IElectronic compiled by
Machine Tool Engineering Foundation in March, 2019
II

ISBN 978-4-909859-01-3© Machine Tool Engineering Foundation 2019
Photographs in front cover: By courtesy of Tungaloy

IIIPreface
Within the machine tool engineering context, there are the four coretechnologies, i.e. light-weighted rigid design for the structural body withlarger damping capacity, enhancement of anti-chatter capability, reductionof thermal deformation and development of innovative NC (Numerical
Control) technology. In viewing from another standing point, the machine
tool technologies consist of those for the “Manufacture (Design and
Production)” and “Utilization”, and the “Utilization” technology depends
upon the skill and qualification of the user to a large extent. Even when wecan purchase the machine tool with the best performance, we cannot run it
efficiently and effectively without having the mature technician.
Of these core technologies mentioned above, the “Anti-chatter Capability”
is closely related to both the manufacturing and the utilization, and thus has
been since 1930s and being investigated vigorously in both the academia
and the industry. Of course, the chatter suppression is at burning issue on
the strength of the chatter vibration theory established in the 1960s, i.e.
Traditional Chatter Theory.
In retrospect, Arnold of University College of Swansea publicized aninteresting report in 1945 and importantly he suggested that Shizuo Doi of
Ryojyun (Lshun) Institute of Technology conducted firstly an academic
research into the chatter vibration in 1937 (1946). After then, many
researchers and engineers involved in R & D (Research and Engineering
Development) activities on the chatter vibration in metal cutting andgrinding, and in the 1960s, Tobias, Tlusty and Merritt contributed to a large
extent to establish the theory of the self-excited chatter vibration ofregenerative type, i.e. major part of the Traditional Chatter Theory.
As well known, this traditional chatter theory consists of the three basic
expressions, i.e. “Uncut Chip Thickness, Cutting Process and Structural
Expressions”. In fact, this theory can facilitate considerably thecontrivance and establishment of a handful of valuable remedies for theIVchatter suppression. One of such outstanding remedies is the “Milling
Cutter of Irregular Tooth Pitch Type and with Differing Helix Angle(Variable Helix Angle of Flute)”, and at present such milling cutters
become very popular while carrying out the heavy-duty cutting.
With the growing importance of a kind of the heavy-duty cutting, e.g.
heavy machining per unit time for the aircraft component made of Al alloy,
however, we can observe a considerable number of the evidences, whichsuggest implicitly the limitation in the application of the traditional chattertheory so far used to some facing problems. Paraphrasing, we rely on the
traditional chatter theory, where the main spindle speed was 2, 000 rev/min
in maximum; however, it is, in general, more than 5,000 rev/min and up to
40,000 rev/min at present. As a result, the chatter vibration induces
unexpected phenomena far beyond from those in the 1960s, and thus, it isnatural to reinvestigate to what extent the traditional chatter vibration
theory is applicable, and also what is the necessary modification to the
traditional chatter vibration theory in order to be compatible it with thefacing problem.
As can be seen from practice at factory floor, furthermore, even now weface often the unknown chatter vibration problem, even when the machine
tool available is in the best quality. In certain cases, we face certain
difficulties in the chatter suppression, even when the mature engineer andtechnician endeavor to solve it. Such present perspectives imply also thecapability limitation of the traditional chatter theory and concerns.
In contrast and in due course, we can observe recently some interesting
symptoms through a relatively few noteworthy and challenging research
activities of time- and manpower-consumption. Importantly, such
symptoms range from the meticulous observation of chatter vibration,
through modification of the chatter theory, to further engineeringdevelopment for skillful chatter suppression. Of course, such a challenging
research includes a considerable number of the controversial points;
however, people in the traditional chatter vibration sphere seem not to have
paid any attentions to these new waves. More specifically, they eye onlyVthe machining point, and do not pay any attention to its surrounding
environments, i.e. machining system, although the chatter vibration isclosely related to the machine-attachment-tool-work system.
Concretely saying, we must discuss to what extent the stability chart
applicable, in which the parameters of the machine-attachment-tool-worksystem are measured while the system is in still-stand and also assumed as
to be linear characteristic. For example, a handful of researchers havealready suggested the poor applicability of the stability chart in thehigher-speed milling for Al alloy in the aircraft component. On the strengthof valuable and innovative findings within some challenging researches,
and also in full consideration of changes in machining environments, thus,
it is natural to reinvestigate and reconsider the traditional chatter theory
having been relied on, and we must deploy to a new horizon. In the mostdesirable case, these new waves may provide us with a clue for unveilingthe further essential features of the chatter vibration and also some tools tocontrive an innovative remedy for the chatter suppression.
This book consists of two parts, and Part I describes first the fundamentalsof the chatter vibration theory and suppression technology by especiallyplacing the stress on the self-excited chatter vibration of “Regenerative
Type”. Then, Part I proposes “A Concept for Advanced Chatter Theory” tomodify the traditional chatter theory to be compatible with the machining
environments at present together with suggesting the leading-edge issues tobe discussed. In nearly all technical papers, reports and books, the chattervibration deals with in fact only eying the machining point, although thesepublications state nominally the necessity of considering themachine-attachment-too-work system. Against this context, Part I can becharacterized by its wider scope, which ranges from the machining point,
through the machining space, to the structural body.
Following Part I, Part II displays first the present perspective of theresearch into the chatter vibration and then shows interesting behavior in
the chatter vibration at present, when machining the component withhigher-speed cutting. It is, in principle, noteworthy that the twoVIrepresentative leading-edge researches are demonstrated to understandwhat are facing issues in consideration of a modified block diagram ofchatter loop. Such a block diagram of chatter loop is, of course, produced
by referring to the traditional one, but integrates satisfactorily symptoms,
which imply the necessity to enhance the traditional chatter theory. In thiscontext, the Editor gives especially a quick note for the dynamic behaviorof the main spindle with higher-rotational speed, which is one of the fatal
issues when discussing the applicability of the stability chart.
Importantly, thought-provoking quick notes are furthermore publicized forthe chatter vibration in metal grinding in consideration of inactive research
activities during these several decades. In fact, we cannot find any valuable
and noteworthy research papers in metal grinding field, and thus the chattervibration theory for metal grinding remains its states as that in 1960s.
Eventually, Part II in this book is in the form of “Omnibus-like Style” toprovide related people with a map maneuver-like meeting spot in order toexchange various differing proposals, suggestions and discourses, throughwhich we may approach to the authentic and reliable information. To thisend, the Editor is very proud of the valuableness of the book in the practice,and extremely the role of the book facilitating the guide for leading-edge
research for young people in the academia.
Arnold R N. (1946) The Mechanism of Tool Vibration in the Cutting of Steel, Cutting
Tools Research: Report of Subcommittee on Carbide Tools. Proc. I Mech E., 261-284.

Professors Drs. Cornel-Mihai NICOLESCUAndreas ARCHENTIKTH Royal Institute of Technology, Chapter 5VIIProfessor Dr. Tony SCHMITZUNC Charlotte, Chapter 6

Professor Dr. Hwa-Soo LEE
Nihon University, Chapter 8

Editor Yoshimi ITO, Tokyo Institute of Technology
Professor Emeritus Dr., Part I Chapters 4 and 7

VIII

Contents
Preface

PART IFUNDAMENTALS OF CHATTER VIBRATION THEORYAND SUPPRESSION TECHNOLOGY

Headnotes ...................................................................................................... 1
Chapter 1 Basic Mathematical Model, Three Basic Expressions andStability Chart
1
1.1 Basic Knowledge about Self-excited Chatter Vibration of Regenerative
Type in Metal Cutting................................................................................ 5
1.1.1 Lower-speed Stability ............................................................. 9
1.1.2 Rational Determination of Chatter Onset in Experiment...... 15
1.1.3 Influences of Attachment on Chatter Stability ...................... 18
1.2 Characteristic Facets of Drilling, Milling and Grinding in Analysesof Chatter .............................................................................................. 22
1.3 Basic Knowledge about Self-excited Chatter of Regenerative Typein Metal Grinding .................................................................................. 31
1.3.1 Theory for Chatter of Regenerative Type ............................. 32
1.3.2 Research and Engineering Development Subjects ............... 38
References ................................................................................................... 41
Chapter 2 First-hand View for Chatter Suppression Technology
2
2.1 Milling cutter of Irregular Tooth Pitch Type and with Differing
Helix Angle ........................................................................................... 48
2.1.1 Milling Cutter of Irregular Tooth Pitch Type ........................ 53
IX

2.1.2 Milling Cutter with Different Helix Angle ........................... 57
2.2 A New Horizon - Integrating Damping Entities within Machine
Bodies, Machine Elements, Attachments and Tools ............................. 61
2.3 Elastic Grinding Wheel and Others .................................................... 66
References ................................................................................................... 69
Chapter 3 Symptoms of Needs to Advanced Chatter Theory
3
3.1 A New Proposal for Classification of Chatter - Reinvestigation
into Validity of “Forced Chatter” ......................................................... 72
3.2 A Proposal for “Modified Block Diagram of Chatter Loop” ................ 77
3.3 Some Representative Issues within Three Basic Expressions .............. 80
3.3.1 Issues in Uncut Chip Thickness Equation ............................ 81
3.3.2 Issues in Cutting Process Equation ....................................... 83
3.3.3 Issues in Structural Equation ................................................ 87
3.4 Drooping Characteristics in Real Dynamic Component of Cutting
Force and Self-excited Chatter.............................................................. 94
References ................................................................................................... 98

PART IILEADING-EDGE RESEARCH AND ENGINEERINGDEVELOPMENTWITHIN ADVANCED CHATTER VIBRATION THEORY

Headnotes .................................................................................................. 101
Chapter 4 Observation of Various Chatter Marks and a Proposal of Trouble
Shooting for Identification of Chatter Patterns
4
4.1 Case Studies on Observation of Chatter Marks .................................. 107
4.2 A Proposal for Troubleshooting to Identify Chatter Patterns.............. 112
References ................................................................................................. 117

