A smooth cutter exit
Small component overhangs
Offset position close to the down milling position
Cutter exit in the least flexible direction possible for the workpiece
In addition, the following needs to be considered:

The stability changes during cutter pass
Considering these guidelines will assure a more robust process that is less prone
to generate vibrations. Large force amplitudes during the exit phase would be
avoided and the post-exit hump eliminated. In addition, burr formation would
be minimized.
6.4.
The Diffuser Case – a Recap
In Section 1.2, Practical Example – Jet Engine Diffuser Case, it was explained
that some milling difficulties had been experienced for this type of component.
From what has been observed in this work, the following milling strategies
could be recommended: When milling around and into the holes of the fuel
injector mounting clamps or flange bosses, down milling should be used. In
addition to tool-to-workpiece position, also tool diameter and pitch have to be
taken into account. Although it is preferable for the cutter entry to be smooth,
it is essential for the cutter exit to be as smooth as possible. This will inhibit
vibrations in the system and call for a cleaner surface condition and tighter
dimensional accuracy. It would also reduce burr formation and subsequent
surface reworking. The axial rake angle should be chosen in such a way as to
provide a smooth cutting surface and a limited heat input into the material.
The lead angle can be increased so that the cutter exit is smooth and adequate
amount of material is removed for each cut.
49
7. Future Work
For a more complete picture of the vibration issues encountered during milling
of thin-walled components, some specific aspects have to be analyzed in-depth.
Some suggestions for future work are provided here. The Licentiate-andBeyond Research Circle in Figure 38 depicts some avenues for continued
research. As opposed to a real-life traffic circle where only one choice can be
done once inside the circle, the research circle allows for some simultaneous
work. The milling stability prediction model, for example, can be developed in
parallel with the effective exit angle sensitivity research.
The sensitivity of the effective exit angle needs to be considered for a
complete realization of the behavior of the force and vibration response. This
requires an analysis of the micro geometry of the cutting process. Also the
other regions of the cutting process needs to be considered, including entry, inprocess, and post-process. This will give insight into what options the process
planners and operators have when choosing machining parameters
According to Figure 37, there is a region from about -25 to +25 degrees
effective exit angle where vibrations are especially noticeable in the machining
system. This region could be examined with smaller angular increments using a
variety of cutting tools, on both macro and micro levels. This could be done by
utilizing a turning holder mounted in a milling holder. Inserts with various
macro and micro geometries could then be used for a complete study of the
vibration behavior. On the macro level, lead angle could be varied, and on
micro level, chip breaker and edge radius could be varied. As the lead angle is
increased, the exit path and exit time are extended. This, in turn, will influence
the cutting forces and vibrations on the system. This would determine which
areas are unsuitable for milling and which areas are feasible although not
optimum.
The dynamics of the stability lobe diagram could be investigated further
taking into account the wall thickness of the workpiece material. Also the
direction of vibrations developed during the cut could be taken into account.
Changes in material properties such as Young’s Modulus, density, and the
damping ratio affect the natural frequency of the workpiece and this, in turn,
51
will affect the shapes of the stability lobes. Also how the stability lobes change
with spindle speed as these properties change would be an area of interest.
The variation in chip thickness as a function of offset could be taken into
account when analyzing the dynamics of the machining system (see Figure 26).
Also, the chip thickness varies together with the component stiffness during the
cut as depicted in Appendix 1, Figure 1.
Damping layers may be applied at various places of the machining structure,
such as the tool/insert, a rotating force sensor/tool, and the tool holder/tool
interfaces. Such layers would play a critical role in structural dynamics and
resonant amplitude control thus enhancing durability, life cycle and cost
reduction.
Laser Vibrometry could be used to determine the stability of a machining
system in real time during the actual machining. The cutting parameters are
then varied instantaneously through a control system as the dynamics of the
system changes.
A comparison with Euler buckling could be done to determine the maximum
overhang a workpiece may have without risk for chatter vibrations. The study
would include a cutting parameter matrix for a complete analysis of the
phenomenon.
The Milling Stability Prediction Model presented in this thesis does not
suggest cutting parameters from a known location in a given stability lobe
diagram. The development of this feature will make the model highly useful
during component redesign. The model may also be expanded to include more
advanced geometries such as workpieces with hollows. It could also include the
dynamics of the stability lobes as noted in Sections 3.3, The Dynamics of
Stability Lobes and Section 5.3, Dynamics of Stability Lobes Experiments.