XChapter 5 Statistical Dynamics - Analysis of Machining Systems
Operational Conditions(by Professors Andreas Archenti and Cornel-Mihai
Nicolescu: KTH Royal Institute of Technology)
5
5.1 Introduction ......................................................................................... 119
5.2 Statistical Dynamics............................................................................ 122
5.3 Random Processes .............................................................................. 127
5.4 Markov Process ................................................................................... 128
5.5 Non-parametric Models ...................................................................... 132
5.6 Nonparametric Models Based on Energy Method .............................. 144
5.6.1 Stiffness Estimation ............................................................ 146
5.6.2 Damping Estimation ........................................................... 148
5.6.3 Identification of Machining System ODPs ......................... 151
5.6.4 Operational Stiffness Identification .................................... 153
5.6.5 Operational Damping Identification ................................... 154
5.7 Parametric Models for Machining System Identification ................... 156
5.7.1 Gauss-Markov Sequence .................................................. 157
5.7.2 Parametric ARMA Models ............................................... 160
5.7.3 Power Spectral Density of ARMA Models ....................... 162
5.7.4 Operational Dynamic Parameters (ODP) ......................... 164
5.7.5 Stability Analysis .............................................................. 166
5.7.6 ARMA Model-based Off-line Identification .................... 168
5.7.7 Parametric Modeling for a Milling Operation .................. 168
5.7.8 Evaluation of the Operational Modal Parameters ............. 170
5.7.9 Real-time Identification of Machining Systems ............... 178
5.7.10 Model Structure Selection and Model Validation ............. 180
5.7.11 Model Structure Selection ................................................ 181
5.7.12 Model Validation............................................................... 182
5.7.13 On-line Identification –Turning Case ............................... 182
5.7.14 Discussion and Conclusions ............................................. 184
References ................................................................................................. 186

XI

Chapter 6 Milling Dynamics: Forced Vibration, Self-excited Vibration, andPeriod-n bifurcations(by Professor Tony Schmitz: UNC Charlotte)
6
6.1 Introduction ......................................................................................... 192
6.2 Time Domain Simulation .................................................................... 194
6.3 Stability Maps ..................................................................................... 198
6.4 Bifurcation Diagrams .......................................................................... 204
6.5 Poincaré Maps ..................................................................................... 206
6.6 Surface Location Errors ...................................................................... 210
6.7 Conclusions ......................................................................................... 215
References ................................................................................................. 215
Chapter 7 Quick Notes for Rigidity and Damping Capacity of Main
Spindle Supported by Rolling Bearing
7
7.1 Introduction ......................................................................................... 222
7.2 Fundamentals of Main Spindle Supported by Rolling Bearing .......... 223
7.3 Difficulties in Measuring Rigidity and Damping Capacity of
Rolling Bearing While Rotating Main Spindle ................................... 230
7.4 Concluding Remarks ........................................................................... 232
References ................................................................................................. 234
Chapter 8 Thought-provoking Quick Notes for Characteristic Factors in
Metal Grinding(by Professor Lee Haw-Soo: Nihon University)
8
8.1 Contact Compliance of Grinding Wheel to Workpiece ...................... 235
8.2 Grinding Stiffness ............................................................................... 239
8.3 Unbalance of Grain Distributions in Grinding Wheels ....................... 241
8.4 Grinding Damping .............................................................................. 244
8.5 Effects of Parameters Being Current Issues on Chatter ...................... 244
References ................................................................................................. 246
XII

PART IFUNDAMENTALS OF CHATTER VIBRATION THEORYAND SUPPRESSION TECHNOLOGY

Headnotes
In Chapter 1, we describe first the fundamental knowledge about the
chatter in turning together with the three burning issues, i.e. (1)Lower-speed Stability, (2) Rational Determination of Chatter Onset inExperiment and (3) Influences of Attachment on Chatter Stability. Of these,
the lower-speed stability is one of the well-known research subjects andhas been investigated vigorously without considering its essential features,
which will be suggested within Item 1.1.1. For the latter two of three, wehave been aware of their great importance; however, it is regrettable thatpeople in chatter concerns are not interested in such research subjects. In
fact, we can observe several forerunning researches at present as
exemplified in Items 1.1.2 and 1.1.3.
On the strength of the chatter theory in turning, then, we describe thechatter vibration theories in drilling, milling and grinding in Section 1.2 byextremely revealing their characteristic and differing features as comparedwith those in turning. Importantly, the grinding mechanism may be
regarded as one of the variants of milling; however, we can observe thevery differing behavior of the chatter vibration in grinding. Such differing
behavior are derived, for example, from the non-uniform elasticity of thegrinding wheel, grain with negative rake angle and grain self-edging
function. In consequence, we discuss the self-excited chatter ofregenerative type in grinding in Section 1.3 and suggest that the chattertheory for grinding does not progress since 1960s. More specifically, we
can suggest what are at burning and facing issues for the chatter vibration
in metal cutting, whereas we cannot unveil such issues in grinding, becauseof not conducting any forerunning and challenging researches so far. Thus,
the Editor suggests finally some representative research subjects like
“Determination for Contact Stiffness of Grinding Wheel in Consideration1

of Grain Self-Edging Function”.
On the strength of the block diagram of chatter loop, we classify first theremedies for the chatter suppression, and give some quick notes for each
representative remedy in Chapter 2. In addition, we show a first-hand viewfor R & D activities in the form of “Puttick Grid”, which may play the roleof seeking a forerunning research subject. Of course, there have been a
considerable number of remedies for the chatter suppression, which ranges
from the contrivance of the cutting edge, through the employment of thedamper, to the application of the vibration-proof material. Of these, theutmost excellent remedies are credited to both the milling cutter ofirregular tooth pitch type with differing helix angle, and the elastic grinding
wheel. Thus, the theory and practice of such milling cutters and grinding
wheel describe also in Chapter 2. Importantly, we clarify the differing
aspects in the effectiveness of the chatter suppression between those of
irregular tooth pitch type and differing helix angle. In Chapter 2, we givefurthermore some quick notes for the positive use of the machine tool joint
to increase the damping capacity in the machine-attachment-tool-worksystem together with suggesting the difficulty to measure correctly thedamping capacity within this system.
As can be readily seen from some controversial points suggested in
Chapters 1 and 2, the dire necessity is to conduct the further research into
the chatter from an innovative viewpoint, so that we may clarify the newer
essential feature of the chatter than ever before. On that occasion, we mustbe first aware what are at burning issues to be the traditional chatter theory
applicable to the machining state at present. Paraphrasing, a crucial issue isto establish an advanced chatter theory, which is fully available for thehigher rotational speed of the main spindle at present, and which can befacilitated the chatter including the forced excitation caused by the cutting
mechanism itself. Thus, Chapter 3 deals first with the influence of theforced excitation on the stability chart, and then discuss the major issues inthe three basic expressions for the chatter theory after proposing a modified
block diagram of chatter loop. In the latter case, it is extremely notified that2

we must measure the coefficients of the structural equation while the main
spindle rotates more than 2,000 rev/min, if possible 5,000 rev/min andbeyond together with considering the effects of the joint like those between
main spindle and collet chuck, and between collet chuck and end mill.

3

Chapter 1 Basic Mathematical Model, Three
Basic Expressions and Stability Chart
In machining, the chatter vibration (hereafter call it “Chatter” only) is at
facing issue to increase the productivity and to ensure the part quality.
Although we have had the long history in R and D (Research andEngineering Development) for the chatter since 1930s, we cannot suppress
the chatter completely in certain machining cases even now. A root cause ofits difficulties lies in the complex and synergistic behavior of the chatter. In
fact, we must deal with the chatter from the viewpoint of the dynamic
behavior in the machine-attachment-tool-work system. More specifically, R
and D for the chatter ranges from the clarification of essential features of
the chatter, through the establishment of the chatter theory, to thecontrivance of various suppression remedies. Obviously, we may observe
influences of each entity’s own, and in certain cases the synergistic
influences of some entities in the above-mentioned system on the chatter.
Reportedly, we have, in nearly all cases, conducted the research into theself-excited chatter of regenerative type so far, although there are aconsiderable number of the unknown types in the chatter (refer to Chapter3). It is furthermore worth suggesting that the chatter while milling anddrilling have recently been investigated vigorously because of theirenchanting features from academic research point of view, e.g. influencesby the continuous change of the wedge angle and thinning of chisel point
on chatter in drilling. In fact, we need to discuss the sophisticatedmathematical model in such machining methods as will be discussed in
Section 1.2, and the advanced simulation software and dynamic
information processing available at present can facilitate considerably tosolve such a complex model.
As will be clear from the above, we must first understand the utmost
simplest case in the chatter, i.e. regenerative type in turning, and as widely
known, Tobias (1961), Merritt (1965) and Tlusty (1970) contributed to the4establishment of the corresponding chatter theory to a large extent as will
be described in Section 1.1 (hereafter call it “Traditional Chatter Theory”).
Importantly, nearly all researchers and engineers rely on this traditional
chatter theory, although there remain still some engineering problems, e.g.
“Lower-speed Stability”, “Rational Determination of Chatter Onset(Commencement) in Experiment“ and “Effects of Attachment on Chatter“.
Of special note, we must be aware that the chatter in metal cutting has been
so far investigated to a large extent as compared with that in metal grinding.
In addition, it is notable that the leading objective is the chatter for bending
mode, but not for torsional mode. In general, we may assume that the
chatter in torsional mode activates at the higher rotational speed of main
spindle as compared with that for bending mode, and thus for the time
being, such a chatter is not crucial issue in metal cutting. In contrast,
Entwistle and Stone (2013) have suggested a preferable effect of thetorsional movement in the grinding wheel on the chatter.

1.1 Basic Knowledge about Self-excited Chatter Vibrationof Regenerative Type in Metal Cutting
In many respects, it is, in general, very convenient for the ease ofunderstanding to use the “Block Diagram of Chatter Loop” shown in Fig.
1.1. As widely known, Merritt first proposed it in the 1960s based on thethree basic expressions so far established by Tobias and Tlusty. We canbenefit considerably from the block diagram of chatter loop especiallywhen estimating the stability limit for the chatter by relying on theautomatic control theory. In addition, the chatter suppression technology
has so far been developed on the strength of the block diagram of chatterloop to a large extent as typically exemplified by the milling cutter ofirregular tooth pitch type and with differing helix angle.
For the simplified model of the machine-attachment-tool-work system
shown together in Fig. 1.1, the three basic expressions can be written as
5

Fig. 1.1 Mathematical model and block diagram of chatter loop for selfexcited chatter vibration of regenerative type (by Merritt)“Uncut Chip Thickness Equation”
u(t) = u0(t) – y(t) + μy(t-T) ----------- (1.1)“Cutting Process Equation”
F(t) = kcu(t) --------------------(1.2)“Structural Equation”

where,
c = Damping coefficientF(t) = Resultant cutting force or vector force exciting structurekc = Specific cutting force (Static cutting stiffness)
k = Spring constant of structurem = Equivalent mass of structure
N = Spindle speed
T = Delay time (1/N for lathe)t = Timeu(t) = Instantaneous uncut chip thickness
u0(t) = Steady-state (initial) uncut chip thickness
6y(t) = Relative displacement between tool and work
μ= Overlap factor

To produce the block diagram of chatter loop, then, we should convert
these expressions into the following by using the Laplace transformation.
u(s) = u0(s) – y(s) + μe-Tsy(s) ----- (1.4)
F(s) = kcu(s) ----------------------(1.5)y(s)/F(s) = (1/k)G(s) ------------------- (1.6)
where,
G = Normalized dynamic compliance of structures = Laplace operator
From the three basic expressions, we can first determine the “Characteristic
Equation (Transfer Function)”, and then produce the stability chart asshown in Fig. 1.2, where a representative index is either “Limit Depth of
Cut or Limit Width of Cut” in the vertical axis. In producing the stability
chart, we must assume the utmost critical condition, i.e. μ = unity.

Fig. 1.2 Typical stability chart (by Merritt)

7

Within the allowable magnitude of this index, i.e. less magnitudes thanabsolute stability limit, we have not any chatter across the whole spindle
speed. To ease of use, we define the three borderlines as shown together in
Fig. 1.2, i.e. asymptotic, tangent and lobed borderlines 1.
As can be readily seen, the asymptotic borderline indicates the absolutestability limit, under which no chatter occurs across the whole spindle
speed. Reportedly, it is desirable to machine the component by using thespindle speed, which corresponds with the “Cusp-like Region” of lobed
borderline, and duly such a remedy is being prevailed as exemplified bysome merchandized software.

Fig. 1.3 Stability chart in general, lower-speed stability and effect of end
mill with differing helix angle
Obviously, the stability chart can facilitate the prediction of the chatter
onset, whereas we have two fundamental problems to be solved since
1960s: one is the existence of “Lower-speed Stability”, and another is the1

In the stability chart, we employ often some indexes equivalent to either the limit depth of cut or thelimit width of cut. For example, Tobias employed the “Vibration amplitude enlargement coefficient Q”,
which is given by Q = 1/(2D), where D is the ratio of equivalent damping coefficient to critical damping
coefficient. In addition, Tlusty and Merritt employed “Ratio of chip width to limited chip width” and“Ratio of cutting stiffness to machine stiffness”, respectively.

8

“Rational Determination of Chatter Onset in Experiment”.
For the sake of further understanding, Fig. 1.3 illustrates the ranges of thelower-speed stability and the stability limit obtainable from the milling
cutter with differing helix angle, which is very beneficial especially in thelower-speed range by enlarging the stability limit.

1.1.1 Lower-speed Stability

As widely known, there is a discrepancy between the theoretical andexperimental stability charts at the lower-speed range as already shown in
Fig. 1.3. More specifically, the limit depth of cut increases considerably at
the lower-speed range in practice, although the chatter should occur at thesame speed range in accordance with the chatter theory. This is the“Lower-speed Stability”, and in general the heavy-duty cutting is carriedout in such speed ranges, where we face considerable danger of chatter
onset, and thus, the turner welcomes the lower-speed stability. As suggested
by Tobias, the lower-speed stability is derived from the interaction between
the relief angle (flank) in the cutting tool and the work, i.e. “Penetration
Effect”, which can facilitate the increase of damping. In fact, Tobias
verified theoretically the validity of the lower-speed stability by including
the penetration effect in the (nominal) dynamic cutting force. Of special
note, now we use the term, “Process Damping” instead of the penetration
effect, and importantly such a hypothesis of the process damping has been,with no doubt, accepted as the utmost leading causality for the lower-speed
stability by nearly all people in chatter concerns.
From the empirical point of view, such an interpretation is acceptable. In
fact, the turner increases often the contact between the relief face of cutting
tool and the work as a factory floor-based remedy to suppress the chatter 2.
Of note, we can also suppress the chatter with higher frequency by placing
the structural component with large damping, e.g. tool slide made of
2 Following the proposal of Tobias, Kegg conducted a series of experiments for the lower-speed stability
in 1969, and he verified the validity of the penetration effect. In the experiment, he carried out cylindricalturning, parting-off and milling for the workpieces made of cast iron, steel, Al alloy and plastics.
Kegg R L. (1969) Chatter Behaviour at Low Cutting Speeds. Annals of CIRP; 17: 97-106.

9

concrete, as close to the cutting point as possible.
In this context, Sellmeier and Denkena (2010) conducted recently aninteresting research into milling of the Al alloy. Fig. 1.4 reproduces theexperimental data, and as can be readily seen, they reported the amazing
increase of the depth of cut without the chatter by increasing the chamfer
length of cutting edge. In fact, the depth of cut increases around four times
and three times in the lower and higher speed ranges, respectively.
Importantly, they assert that the chamfer of the cutting edge can increase
process damping, and that process damping is effective even in the higher
spindle speed range.

Fig. 1.4 Changes of stability limit for various chamfer length(by courtesy of Denkena, 2015)
In consideration of the average magnitude of damping capacity in themachine tool as a whole, is it possible to be attributed such an amazing
increase of the depth of cut to only the process damping? Although notdenying the effectiveness of the factory floor-based remedies, we must eyeother sources of damping possible within the machine-attachment-tool-work system. In fact, we have not verified obviously so far that
damping at the cutting point is dominant in the determination of the10

damping coefficient within the “Structural Equation”. More specifically,
process damping has not been experimentally and quantitatively verified its
validity; however, some researchers and engineers assert the following.
(1) Process damping is derived from the elastic and plastic deformation of
the interference volume at the joint between the flank of cutting tooland the wavy surface of the work. In short, the plowing force causes
process damping.
(2) Process damping is subjected to the “Frictional Energy Loss” andviscous-like.
Even admitting that the plowing force is one of the leading causalities ofprocess damping, a burning issue is the “Strain Rate”, when theinterference volume deforms elastically and plastically. In general, we cancompute such deformation by using the material constants, which can beobtained by the material testing. As well known, the strain rate at thematerial testing is less than 10-3/s, whereas the strain rate at the rake face
while machining is more than 103/s. Although the concrete data for thestrain rate at the flank is not clarified as yet, at least, we must be aware ofdiffering features of deformation mechanism between at the flank and at
the material testing.
It can be furthermore suggested the necessities for conducting thecorresponding research with much wider scope as follows (Ito, 2014).
(1) In accordance with the knowledge in the tribology and machine tool
joint, viscous damping appears at the dry joint, provided that the“Tangential Force Ratio”, i.e. microscopic coefficient of friction,
regulates the sliding condition to large extent, but not the “Coulomb’s
Friction”. We must thus investigate once the availability of thetangential force ratio at the cutting point.
(2) The tangential force ratio varies with the magnitude of sliding length,
i.e. “Displacement-dependence Characteristics”. In general, the shorter
the sliding length, the more effectiveness of the tangential force ratio.
In this context, Ito and Matsumura (2017) suggested that the contact
between the relief angle of cutting tool and the work is one of the machine11

tool joints, and they asserted a doubt for the effectiveness of process
damping from viewpoint of its magnitude, i.e. process damping being too
small to suppress the vibration 3. In addition, they suggested the difficulty
for measuring correctly process damping, although some researchers
publicized the measured value of process damping. Of special note, thedamping capacity of tooling with HSK (in German; der Hohle Schaft Kegel,in English; Hollow Shank Taper) is 0.01 ~ 0.03 in logarithmic damping
decrement in accordance with the private information from professor
Tsutsumi of Tokyo Agricultural University.
More specifically, Saljé and Isensee (1976) measured, for example, thedamping capacity of the center by exciting the mid-way of the slender work,
which was supported at both the ends by centers. In many respects, thedamping capacity of surrounding components could include in themeasured values, and of course, it is very difficult to eliminate suchunfavorable noises without using, at least, the equivalent monolithic centersystem. Thus, Saljé and Isensee reported some comparative damping
capacity, but not the real one. Of these measured values, the damping
capacity for the axial force of 1,000 kgf is 0.005 in damping rate, which isequivalent to 0.03 in logarithmic damping decrement by the simple
conversion. This magnitude is only several times of the material dampingof cast iron. It is thus natural to estimate relatively small damping force at
the cutting edge.
Against to this context, someone asserts that the contact condition of
cutting point differs from the machine tool joint; however, it may be said
that the sliding condition at the tool flank is not severe as compared withthat at the rake face, where the apparent contact area seems to be the sameas the real contact area.
3

Jochem et al suggest that process damping increases with lowering the speed, where the pitch of wavy
surface generated by the chatter is shorter, resulting in the density contact at the relief face of the cutting
tool. From the viewpoint of the joint stiffness and damping capacity in machine tools, this suggestion isincredible and seems to be wrong. In fact, this suggestion is completely in reverse to the fundamental
knowledge about the joint characteristics, where the density contact results in more rigid and lessdamping capacity.
Jochem C, Roukema, Altintas Y. (2006) Time domain simulation of torsional-axial vibrations in drilling.
Inter. Jour. of Machine Tools & Manufacture; 46: 2073-2085.

12

Of special note, we must eye furthermore the excitation energy aspect in
the chatter, and for example, Sato (1973) conducted an interesting research
into the mechanical impedance for the turning machine-tool-work system.
In this experiment, the system was excited by the small-sized hydraulicvibrator, which was placed between the tool slide and the work and input
the cutting force-related signal, while the work is still stand. He showed an
obvious difference of the mechanical impedance in the lower frequency
from that in higher frequency as shown in Fig. 1.5.
In short, the system cannot be excited in the lower frequency range lessthan 60 Hz, because of shortage of excitation energy, and intuitively, this
implies another causality of the lower-speed stability.

Fig. 1.5 Change of mechanical impedance in turning machine-tool-worksystem (by courtesy of Sato)
Summarizing, process damping has been theoretically investigated byassuming some damping sources at the contact between the relief face of
cutting tool and the work in nearly all research reports. A “Must” at present
13

is the verification of such an assumption so far employed, and thus it isnecessary to visualize such contact conditions. In this context, people assert
the difficulty for the visualization; however, Itoh et al (1991) tried an
in-process measurement of the tool wear by using the ultrasonic waves. Fig.
1.6 reproduces the single-point cutting tool integrated the ultrasonictransducer, and they publicized some fruitful results.
More specifically, prior that of Itoh et al, Spur and Leonard (1975) once
tried an in-process measurement of the flank wear also by means ofultrasonic waves. Spur and Leonard measured the flank wear by using the
response time of ultrasonic waves, whereas Itoh et al used the change of thesound pressure, i.e. echo height on CRT (Cathode Ray Tube), to improvethe measuring accuracy by eliminating the thermal elongation of the tool
shank. As will be clear from the above, we will be able to verify
experimentally that the lower-speed stability increases when the contact
area of flank face becomes larger.