By reducing system vibrations, ceramic inserts may be used without failing
and the noise may be reduced to acceptable levels in the workshop. Ceramic
inserts can be said to be a common goal for all the various avenues out of the
research circle.
52
Figure 38: The Liceentiate-and-B
Beyond Researcch Circle.
53
8. References
1. F.W. Taylor. On the Art of Cutting Metals. ASME 1907.
2. S.A. Tobias and W. Fishwick, Theory of Regenerative Machine Tool
Chatter. The Engineer, London, 1958.
3. S.A. Tobias, Machine Tool Vibration. Blackie and Sons Ltd., 1965.
4. F. Koenigsberger and J. Tlusty, Machine Tool Structures Vol. 1:
Stability Against Chatter. Pergamon Press, 1967.
5. R. Shridhar, R.E. Hohn and G.W. Long, A Stability Algorithm for
the General Milling Process. Contribution to Machine Tool Chatter
Research-7. Transactions of the ASME Journal of Engineering for
Industry 90:330-334, 1968.
6. I. Minis, R. Yanushevsky, R. Tembo and R. Hocken, Analysis of
Linear and Nonlinear Chatter in Milling. Annals of the CIRP
39:459-462, 1990.
7. Y. Altintas and E. Budak, Analytical Prediction of Stability Lobes in
Milling, Annals of the CIRP, Vol. 44, No. 1, pp. 357-362, 1995.
8. E. Budak and Y. Altintas, Analytical Prediction of Chatter Stability
in Milling – Part I: General Formulation. Journal of Dynamic
Systems, Measurement and Control 120:22-30, 1998.
9. E. Budak and Y. Altintas, Analytical Prediction of Chatter Stability
in Milling – Part II: Application of the General Formulation to
Common Milling Systems. Journal of Dynamic Systems,
Measurement and Control 120:31-36, 1998.
10. G. Quintana and J. Ciurana, Chatter in Machining Processes: A
Review. International Journal of Machine Tools and Manufacture.
International Journal of Machine tools and manufacture, Vol 51,
No 5 pp 363–376, 2011.
55
11. G. Quintana, J. Ciurana, I. Ferrer and C. Rodriguez. Sound
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57
Appendix 1
Regeneration Stability Theory
This information is based on the theory presented in Y. Altintas.
Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations,
and CNC Design. Cambridge University Press, 2000.
The general dynamic chip thickness
is described by:
Τ
t
Τ
Eq. 1
where y(t) is the modulation of the surface of the workpiece during the current
pass of the milling cutter at time t and y(t-T) is the modulation of the surface
of the workpiece during the previous pass at one spindle revolution period (T)
before t.
Τ
Eq. 2
[mm] is the intended chip thickness (equal to the feed rate of the machine)
[N/mm2] is the cutting coefficient in the feed direction
[kg] is the mass
is the damping coefficient
[N/m] is the stiffness
[mm] is the depth of cut
The following Laplace definitions will prove useful:
Eq. 3
0
1
Eq. 4
0
0
Eq. 5
Τ
Eq. 6
By definition, therefore, the dynamic chip thickness in the Laplace domain
becomes
Eq. 7
or
1
Eq. 8
The dynamic cutting force in the Laplace domain thus becomes
Eq. 9
Define Φ
as the transfer function of the workpiece structure (single degree
of freedom).
The current vibration is
Φ
Eq. 10
Φ
Eq. 11
0
0
0
Eq. 12
Set the initial conditions
0
0
0
Eq. 13
⟹
When the damping is constant at
Eq. 14
0, the system natural frequency is
Eq. 15
or
Eq. 16
2
Let the damping ratio be ζ
Eq. 17
Generally for mechanical structural systems, ζ is less than 1. The following is
then valid:
2
2
2
⟹
Eq. 18
Φ
2
Eq. 19
The transfer function of the workpiece structure becomes
Φ
Eq. 20
∴ Φ
Eq. 21
Combining Eq. 8 and Eq. 11 gives:
1
1
1
Φ
Φ
Eq. 22
Eq. 23
Eq. 24
3
The stability of the closed-loop transfer function is determined by the roots (s)
of the characteristic equation, i.e.:
1
1
Φ
0
Eq. 25
Let the root of the characteristic equation be
Eq. 26
If the real part
0, the time domain solution will have an exponential term
with positive power, i.e. | | . The chatter vibrations will grow indefinitely
and the system will be unstable.