Fig. 1.6 In-process sensor of built-in type for flank wear measurement
by ultrasonic waves

141.1.2 Rational Determination of Chatter Onset in ExperimentWith respect to the “Rational Determination of Chatter Onset”, the Editor
publicized already a position paper, and suggested duly that nearly all
research papers and technical reports so far publicized did not pay any
attentions to the identification of the chatter onset in the experiment (Ito,2013). Apparently, these publications believe that the chatter onset can be
automatically and absolutely determined without any problems from theexperimental data. It is however necessary to pay the special attention to
the arbitrary behavior for the output signal obtainable from the chatter. In
addition, we must be aware that the machining space is very badenvironments for the sensor, and often very noisy. As a result, a root causeof difficulties lies in the selection of the robust sensor and suitable signal
processing to detect the reliable output signal.
In retrospect, Braun of Israel Institute of Technology (1975) suggested
already the difficulty in identifying correctly the chatter onset without
processing properly the signal and proposed an index for the absolute
threshold to determine the transition from stable to unstable states.
Following that of Braun, Nakazawa et al (1979), Higuchi et al (1986) andNicolescu (1991) conducted similar researches and reported some valuable
results. For example, Higuchi et al detected sound signal with time-series
analysis and observed that the chatter onset decided by the sound signal isearlier than that decided by the human sensory. In fact, Rahman and Itoreported already such an interesting suggestion as will be stated in detailbelow.
Rahman and Ito (1979) once suggested that the determination of chatter
onset depends considerably upon what is the characteristic index, and how
to detect such an index. In fact, they detected independently (1) thevibrational amplitude, (2) cutting force, (3) work displacement, and (4)surface roughness of the finished work as shown, for example, in Fig. 1.7.
Importantly, they employed the tapered workpiece, which was proposed by
The Machine Tool Industry Research Association, England.
From Fig. 1.7, we can understand the considerable difficulties in the15

identification of the chatter onset in the experiment as follows.
(1) A characteristic behavior can be observed in the work displacement inhorizontal direction (depth of cut direction) at the chatter onset. Morespecifically, the horizontal displacement of the work launches out toreduce at a certain point (as marked with “a” in Fig. 1.7), although thedepth of cut increases continuously.

Fig. 1.7 Determination for onset of self-excited chatter using characteristic
behavior in displacement of work(2) The point, where the horizontal displacement of the work is about todecrease, is identical to the point, where the surface finish of the work
starts to be worse. In contrast, the vibrational amplitude shows any
marked changes at this point.
(3) The detection of the chatter onset by human ears (as marked with “c”
in Fig. 1.7) is later than that by the horizontal displacement of the16

work. For example, we can recognize about 2.5 mm different between
both the depths of cuts.
In 2010, Denkena et al (2010) have quickly stated that they determined thechatter onset in end milling by totally using the information obtained fromthe visual inspection, surface roughness of finished work, sound andvibration while machining, and deflection of tool holder. This implies thateven now we have not had any reliable tools to determine experimentallythe chatter onset, and thus we must be duly aware of the following.
(1) As exemplified in Fig. II.1 (a) in the headnote of Part II, there havebeen a considerable number of the reports regarding the identificationof the chatter; however, it is even now difficult to determine rationallythe chatter onset in practice.
(2) We can detect the chatter by various sensors and methods, e.g. humanears, microphone, accelerometer, cutting force, surface roughness andso on. There are however no guides indicating a suitable and reliable
detection method of the chatter onset.
Of special note, Nicolescu (1991) investigated the chatter onset by using
similar methods to that of Braun by detecting the mean-square value ofvibrational amplitude. In addition, Yeh et al (1992) proposed a method, inwhich signal processing is based on the average value of vibrational
amplitude together with using the two-tiered threshold. By it, the detection
accuracy becomes better than other methods.
As will be clear from the above, it is incredible that some researchers assert
the good agreement between the theoretical and experimental stability
charts without clearly stating the determination method of the chatter onset
in the experiment. It is again emphasized that one of crucial issues in the
self-excited chatter is to define rationally and to determine correctly thechatter onset in the experiment. In this context, we must be furthermore
aware that the machine-attachment-tool-work system is of multiple-degreeof freedom with directional orientation in vibrational parameters.
Conceptually, we must produce a stability chart, in which the stability lobe
should have certain allowances, i.e. stability chart depicting with band-like17

lobe, even when using the simplified mathematical model. Then, we must
compare the agreement between the theoretical and experimental stability
limits.

1.1.3 Influences of Attachment on Chatter Stability

We used to state a stereotyped introduction in the research paper, i.e.
chatter being dynamic behavior within the machine-attachment- tool-worksystem. On the contrary, nearly all research papers assert the good
agreement between the theoretical and experimental stability charts without
stating anything about the work and tool holding devices. In general, such

Fig. 1.8 Superimposing “Scale-like Mark” on self-excited chatter markwork and tool holding devices are non-linear characteristic in their joint
rigidities and affect considerably the magnitude of the spring constant in
the “Structural Equation” (Ito, 2017). In accordance with Editor’s

18

long-standing experience in the factory floor, the stability chart varies
considerably by the work holding condition. For example, the work held by
the chuck with knurled gripping surface increases much more the stability
limit in turning than that with finish-turned.
Typically, M. Doi et al (1982) revealed that the three-jaw chuck induces theparametric vibration mingling within the self-excited chatter ofregenerative type as will be discussed herein. As reported elsewhere, thework holding stiffness of the three-jaw chuck varies three times every one
rotation of the main spindle, resulting in the directional orientation effect of
the work stiffness 4. As a result, the chatter onset differs from each otherdepending upon the jaw position as shown in Fig. 1.8. Eventually, we must
investigate the regenerative type in consideration of mingling theparametric vibration caused by such a directional orientation by solving the“Mathieu Differential Equation”.
More specifically, they showed apparently an evidence, i.e. superimposing
partly “Scale-like” mark caused by the parametric vibration on the
self-excited chatter mark as shown together in Fig. 1.8 with enlarged viewfor the three-dimensional surface roughness. In fact, we can observeobviously the “Scale-like” mark at the slope of the self-excited chatter
mark.
Figure 1.9 demonstrates a comparison of the theoretical asymptotic
borderline of stability for work held by the three-jaw scroll chuck with thatby four-jaw independent chuck. Of note, the chatter stability is evaluatedby the index kc/kw, where kc and kw are the cutting stiffness and average
stiffness of chuck-work system, respectively, and the experimental value ofstability limit is also shown in Fig. 1.9.
From Fig. 1.9, we can observe the very interesting behavior as follows.
(1) In the case of three-jaw scroll chuck, the experimental values areplotted far lower at lower spindle speed and higher at the higher spin4 In retrospect, Tlusty suggested already the directional orientation in the rigidity of the tailstock andalso certain influences of the live center on the stability limit.
Tlusty J. (1970) 4.3 Centre lathe. In: Koenigsberger F, Tlusty J (eds) Machine Tool Structures, Pergamon
Press, pp. 243-265.

19

Fig. 1.9 Decrease of stability limit by mingling with parametric vibration
dle speed than the theoretical asymptotic borderline, respectively. The
experimental value at higher spindle speed corresponds well with thatof parametric vibration.
(2) In the case of four-jaw independent chuck, all the experimental values
can be plotted to be closer to the theoretical asymptotic borderline, andthus it may be concluded that there are no influences of the parametricvibration. Importantly, we can observe such a parametric vibrationeven in light cutting as shown also in Fig. 1.9.
In addition, M. Doi et al (1985, 1990) suggested the larger influence of jaw
traveling mechanism on the chatter stability by comparing the four-jaw
independent chuck with that of scroll type.
After long interval, Monnin (2013) has recently measured the frequency
response in the first bending mode of the main spindle-holding
device-milling cutter system in still stand as shown in Fig. 1.10 and Table

20

1.1. In addition, Monnin identified the specific cutting force for eachmilling cutter.

Fig. 1.10 Receptance functions at tool tip for various tool-holder
assemblies (by Monnin)

Table 1.1 Dimensional specifications of milling cutters (by Monnin)
Monnin conducted the experiment by using the quinaxial-controlled MC
21

(Mikron Agie Charmilles-brand; Type HPM), and the frequency response
was measured by impulse excitation. In the experiment, furthermore, thecorresponding signal was detected with the accelerometer, and the
maximum singular value (MSV) is an index to represent the maximum
amplitude of receptance function. As can be readily seen, we can observe
the remarkable differing feature when varying the milling cutter and its
holding device.
To this end, it emphasizes again, as exemplified these forerunning researchworks, that the chatter should be dealt with from the system point of view;
however, we are not so keen to do so.

1.2 Characteristic Facets of Drilling, Milling and Grindingin Analyses of ChatterIn the year 2000 and beyond, we have placed the main stress on the
research into the chatter for milling and drilling instead of turning. Of
course, the research has been carried out on the strength of the traditional
chatter theory for turning; however, it is necessary and inevitable to pay the
special attention to the differing features in cutting mechanisms among
turning, milling, drilling, and furthermore grinding.
For example, a root cause of difficulties in drilling lies in the uncertainty
derived from the damping capacity of the chisel point. Fig. 1.11 sows the
typical triangular-like shape of the hole, which is generated in the very
beginning of the drill engagement to the work (Abele et al., 2007). Such abehavior is well known from the past, and Shiozaki and Furukawa (1974)
proposed already a simple qualitative analysis regarding why thetriangular-like hole is generated as shown in Fig. 1.12.
In the left-hand side of Fig. 1.12, the simple mathematical model is shownin consideration of the geometrical deviation of both the cutting edges in
the chatter.
From Fig. 1.12, the instantaneous relationships between the drill diameter
and waviness of drilled hole can be written as22

Fig. 1.11 Triangular-like hole shaped by chatter of self-excited type(by courtesy of Abele et al., 2018)[ rw(φc) + rw0 ] + [ rw(φc – π) + rw0 ] = d0 + y(φc) --------- (1.7)
where, d0 = 2rw0, and rw(φc) + rw(φc – π) = y(φc)
Thus, we should first convert φc in Eq. (1.7) by the Laplace transformation,and then the operator “s” is replaced by “jn”. Consequently, the transfer
function between Y(s) and Rw(s) yields to[ Rw(jn)/Y(jn) ] = 1/[ 1+ e-πjn ] --------- (1.8)
The gain of this transfer function is in maximum when n is odd number asshown in right-hand side of Fig. 1.12. In general, the vibration amplitude
becomes larger at the lower odd number, and thus we can observe thechatter mark of triangular-like shape. It is however regrettable that they did
not analyze quantitatively, because of difficulty to estimate the damping
23

capacity at the chisel point. It seems that Abele et al faced also the sameproblem.

Fig. 1.12 Mechanism for generating “Odd-shaped” hole in drilling(by Shiozaki and Furukawa)In the following, thus, some quick notes will be given with respect to thediffering features in drilling, milling and grinding as compared with turningby single-point cutting tool, which is closely related to the “Cutting Process
Equation”.
Figure 1.13 shows the characteristic features in twist drill, i.e. existence ofchisel point and changing wedge angle (equivalent rake angle) along
cutting edge. More specifically, the chisel point has certain thickness (web)and acts like rotating dull chisel resulting in the poor cutting performance,
whereas it may facilitate higher damping to be in realty. Importantly, theequivalent rake angle increases with the distance from the drill center as24

shown in the right-hand side of Fig. 1.13 5. In short, the specific cutting
force changes along with the cutting edge (in radial direction).

Fig. 1.13 Changes of cutting edges from dull to sharp wedges in twist drill
More importantly, we often employ the thinning of the drill point to reduce
the thrust component of the cutting force and to improve the cutting
performance as shown in the right-hand side of Fig. 1.14. In drilling forlarger hole, we used to pre-machine the small hole to eliminate theunfavorable influence of the chisel point as shown in the left-hand side ofFig. 1.14. As can be readily seen, such a drilling method can be replaced byturning with twin-cutting edge of continuously changing rake angles. In
addition, the re-ground drill shows certain deviations from the symmetry inboth the cutting edges, i.e. lips of different lengths, without carefully
re-grinding, which results in the unevenness in the radial component of the
cutting force in practice. Obviously, such an incremental radial component
5 To ease of understanding, the cutting edge is schematically illustrated as like as that of single-point
cutting tool in turning, and thus we call it “Equivalent Rake Angle”. In fact, the rake angle of the twist
drill is indicated in the top of Fig. 1.13.

25

Fig. 1.14 Various thinning methods and preparatory-produced hole toeliminate unfavorable influences of chisel point in twist drilling
may cause either unfavorable, or advantageous influences on the chatterstability, although the twist drill has the body clearance. It is thus
recommendable to investigate the chatter when using the re-ground drill in
consideration of its quality.
Reportedly, the research into the chatter in milling has recently been carriedout vigorously, because of its attractive features as mentioned beforehand.
For example, we need to discuss the sophisticated mathematical model,
whereas we can apply the advanced simulation software and dynamic
information processing to solve the chatter, which can facilitate the ease ofanalytical and experimental activities to a large extent.
The milling cutter rotates and traveling simultaneously, and thus the cuttingedge generates the trajectory represented by Trochoidal curve as shown in
Fig. 1.15. In consequence, the uncut chip thickness is given by the area,
which is surrounded by both the tooth being engaged and the preceding
tooth. In many respects, the characteristic features in milling are thus26

derived from the following three factors.

Fig. 1.15 Tooth trajectory and uncut chip thickness in milling(1) The uncut chip thickness is approximately proportional to the feed rate
per tooth and simultaneously the rotational angle of tooth.
(2) The cutting force depends upon the number of cutting teeth being
engaged in the work.
(3) The impact load acts on the work when the tooth engages in the work,and thus there are considerable differences in the cutting mechanism
between up-cut and down-cut modes.
Now let us determine the (x, y) co-ordinate by taking the deepest positionof the tooth as an origin, and the tooth is now in the position P after rotating
φ as shown in Fig. 1.15, the co-ordinate P can be written as27

where, r = Radius of milling cutter, s = Feed speed of table
N = Rotational speed of milling cutter,ω = Angular velocity of milling cutter.
By substituting Eq. (1.10) to Eq. (1.9),

The cutter trajectory for one revolution before is given by

where,

Because the Trochoidal curve is one of the transcendental functions, it isdifficult to calculate analytically the distance between two adjacent curves,and thus after carrying out the approximation method, the uncut chip
thickness yields to
h = szsinφ ------------------- (1.13)
where,In short, the chip thickness changes continuously with the rotation of themilling cutter, and thus the specific cutting force is not constant, but varies
while the milling cutter rotates. Importantly, we must be aware that the
chatter theory becomes complex in the order of turning, drilling, milling28

and grinding as will be clear in the following.
Figure 1.16 illustrates the grinding wheel mounted on the wheel flange, andfurthermore three core components consisting of the grinding wheel, i.e.
grain, binder and pore.
In grinding, a grain behaves like the tooth (cutting edge) in milling, i.e.
trajectory for form-movement being represented by the Trochoidal curve,
and in due course, grinding can be classified into the up-cut and down-cut
modes. In general, the down-cut mode is dominant and eventually, we canobserve other characteristic features as follows.

Fig. 1.16 Three core elements of grinding wheel and flange
for wheel mounting(1) A grinding wheel is very elastic as compared with the cutter body andtool shank in the milling cutter.
(2) In the case of the vitrified grinding wheel, its rigidity is not uniform
across the whole circumference, resulting in the directional orientationof rigidity. Thus, we must consider the danger of mingling the29parametric vibration like that caused by the jaw chuck in turning as will
be discussed in Section 1.3. In contrast, with the advance of the
production technology for the grinding wheel, we may guess that such arigidity deviation becomes less, although we have not had the
publicized data since the late 1960s.
(3) A grain seems as to be a tooth with negative rake angle, and theuncountable number of grains engages in the work, resulting in a
problem for the quantification of “Grain Distribution”, which is, ingeneral, very difficult.
(4) The grain distribution is very unstable, which is caused by the “Grain
Self-Edging Function (Grain self-regeneration effects, Autogenousedge-generation effects)”. In short, the worn grain cannot stand thegrinding force, and then drops out from the grinding wheel. In due
course, a new grain joins within grinding instead of the worn one.
As a result, we must, at least, consider (1) the “Contact Stiffness of
Grinding Wheel”, (2) “Grinding Stiffness and Damping”, and (3)“Anti-wear Stiffness of Grinding Wheel” in the block diagram of chatterloop for grinding as will be discussed in Section 1.3. Importantly, thegrinding wheel may generate the wavy surface on the work, when thegrinding stiffness is higher than its anti-wear stiffness, and it may beinterpreted that such waviness induces either the forced vibration or chatterof regenerative type.
Based on our long-standing experiences, furthermore, we must pay the
special attention to the flange structure for mounting the grinding wheel. As
can be seen from Fig. 1.16, the wheel flange consists of various interfaces,
which give rise certain damping capacity, and eventually, the paper ring issomewhat helpful to increase the damping capacity effective forsuppressing the chatter vibration with higher frequency.
In addition, it is better to consider that the radial component (axial
component in turning) of the grinding force is larger than the tangential
component (principal component in turning), which is definitely in reverse
relation to turning.

30

To this end, thus, it emphasizes that the chatter in grinding is much more
complex than those in metal cutting, and this is one of the obstacles
regarding the poor research achievement for the chatter in grinding.

1.3 Basic Knowledge about Self-excited Chatter ofRegenerative Type in Metal Grinding
In discussing both the chatters of regenerative types in metal cutting andgrinding, it is first better to understand the differing features in the chatter
onset with machining time.

Fig. 1.17 Representative behavior of chatter vibration of regenerative type
in metal cutting and grinding in general (by Furukawa)
Fig. 1.17 shows the general behavior, and as can be seen, the chatter inmetal cutting occurs at the very short time lapse after the start of cutting
and rapidly increases its amplitude. In contrast, the chatter vibration inmetal grinding occurs after relatively long time lapse, and the amplitude is
not so large. It is also interesting that we can identify two differing31

causalities for the chatter in metal grinding as shown together in Fig. 1.17.
In fact, there are two types: “Type I” is caused by the waviness ofmachined surface, and its frequency is either closer to the resonance stateof the grinding wheel-work system, or a little bit higher than the resonancefrequency. In contrast, “Type II” is caused by the waviness of the grinding
wheel being in contact with the work, and in general, its frequency is muchhigher than that of Type I.
Importantly, we can observe the chatter onset with increasing the depth of
cut in metal cutting, whereas the chatter onset in metal grinding occurs
with decreasing the depth of cut.

1.3.1 Theory for Chatter of Regenerative TypeNow let us discuss the three basic expressions on the strength of those forturning with single-point cutting tool 6.
Figure 1.18 illustrates the correlation between the initial depth of cut u0(t)and the instantaneous depth of cut u(t), and as can be readily seen, theinstantaneous depth of cut is given by

In the cylindrical and internal grinding methods, and furthermore surface
grinding with rotary table type, the period per one revolution of the work
Tw is constant, and thus by substituting rw(t) into rwin,

Thus,

By applying the Laplace transformation, the “Uncut-chip Thickness
Equation” yields to

6

Because of ease of understanding, the theoretical analysis in Section 1.3 is based on that of Furukawa.
Furukawa Y. (June 1975) Chatter Vibration in Grinding and Its Suppression. Handouts in No. 411
Seminar for “Fundamental Technologies in Machining”.