If the real part
0, the time domain solution will have an exponential term
with negative power, i.e. | | . The vibrations will be suppressed by time and
the system will be stable with chatter vibration free vibrations.
If the real part
0, then
and the system is critically stable. The
workpiece oscillates with constant vibration amplitude at chatter frequency .
The characteristic equation of the dynamic cutting process has additional terms
beyond the structures transfer function. Therefore, the chatter vibration
frequency does not equal the natural frequency of the structure. However, the
chatter vibration frequency is still close to the natural mode of the structure.
⟹1
1
Φ
0
Eq. 27
is the maximum axial depth of cut for chatter vibration-free
where
machining.
The transfer function may be divided into real and imaginary parts, i.e.
Φ
Eq. 28
This is not to be confused with the dynamic chip thickness in the Laplace
domain, H(s).
Generally:
Eq. 29
Eq. 30
4
Therefore,
1
1
Τ
Τ
Τ
Τ
1
Eq. 31
0
Τ
Τ
Eq. 32
Τ
Τ
0
Eq. 33
Collecting all real and imaginary terms separate gives:
1
1
Τ
Τ
1
Τ
Τ
0
Eq. 34
Both the real and imaginary parts must be equal to zero.
For imaginary part equal to zero:
Τ
1
Τ
Τ
Τ
0
Eq. 35
1
Eq. 36
tan
Eq. 37
where ψ is the phase shift of the structure’s transfer function.
Using the trigonometric identities
cos 2
cos
sin 2
2 sin cos
and substituting 2
Τ
sin
Eq. 38
Eq. 39
Τ, we get
cos
Τ⁄2
sin
5
Τ⁄2
Eq. 40
sin
Τ
Τ⁄2 cos
2 sin
⁄
tan
Τ⁄2
Eq. 41
⁄
⁄
Eq. 42
⁄
From
cos
sin
cos
Τ⁄2
1
Eq. 43
it becomes
1
Τ⁄2
sin
Eq. 44
Inserting into Eq. 42 yields:
tan
2 sin
sin
Τ⁄2 cos
Τ⁄2
sin
Τ⁄2
Τ⁄2
Τ⁄2 cos
Τ⁄2
Τ⁄2
2sin
2 sin
Τ⁄2
Τ⁄2
cos
sin
cot
Τ⁄2
Eq. 45
Using the trigonometric identities
cot
tan
Eq. 46
tan
tan
Eq. 47
tan
tan
Eq. 48
6
It becomes
tan
tan
tan
⁄2
2
⁄2
2
⁄2
tan
tan
⁄2
tan
⁄2
⁄2
2
2
Eq. 49
Eq. 50
3
Eq. 51
tan
Eq. 52
The spindle speed n and chatter vibration frequency
have a relationship
that affects the dynamic chip thickness. Assume that the chatter vibration
frequency is
or . The number of vibration waves left on the surface of the
workpiece is:
Τ
Eq. 53
is the fractional wave generated. The
is the integer number of waves and
angle represents the phase angle between the inner and outer modulations.
For
0 or
1 the chip thickness will remain constant despite the
presence of vibrations. For other values of , the chip thickness changes
continuously.
2
Τ
2π
Eq. 54
2π
Eq. 55
7
tan
⁄2
tan
Eq. 56
⁄2
3
2
2
2
3
3
Eq. 57
Eq. 58
2
2
Eq. 59
2
Eq. 60
The spindle period is Τ
and the spindle speed
By setting the real part of the characteristic equation to zero, we may now
derive the critical axial depth of cut:
1
1
Τ
Τ
0
Eq. 61
or
Eq. 62
⁄
From Eq. 61 we have that
⁄
The right hand side denominator thus becomes
1
Τ
Τ
Τ
Τ
1
1
Τ
sin
1
Τ
Τ
1
1
Τ
Τ
sin
1
Τ
Τ
8
1
2 cos
1
Τ cos
1
Τ
Τ
2 cos
1
Τ 1
Τ
2 1
1
2
sin
Τ
Τ
Τ
Eq. 63
Therefore:
Eq. 64
must be a positive number. The solution
Because it is a physical quantity,
is therefore only valid for negative values of the real part of the transfer
function
and chatter vibrations may occur at any frequency satisfying
is chosen at the minimum value of
, no
this condition. When
chatter is generated no matter spindle speed. The harder the work material, the
larger the cutting constant , with a reduction in the axial depth of cut as a
consequence. Another factor reducing the axial depth of cut is the flexibility in
the machine tool or work piece structure. This also reduces the productivity.