32

Fig. 1.18 Relationships between initial and instantaneous depths of cuts(by Furukawa)

Obviously, Eq. (1.17) represents the regenerative effect, and in due course
applicable also to the chatter of Type II.
In grinding, we must consider the damping force, because the grinding area
depends upon not only the specific grinding force, but also the relative
approaching speed of grinding wheel to the work. Thus, the “Grinding
Process Equation” can be written as

where, Pn = Radial component of grinding force, kg = Specific grinding
force (grinding stiffness), cg = Grinding damping.
By applying the Laplace transformation, Eq. (1.18) yields to

In both the above expressions, kg and cg are given by33

where, B = Grinding width, V = Rotational speed of grinding wheel, v =Rotational speed of work, R = Radius of grinding wheel, r = Work radius, β= Angle between wheel approaching axis and acting direction of radial
component of grinding force, and σs = Specific grinding force.
Because of the considerable elasticity of the grinding wheel, its contact
deflection to the work cannot ignore in discussing the dynamic
characteristics of the machine-attachment-work-tool system in grinding.
When xn is the deflection along the direction of the radial component Pn, indue course the “Structural Equation” (total dynamic receptance) yields towhere, kcon is the contact stiffness of the grinding wheel, which is the ratioof the radial component of grinding force to the due wheel deflection.
It is said that the contact stiffness shows the non-linear springcharacteristics, i.e. hard spring, depends upon the grain size, hardness andso on, and can be estimated on the strength of Herz theory. From theviewpoint of the machine tool joint, however, it is doubtful that the contact
stiffness of the grinding wheel can be estimated by Hertz theory as will be
discussed in Item 1.3.2.
In discussing the chatter in metal grinding, another primary concern is thewear of the grinding wheel, i.e. increasing wear of grinding wheel withmachining time. Thus, we must consider the “Wear Stiffness of Grinding
Wheel”, which is the ratio ka of the radial component of the grinding force34

to the wear thickness of the grinding wheel as follows.

where, q is the grinding ratio 7.
In addition, we must eye that the waviness of ground surface rw is not
identical to the vibration amplitude xn between the grinding wheel and thework in radial direction, and these two are interrelated bywhere, ωcr is the critical angular frequency and given by(+ sign is for cylindrical and surface grinding, and – sign is for internal
grinding)
From Equations (1.17), (1.19), and (1.21) ~ (1.23), we can produce theblock diagram of chatter loop as shown in Fig. 1.19, which consists of twoloops corresponding with Types I and II as already mentioned beforehand(see Fig. 1.17) 8. In fact, we can discriminate these two clearly without any
trouble. For example, we can observe a cycle of chatter onset-growth-saturation for every work in Type I, whereas Type II disappears by dressing thegrinding wheel whenever.
7

In the past, q was more than 20 and V/v was between 50 and 100 in carbon steel grinding with standard
grinding conditions, an thus
In short, the “Wear Stiffness” was more than 1,000 larger than the “Grinding Stiffness”.
8 In Eq. (1.23), the interference effect is larger when ωcr is smaller than the chatter frequency ω, resulting
in the smaller rw. In general, the interference effect becomes larger in the order of cylindrical, surface andinternal grinding, and along with in the lower rotational speed of the work.

35

Fig. 1.19 Block diagram of self-excited chatter loop for regenerative type(by Furukawa)In general, at issue in grinding chatter is to suppress the onset of Type I,
which is caused by the waviness of the machined surface, and thus we may
suggest several guides as follows.
(1) Ease of chatter onset in cylindrical and center-less grinding, andfurthermore in surface grinding with rotary table and horizontal main
spindle.
(2) Plunge grinding is the utmost dangerous to chatter onset.
(3) High anti-chatter capability in internal grinding.
(4) The lower hardness of a grinding wheel, the more anti-chatter capability
is its property. In general, the elastic grinding wheel is recommendable
to suppress the chatter.
For the sake of further understanding, Fig. 1.20 shows the block diagram ofchatter loop for cylindrical grinding of plunge type (g = unity), and fromthis diagram, the “Characteristic Equation (Transfer Function)” can be
written as36

Fig. 1.20 Block diagram for self-excited chatter caused by waviness ofmachined surface

By solving the above expression, we can determine the stability limit, andthus the characteristic equation for the chatter of Type II is also shown
below.

Fig. 1.21 Stability charts for plunge grinding (by Inasaki)
37

Figure 1.21 shows simultaneously two stability charts for Types I and II inplunge grinding together with varying the grinding width (Inasaki, 1977).
As can be readily seen, there are apparently different behavior between theTypes I and II. More specifically, Type I does not occur at the lower work
speed, i.e. larger absolute stability limit nearly all grinding conditions,whereas Type II occurs easily across the whole grinding conditions.

1.3.2 Research and Engineering Development Subjects

As will be clear from the above, there are even now a handful of fatal
problems being unsolved, which are in closer tie with the grinding wheel’s
own characteristic features. Such features are obstacle to establish much
more reliable chatter theory for grinding than ever before, and in contrast
result in some hindrances for the inactive research into the chatter behavior
in metal grinding.
Having in mind the achievements so far reported and fatal problems to beinvestigated hereafter, the Editor suggests first the necessity of the
meticulous observation of the chatter behavior again. In the observation,
primary concern is the discrimination of the torsional chatter as suggestedby that of Entwistle and Stone (2013). Obviously, we must reinvestigate
duly the availability of the chatter theory so far established for grinding andconcerns at present as same as the traditional chatter theory for metal
cutting.
Paraphrasing, we must mind why we cannot find the stability chart recently
publicized instead of that of Fig. 1.21, which was produced in the 1970s. In
fact, this evidence indicates the complexity in the chatter of regenerative
type in metal grinding. After conducting the research into the availability of
the chatter theory from total and wider scopes, we must thus delve into theindividual R & D subject, for which some quick notes will be given in thefollowing. Of course, it is desirable to discuss in detail what are necessary
to establish an advanced chatter theory as same as that for metal cutting;however, we have not had enough reference materials to do so (see Chapter
3).

38

Contact stiffness of grinding wheel

As reported elsewhere, some researchers assert that the contact stiffness ofthe grinding wheel can be estimated by Hertz theory. In short, Hertz theory
is available for the two-surface in contact problem, where both the surfaces
are perfect planes without any roughness and waviness; however, thegrinding wheel has the very rough surface. Importantly, it is common sense
now that the contact between the grinding wheel and the work is one of themachine tool joints, where we can apply Hertz theory in very special cases,
i.e. joint consisting of nearly perfect surfaces.
In fact, the contact stiffness depends strongly upon the “3-Dimensional
Distribution of Grains”, and thus changes almost every second while
grinding. In consequence, we need now to establish an innovative methodfor determining the contact stiffness in real time; however, as well known,
primary concern in grinding mechanism is to measure and visualize the3-dimensional grain distribution at present.

Grinding Damping

As can be readily seen from Eq. (1.20), “Grinding Damping” is calculated
by only considering the geometrical relationship while the grinding wheel
approaches to the work. Is this assumption acceptable, when we discuss thevibration problem?
Supposedly, the theoretical analysis is good agreement with theexperimental result by considering the grinding damping, which plays
similar roles of the process damping in metal cutting, i.e. wider absolutestability limit in lower rotational speed. Against to this context, we have notclarified yet what is the essential feature of the grinding damping including
its causality. It seems that the binder behaves like a viscous damper;
however, we have not had any evidences to support such an assertion yet.

Regenerative Chatter Mingling with Parametric Vibration

The grinding wheel of vitrified type seems to show certain fluctuation in its
contact stiffness around the wheel, and such a fluctuation induces a39

parametric vibration as similar as that shown in Fig. 1.8. In fact, Hahn andPrice (1969) measured the variation of the resonance frequency when thegrinding wheel contacts with the cantilever, i.e. one of the equivalentindexes for the rigidity.
As shown in Fig. 1.22, we can see a considerable variation in the resonancefrequency, and other researchers also suggested such an interesting
behavior on that occasion. In short, the contact stiffness can be calculated
from the resonance frequency, and we can understand that the vibrational
amplitude becomes larger with large deviation of the contact stiffness.

Fig. 1.22 Directional orientation in rigidity of grinding wheel Measurement of contact resonance frequency (by Hahn and Price)
It is regrettable that we have not had such valuable data since then, andthus a facing problem is to investigate the present state in the unequal of
the contact stiffness of the grinding wheel. In addition, we must examine
the magnitude of such an unequal is enough to excite the parametricvibration. In general, we may believe that the grinding wheel shows
uniform contact stiffness with the advance of the related technology,
whereas the directional orientation in the contact stiffness could affectsomething definite with enhancement of the finished quality of the work.
40References
Abele E, Tschannerl M, Kulok M. (2007) Produktivitätsfaktor
Bohrungsqulität: Dreieckige Bohrungen Entstehung, Identifikation,
Parametereinfluss. ZwF; 102-6: 340-344.
Braun S. (1975) Signal Processing for the Determination of Chatter
Threshold. Annals of CIRP; 24/1: 315-320.
Denkena B, De Leon L, Grove T. (2010) Prozessstabilität eines
kordelierten Schaftfräsers, ZwF, 105/1/2, 37- 41.
Doi M, Masuko M, Ito Y. (1982) Re-observation of the Chatter Vibration in
Chuck Works - Effects of Jaw Number on the Appearance of a Scale-like
Chatter Mark. In: Proc. of 10th NAMRI of SME, 409-416.
Doi M, et al. (1985) Study on Parametric Vibration in Chuck Work. Trans.
of JSME (C); 51-463: 649-655.
Doi M, Muroya Y. (1990) Development of Dynamic Performance Test forLathe. Trans. of JSME (C); 56-529: 2521-2526.
Entwistle R, Stone B. (2013) Position Paper - Fundamental Issues in
Self-excited Chatter in Grinding. Jour. of Machine Eng.; 13-3: 26-50.
Hahn and Price (1969) ASTME Technical Paper.
Higuchi M, et al. (1986) Study on Determination of Chatter
Commencement in Turning. Trans. of JSME (C); 52-477: 1697-1700.
Inasaki I. (1977) Regenerative Chatter in Grinding. Proc. of 18th MTDR
Conf., p. 423-429.
Ito Y. (2013) Position Paper – Fundamental Issues in Self-excited Chatterin Turning. Jour. of Machine Eng.; 13-3: 7-25.
Ito Y. (2014) Chapter 1 Tangential Force Ratio and Its Applications to
Industrial Technologies: Anti-Vibration Steel Plate for Refrigerator andDerailment of Rolling Stock. In: Ito. Y (ed) Thought-Evoking Approaches
in Engineering Problems, Springer.
Ito Y (2017) Chapter 7 Concept of “Platform”, Its Application andModular Tooling System. In: Ito Y, Matsumura T (eds) Theory and Practice41

in Machining Systems, Springer Nature.
Itoh S et al. (1991) Ultrasonic Waves Method for Tool Wear Sensing In-process and Built-in Type. In: Proc. of Inter. Mech. Engg. Congress,
Sydney, pp. 83-87.
Merritt H E. (Nov. 1965) Theory of Self-Excited Machine-Tool Chatter,
Contribution to Machine-Tool Chatter Research-1, Trans. of ASME Jour.
of Engg. for Industry, 447-454.
Monnin J. (2013) Active Structural Methods for Chatter Mitigation in
Milling Process. Dissertation of ETH Zürich.
Nakazawa H, et al. (1979) Detection of Stability Limit – 1st report
Detection software. Jour. Precision Eng.; 45-11: 1353-1358.
Nicolescu C M. (1991) Analysis, Identification and Prediction of Chatter in
Turning. Doctoral Thesis No. 18, Department of Production Engineering,
Royal Institute of Technology, Stockholm.
Rahman M, Ito Y. (1979) A Method to Determine the Chatter Threshold.
In: Proc. of 19th Inter. MTFR Conf., MacMillan, 191-196.
Saljé E, Isensee U. (1976) Dynamische Verhalten schlanker Werkstücke bei
unterschiedlichen Einspannbedingungen in Spitzen. ZwF; 71/8: 340-343.
Sato H. (1973) Research perspective for Machine Tool Vibration. Jour. of
JSME; 76-658: 1105-1113.
Sellmeier V, Denkena B. (2010) High Speed Process Damping in Milling.
Proc. of 4th CIRP Inter. Conf. on High Performance Cutting.
Shiozaki S, Furukawa Y. (1974) Vibration in Drilling Machine. In: Reportsof Research Unit for Rigidity and Cutting Performance Improvement in
Machine Tools. JSME, p.249.
Spur G, Leonards F. (1975) Sensoren zur Erfassung vonProzesskenngrößen bei der Drehbearbeitung. Annals of CIRP; 24/1:
349-354.
Tlusty J. (1970) Chapter 2 The Theory of Chatter and Stability Analysis.
In: Koenigsberger F, Tlusty J (eds) Machine Tool Structure Vol. 1,
Pergamon Press, 133-177.
Tobias S A. (1961) Schwingungen an Werkzeugmaschinen. Carl Hanser

42Verlag, München.
Yeh L J, Lai G L, Ito Y. (July 1992) A Study to Develop a Chatter Vibration
Monitoring and Suppression System While Turning Slender Workpieces. In:
Proc. of 2nd Inter. Conf. on Automation Tech. Vol. 2, 337-343, Taipei.

43

Chapter 2 First-hand View for Chatter
Suppression Technology
Along with the establishment of the chatter theory, we have endeavored todevelop various remedies to suppress the chatter since 1930s. Fig. 2.1
exemplifies some leading remedies so far widely used, which range fromthe technician’s contrivance at the factory floor, through engineeringknowledge-based method like the damper and end mill with differing helix
angle, to the development of the new vibration-proof material.

Fig. 2.1 First-hand view of remedies for chatter suppression in general
The vibration suppression and absorption are one of the conventional
technologies across the whole engineering field, and thus the damper andvibration-proof material shown in Fig. 2.1 are specified to be applicable tothe machine tool sphere. More specifically, we have so far contrived amyriad of passive dampers, e.g. ball screw stuffing with damping material,
damping chuck, milling cuck stuffing with damping material such as a
44

TRIBOS (see Fig. 2. 19) and also active dampers to suppress the chatter;
however, nobody claims systematically the facing limitation in theircapabilities for the chatter suppression and also the involved engineering
problems.

Fig. 2.2 Gear chuck body made of cast iron to decay chatterwith higher frequency
Importantly, the vibration-proof material has not been applied to themachine tool so far, apart from the mounting device for the main motor,
whereas we can reduce the chatter with higher frequency by placing theunit or component with larger damping near the cutting point, e.g. chuck
body made of cast iron in crown gear cutting as shown in Fig. 2.2, tool post
made of concrete in turning and quick changing cutting head with buttressthread in milling (see Fig. 2.16). In fact, the blur-like chatter mark can bedisappeared by replacing the chuck body made of steel to that made of castiron in Fig. 2.2. Consequently, we must be aware of the increasingnecessities to investigate, with wider scope, the application of the materials
with higher internal damping.
45

More importantly, Fig.2.1 is one of the traditional classification systems,
and against this context, the purpose and aims of this chapter are to suggest
and discuss the remedies being used and possible to contrive on thestrength of the block diagram of chatter loop shown already in Fig. 1.1. As
can be readily seen from Fig. 1.1, we can suppress the chatter ofregenerative type by the following three remedies.

Fig. 2.3 An example of active damper - Adaptive tool holder(by Weinert and Kersting, and courtesy of Carl Hanser)(1) Control of “Cutting Process Dynamics”, i.e. specific cutting force(cutting stiffness): the end mill with serrated-tooth is one of therepresentatives, and this end mill can simultaneously control the “TimeDelay”.
(2) Control of “Structural Dynamics”, i.e. dynamic compliance ofmachine-attachment-tool-work system: as well known, nearly all
prevailed remedies are to increase damping capacity within the system,and obviously the damper is the utmost representative. For example,
Fig. 2.3 reproduces one of the active dampers for gun drilling withsingle cutting edge, in which the gun drill is mounted on the rotor andthe rotor is driven through MRF (Magnetic Rheology Fluid) coupling.
46

More specifically, the chatter suppression can be facilitated by thecontrol of the magnetic fields (Weinert and Kersting, 2005).
(3) Control of “Time Delay”: this remedy is exemplified by the milling
cutter of irregular tooth pitch type and with differing helix angle as will
be discussed in Section 2.1. In the case of turning by single-point
cutting tool, we may use this idea by means of the main spindle withvariable speed.
In consideration that the block diagram of chatter loop yields to thestability chart, it is furthermore notable that we can use the stability chartwhen investigating the synthetic suppression remedy to the chatter ofregenerative type as follows.
(a) Increase of the “Absolute stability limit”.
(b) Positive use of the “Cusp-like region” within lobed borderline ofstability.
Of these two, the latter remedy is preferable to be the heavy-duty cutting in
realty, provided that the stability chart is fully reliable, and in fact, thecorresponding software has been merchandized. It is however difficult to
get the reliable stability chart over wider cutting conditions (see Section
3.3.3), and in general, the stability limit should be indicated by band-width
way, because of considering uncertainties while producing the stability
chart. For example, the thin-walled work changes its dynamic characteristic
drastically with the progress of cutting.
Figure 2.4 is a first-hand view for R & D activities in the form of “Puttick
Grid”, simultaneously indicating a time series-like development history forthe remedies with respect to the time delay control. More specifically, theremedies are for turning and milling, and having in mind theseachievements, we have recently established some new remedies, e.g.
variable speed broaching and broaching with irregular tooth pitch type. As
will be clear from Fig. 2.4, an expectable remedy will be the “Turn-Millingwith Differing (Variable) Helix Angle”. In many respects, we may
recommend the use of the milling cutter of irregular tooth pitch type andwith differing helix angle on the basis of the practical achievement.

47

In the case of turning, we must use the variable speed method as will beclear from the above; however, in such a remedy, the spindle speed should
be fluctuated around 100 rev/min, while the average speed maintains at 305
rev/min as reported elsewhere. These values imply the difficulty of thepractical application of the variable speed method.

Fig. 2.4 First-hand view of R & D in chatter suppression based on
Time-Delay Control

2.1 Milling Cutter of Irregular Tooth Pitch Type and withDiffering Helix Angle
As can be seen from the headnotes in Part II, nearly all researchers place
the stress on the self-excited chatter of regenerative type in their works.
Importantly, they rely on even now the chatter theory established in the
1960s (traditional chatter theory), although a handful of the questioned
points have been suggested elsewhere. In contrast, it is very interesting that48

such a theory is very effective even now to contrive some outstanding
remedies for the chatter suppression as exemplified by the end mill ofeither irregular tooth pitch type or with differing helix angle. We will thus
delve into its past and present perspectives in the following.
In this context, we must first be aware that both the milling cutters of
irregular tooth pitch type and with differing helix angle are effective at thehigher-speed range in practice, where the main spindle rotates more than
2,000 and up to 40,000 rev/min.

Fig. 2.5 Example of face milling cutter of irregular tooth pitch type(by Sisson)
As reported elsewhere, Slavíček (1966), Opitz et al (1967) and Sisson(1969) investigated the milling cutter of irregular tooth pitch type, andVanherk (1968) and Stone (1970) investigated the milling cutter withdiffering helix angle in the past. Of these, Sisson investigated such a
milling cutter shown in Fig. 2.5 and suggested that the chatter can besuppressed when the succeeding cutting edge is shifted (1/4) λ (λ: wave
length of chatter mark). Importantly, Sisson suggested already on thatoccasion that the irregular tooth allocation could first facilitate the49relaxation of the impact load caused by the engagement of cutting tooth.
It is extremely notable that we clarified already the essential differing
effects of the irregular tooth pitch type from those with the differing helix
angle in the 1960s. In short, the former is effective within limited spindle
speed range as will be shown in Fig. 2. 10. In contrast, the latter is very
effective across the whole lower spindle speed range. These characteristicfeatures maintain even now as exemplified by the merchandized product aswill be shown in Fig. 2.6 and in that of Denkena et al (2010). We must thus
credit these outstanding theoretical analyses and experiments to theforerunning researchers, for example, Opitz et al and Stone. Eventually, wemust reinvestigate why the theories established by Opitz et al and Stone areapplicable to the machining conditions with the higher-spindle speed atpresent, which is far beyond expectation of the researcher in the era of1960s.
Figure 2.6 shows first the merchandized end mill of irregular tooth pitch
type and with differing helix angle, and then the marked effect of thediffering helix angle type on the critical depth of cut. At present, both types
are applied furthermore to (1) the band and metal saws, (2) hole saw, (3)side milling cutter, and (4) broach in practice.
In the academia, Denkena et al (2010) investigated the critical depth of cut

50

(a) Schematic view of cutting edge(b) Increase of critical depth of cut by end mill with differing helix angle
Fig. 2.6 End mill with cutting edges of irregular pitch type and withdiffering helix angle (by courtesy of OSG, 2018)of 3-flute end mill, which is of irregular tooth pitch along the flute, andwith nicked cutting edge (serration type), but the constant lead angle, whilemachining the monolithic complex component for the aircraft made of Al
alloy 1 . In due course, such an end mill improves considerably its
anti-chatter capability as shown in Fig. 2.7. In fact, the chip removal
volume per unit time is up to 8.85 l/min at 27,000 rev/min in main spindle
rotational speed, which is around 50 % larger than that removed by theconventional end mill. Importantly, Denkena et al suggested that the nicked
cutting edge contributes furthermore the reduction of the depth of cut,
resulting in the superimposing effect on the chatter suppression. Of special
note, we can observe the remarkable anti-chatter capability at certain speed
ranges only in Fig. 2.7. As already stated, this evidence can be reconfirmed
the essential feature of the milling cutter of irregular tooth pitch type.
As a matter of course, we can find another trial in this field, and Fig. 2.8
1

The research was carried out in corporation with Heller, MAG, Waldrich Coburg and Walter.