By means of the following five steps, we may plot the stability lobe diagram:
For known transfer function Φ of the structure at cutting point and Kf cutting
constant,
1. Select a chatter frequency
at the negative real part of the transfer
function.
2. Calculate the phase angle of the transfer function at
from
tan
(Eq. 37)
9
3. Calculate the critical depth of cut from
(Eq. 64)
4. Calculate the spindle speed for each stability lobe number k=0,1,2….
Eq. 65
From
Τ
Eq. 66
5. Repeat the procedure by scanning the chatter frequencies around the
natural frequency of the structure
10
Dynaamic Milling Modeel
Figure 1: Regenerattion
Assume two orthoggonal Degreess of Freedom , N teeth, and zero degreee helix
angle D
Define dynam
mic displacem
ments as x andd y where x is
i the feed dirrection
and y iis the normal direction.
A coorrdinate transfo
formation in the
t radial (chhip thickness)) direction givves:
sin
cos
Eq. 67
is th
he instantaneeous angular immersion
i
off tooth j.
If Ω is the spindle angular
a
speed
d in rad/s, theen
Ω
Eq. 68
The resulting chip thickness
t
sin
Eq. 69
,
11
is the feed rate per tooth and
cutter.
sin
,
is the static part and
Define
,
is the dynamic displacements of the
is the dynamic part.
,
as a step function
1,
0,
Since
sin
Eq. 70
is staic, it does not contribute to the dynamic chip load.
sin
cos
Eq. 71
,
sin
cos
sin
sin
Δ sin
, and
structure.
,
cos
cos
Δ cos
Eq. 72
represent the dynamic displacements of the cutter
Define as the tangential and radial cutting forces acting on tooth j.
They are proportional to the axial depth of cut
and to the chip thickness
.
Eq. 73
Eq. 74
and
are constants.
12
Resolving cutting forces into x,y directions gives
cos
sin
sin
cos
Eq. 75
The cutting forces contributed by all teeth will be
∑
Eq. 76
∑
, where
is the pitch angle.
cos
sin
cos
Δ sin
Δ cos
Δ sin
Δ sin
Δ
cos
sin
cos
sin
cos
Δ cos
Δ cos
Δ sin
cos
sin
sin
Δ cos
sin
sin
Δ
cos
Eq. 77
Using
sin
/2
cos
/2
Eq. 78
13
With
and sin
/2
sin
cos
2 sin cos , it yields:
sin
1
sin 2
cos
cos
cos
2
cos 2
Eq. 79
1
2
cos 2
cos 2
1
cos 2
2
2
cos
cos
sin
sin
sin 2
Eq. 80
sin
cos
sin
Δ sin
cos
Δ cos
Δ sin
Δ sin
Δ
Δ
2
cos 2
sin
1
1
2
1
sin
sin
Δ cos
Δ cos sin
Δ cos sin
sin cos
Δ cos
sin
cos
cos
cos
sin
cos
Δ cos
sin
Δ
cos
cos
sin
cos sin
1
1 cos 2
2
1
1
2
1
cos 2
2
2
cos 2
14
cos
cos
sin
Eq. 81
sin
sin
sin 2
Eq. 82
cos
sin
cos
cos
1
2 cos
2
sin
sin
1
Eq. 83
sin 2
1
cos 2
Δ
sin 2
Δ
1
cos 2
yields
Δ
cos 2
cos 2
cos 2
sin 2
and
2
cos 2
1
sin 2
2
Inserting into
2
1
Eq. 84
1
cos 2
sin 2
Δy sin 2
cos 2
Eq. 85
Define time-varying dimensional dynamic milling force coefficients:
∑
sin 2
∑
1
∑
1
∑
sin 2
1
cos 2
cos 2
sin 2
cos
1
15
Eq. 86
Eq. 87
sin
Eq. 88
cos 2
Eq. 89
Resulting expressions in matrix form:
Δ
Δ
Eq. 90
The angular position of parameters changes with time and angular velocity. In
time domain in matrix form:
Δ
Eq. 91
For milling, the direction of the force is not constant but varies with time.
is periodic at tooth passing frequency
Ω
2 /
or tooth period .
Eq. 92
Expansion into Fourier series gives:
∑
Eq. 93
Eq. 94
is the number of harmonics of the tooth passing frequency . depends on
the immersion conditions and on the number of teeth. For the most simplistic
approximation,
0.
Eq. 95
16
is valid only between the entry and exit angle angles, i.e., where
1. Also,
Ω
Ω
Eq. 96
at
becomes equal to the average value of
0
.
Eq. 97
are integrated functions defined as:
cos 2
2
sin 2
2
sin 2
2
cos 2
2
sin 2
Eq. 98
cos 2
Eq. 99
cos 2
Eq. 100
sin 2
Average directional factors are dependent on
on width of cut
,
.
Eq. 101
(redial cutting constant) and
The dynamic milling expression:
Δ
(Eq. 92)
is reduced to:
Δ
Eq. 102
17
is a directional cutting coefficient matrix (time invariant but immersion
dependent).
Average cutting force per tooth period is independent of the helix angle.
Therefore,
is valid also for helical end mills.
The transfer function matrix:
and
and
and
Eq. 103
are the direct transfer functions in the x and y direction
are the cross transfer functions.
is the vibration vector at the present time
at previous tooth period
.
and
In frequency domain using harmonic functions, the vibrations at the chatter
frequency
will be:
Eq. 104
Eq. 105
Substituting Δ
gives
Δ
1
Eq. 106
is the phase delay between the vibrations at successive tooth periods T.
18
Substituting Δ
into the dynamic milling equation
Δ
(Eq. 103)
1
Eq. 107
yields
This has a non-trivial solution if its determinant is zero:
1
0
Eq. 108
This is the characteristic equation of the closed-loop dynamic milling system.
To simplify, define the oriented transfer function matrix:
Eq. 109
The eigenvalue for the characteristic equation is:
Λ
1
Eq. 110
Resulting characteristic equation becomes:
Λ
0
Eq. 111
Its eigenvalue can be solved for a given chatter frequency
, static cutting
,
,
factors
, radial immersion
, and transfer function of the
structure.
19
Consider two orthogonal degrees of freedom in the feed (x) and normal (y)
0 .
directions
Then the characteristic equation becomes a quadratic function.
Λ
Λ
1
0
Eq. 112
Eq. 113
Eq. 114
The eigenvalue becomes:
4
Λ
Eq. 115
For the plane of cut (x,y), the characteristic equation is quadratic.
The transfer functions are complex:
Λ
Λ
Λ
Eq. 116
cos
Substituting the eigenvalue and
Λ
1
sin
into
gives the critical depth of cut at chatter frequency
:
Eq. 117
20
is a real number, the imaginary part is zero.
Since
Λ 1
Λ sin
cos
0
Eq. 118
into the real part of
Substituting
, the final expression
for chatter free axial depth of cut is
1
Eq. 119
/
tan
/
tan /2
/2
Eq. 120
tan
is the phase shift of the eigenvalue.
2 is the phase shift between inner and outer modulations.
2
where is an integer number of full vibration waves (lobes)
imprinted on cut arc.
2
Eq. 121
The spindle speed is given by
Eq. 122
The transfer functions are identified and the dynamic cutting coefficients are
evaluated. Then, the stability lobes are calculated as follows:
i.
ii.
Select a chatter frequency from transfer functions around a dominant
mode.
Solve the eigenvalue equation
Λ
Λ
1
0
(Eq. 113)
21
iii.
Calculate the critical depth of cut
1
iv.
(Eq. 120)
Calculate the spindle speed for each stability lobe, k=0, 1, 2, ….
(Eq. 123)
v.
Repeat the procedure by scanning the chatter frequencies around all
dominant modes of the structure evident on the transfer function.
For a thin-walled component, flexibility is assumed only in the y-direction (see
Figure 1). For such a case,
0
Eq. 123
becomes
00
0
Eq. 124
Solve:
Λ
Λ
1
0
(Eq. 113)
0
Λ
Eq. 125
Eq. 126
Therefore:
Λ
Λ
Λ
Eq. 127
22
Using
Eq. 128
yields
1
Λ
1
Eq. 129
Λ
Eq. 130
Λ
cos 2
2
sin 2
Eq. 131
and
that denote start and exit of cut are derived
The immersion angles
from the tool to workpiece position.
and
are constants and are the slopes
of the force curves for
and
respectively.
23
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