51

reproduces a patented idea (US 2005/0084341 A1, 2005), which can be

Fig. 2.7 Anti-chatter capability of end mill of irregular tooth pitch type
and with serrated cutting edge while slotting(by courtesy of Denkena)
characterized by integrating either two or three spacing irregularities within
a whole cutter body. More specifically, three irregularities are as follows.
(1) In the first irregularity, the columns of pockets typically associatedwith flutes are staggered circumferentially, and in preferable case, thecutter body has three flutes with different dividing angles.
(2) In the second irregularity, at least, some pockets are arranged at
different radial rake angles.
(3) In the third irregularity, at least, some pockets are arranged at different
axial rake angles.
In consideration of the better applicability of the milling cutter of “TimeDelay Control” type to the practice even now as mentioned above, we willnote quickly its theoretical background and concerns based on those of52

Opitz et al and Stone in the following.

Fig. 2.8 A further development of milling cutter of irregular tooth pitch
type (US Patent, US 2005/0084341 A1, 2005)

2.1.1 Milling Cutter of Irregular Tooth Pitch Type

Opitz et al (1967) aimed at the analysis of face milling and employed amathematical model shown in Fig. 2.9, where the cutter has a couple of thedifferent teeth pitches, and both the trajectories (Trochoidal curves) of teeth
are replaced by the concentric double circles for simplicity. Morespecifically, such an irregular tooth pitch can facilitate the relaxation for thesuperimposing effect of the cutting force by the succeeding tooth as alreadymentioned. Obviously, such a phase shift in cutting force is effective tosuppress the chatter.
By modifying the three basic expressions in the case of turning to becompatible with milling, the cutting force Fx for instantaneous uncut chip
thickness u(t) can be written as53

Fig. 2.9 Mathematical model for face milling cutter of irregular tooth pitch
type (by Opitz et al)

where, k1: Specific cutting force,x(t): Vibration amplitude of work at time t,
T: Time for one revolution,
U: Circular length of cutting teeth,z: Total number of cutting teeth,
ε: Angle between leading direction of vibration and feed direction,and in case of Fig. 2.9, ε = 0.
i = 1, 3, 5---z-1.
By substituting the cutting force into the structural equation, and carryingout the discrimination of the chatter onset, the stability charts are duly
produced as shown, for example, in Fig. 2.10 (because of difficulties in theanalytical calculation, they used the analog computer). In Fig. 2.10,
experimental values are not plotted for the simplification and it isnoticeable that Opitz et al asserted that the theoretical stability limit is in54good agreement with that obtained by the experiment.

Fig. 2.10 Effects of face milling cutter of irregular tooth pitch type(by Opitz et al)
Importantly, the stability limit increases considerably by employing theface mill of irregular tooth pitch type, provided that the speed range islimited between 20 and 70 rev/min as can be seen from Fig. 2.10. In
addition, it is interesting that the absolute stability limits are equal in boththe face milling cutters.
After conducting a series of investigations, they provided us duly with thesimplified formula to select the preferable pitch ratio as follows.

where, f0 (cps) and n (rev/min) are the natural frequency of system andmain spindle speed, respectively.
In short, they suggested that the limitation of the milling cutter of irregular
tooth pitch type depends upon the natural frequency of the system.
55

Fig. 2.11 Differences in stability charts depending upon cutting modes- In face milling with cutter of irregular tooth pitch type
Importantly, they investigated furthermore the difference of the stability
charts between for up-cut milling and for down-cut milling from both thetheoretical and experimental aspects. Fig. 2.11 reproduces the theoreticalcomparison, where the evaluation index of the chatter onset is the “ChipThickness Coefficient k1”. In the experiment, the evaluation index is thewidth of cut; however, both the stability charts are qualitatively in good
agreement, and thus it is very interesting that the stability limit for
down-cut milling increases drastically as compared with that for up-cut
milling, although the shock load is considerably large in down-cut milling.
This interesting behavior is caused by the differing relationships among φ,ε and β in Eq. (2.2) while either down-cut milling or up-cut milling.
56

2.1.2 Milling Cutter with Differing Helix Angle

This milling cutter was first developed to improve the shortcoming in theface mill of irregular tooth pitch type and aimed at the application to thecircular (cylindrical, plain, or slab) mill. Thus, this cutter seems as toincrease the number of the adjacency teeth up to infinity and also vary the
tooth pitch continuously along the cutting edge (flute).
Stone (1970 a and b) employed the mathematical model, which is a
developed plan view of cutter with successive teeth at different approachangles, as shown in Fig. 2.12. Of course, he applied his theory to broachingand obtained the fruitful result.

Fig. 2.12 Developed plan view of cylindrical milling cutter with differing
helix angle (by Stone)
When assuming the original point of x axis at the constant pitch line, andconsidering an element of width dx at distance x, the cutting force dP
acting on the element dx yields to57

where, k = Specific cutting force
y0 = Vibration amplitude
ω = Frequency of vibrationt = Time variableτ = Time interval between successive teeth
Obviously, the second term in Eq. (2.4) expresses the regenerative effect.
Then, in consideration of the time interval for one revolution of the cutter T,
τ = T/z on the constant pitch line, where z is the total number of teeth oncircular cutter.
Thus, τ at x can be written as

Fig. 2.13 Stability charts for circular milling cutterswithout and with differing helix angle (by Stone)58

In short, we can control continuously the regenerative effect along thewhole cutting edge by the difference between m1 and m2. In short, thecharacteristic index αmd is given by

where, d = Length of arc of cut
tan-1m = Mean helix angle
As can be readily seen, we can substitute Eq. (2.4) into the structural
equation, and duly obtain the stability chart, in which the vertical andhorizontal axes indicate the indexes equivalent to the limit width of cut andmain spindle speed, respectively. Fig. 2.13 shows the extreme effectivenessof the circular milling cutter with differing helix angle on the chatter
suppression, where β is “Damping Ratio”.

Fig. 2.14 Stability charts for milling with successive teeth
at different angles (by Stone)
59More specifically, the milling cutter with differing helix angle shows larger
stability limit in the lower spindle speed range than that in the higher
spindle speed range, which seems the lower-speed stability-like behavior,
and which differs absolutely from that for the milling cutter of irregular
tooth pitch type. In due course, we can benefit considerably in practice, andas a result, the end mill with differing helix angle has been mushroomed asalready shown in Fig. 2.6.
Figure 2.14 shows the effect of characteristic index αmd, and as expected
the stability chart depends upon it to a large extent. Importantly, Stone
conducted a series of experiments to clarify the effect of the helix angle on
the limit width of cut by using the tapered testpiece as shown in Fig. 2.15
and suggested some valuable design data as follows (in case of 122 rev/min,data are not shown).

Fig. 2.15 Widths of cut at stability boundary for various differencesin helix angle (by Stone)60(1) The chatter suppression capability increases with the helix angle, and it
is extremely improved when the helix angles are between 2 ~ 4 deg.
(2) Similar chatter suppression effect can be obtained when the helix angle
varies continuously within a tooth 2.
To this end, it emphasizes that Opitz et al and Stone proposed the valuable
theories for the milling cutter of irregular tooth pitch type and withdiffering helix angle, respectively, and their validities have been verified bya considerable number of the merchandized milling cutters. In this context,
we must reinvestigate to what extent in the main spindle speed those of
Opitz et al and Stone are applicable at present.

2.2 A New Horizon - Integrating Damping Entities within
Machine Bodies, Machine Elements, Attachments andTools
With prevailing the machining function-integrated kind like “Mill-turn”,
both the modular-designed chuck and the cutting tool become gradually
popular. In the sphere of machine tool joint, it is common knowledge thatthe jointed body shows higher damping capacity, but lower rigidity ascompared with those in the monolithic body component.
Typically, we may expect the preferable capability of chatter suppression inmodular tooling, in which the cutting-edge module is jointed with the tool
shank module by the thread of buttress type as shown, for example, in Fig.
2.16. The buttress thread has a saw tooth-like shape, and thus ease oftightening, but hard for loosening. It emphasizes again that the joint placed
closer to the cutting point can facilitate the suppression of the chatter markwith the higher frequency (see Fig. 2.2). Of course, we cannot expect thehigher damping capacity when the joint is in the node of the vibrationmode.
As will be clear from the above, it is preferable to suppress the chatter2

β = h/k, where h is “Hysteretic Damping Coefficient” showing the viscous damping-like effect when
its magnitude is small, and k is “Spring Stiffness”.

61Fig. 2.16 End mill of modular tooling system (Type PXM, by courtesy ofOSG, 2018)
without using the additional device like the damper. Conceptually, we must
develop an innovative remedy either capable of integrating within themachine body, attachment and tool compactly, or by contriving thestructural constitution of the machine body itself. In fact, the modular tool
is one of the representatives of the former, and regarding the latter, forexample, Watanabe and Sato (1988) clarified the effects of the fitting
tolerance in the bearing, i.e. non-linearity in the spring constant, on the
dynamic characteristics of the main spindle. More specifically, they
employed NLBBA (Nonlinear Building Block Approach) to obtain thefrequency response of the main spindle. Importantly, we can see theconsiderably large real number in the loci of the spindle compliance on thecomplex plane, when the exciting force and bearing clearance are 25 N and1.0 μm, respectively. Conceptually, the bearing clearance is large in thiscase from the viewpoint of the structural design; however, that of Watanabe
and Sato suggests the importance for the positive use of the non-linearity inthe structural design to suppress the chatter.
62

As will be clear from the above, it is desirable that the structural bodycomponent itself is either with higher damping capacity together withacceptable rigidity, or integrates compactly the higher damping element. It
is however regrettable that we have not had such engineering design data,
apart from a very few reports. In fact, we have not measured the damping
capacity of the main bearing while rotating since 1980, because of extreme
difficulty in the measurement of the damping capacity (for detail, refer to
Chapter 8 of Ito, 2008).

Fig. 2.17 Experimental set-up to measure damping of bearing
while rotating (by courtesy of Tsutsumi)
Figure 2.17 reproduces the well-contrived test rig and some damping ratios
for the tapered roller bearing while rotating the outer ring, which were
measured from the magnification factor in second vibration mode
(Tsutsumi et al, 1980). Importantly, damping of the tapered roller bearing isin strong dependence upon the pre-load and rotational speed,63simultaneously showing the maximum value at certain rotational speed.
It is notable that the measurement of the damping capacity is very difficult
work as exemplified the test rig shown already in Fig. 2.17. To carry out
the accurate measurement, Tsutsumi et al were very keen to eliminate thedisturbance from the driving system and to avoid the unfavorable
disturbance from surroundings. In fact, the shaft is fixed using a flexible
bar, the bearing housing is supported by the air bearing to minimize theinfluence of damping of surroundings, and the housing is driven by the
motor through the gear-toothed belt and thin tube. In addition, the bearing
and its surroundings are cooled by the air and oil bath lubrication, so thatthe temperature of the shaft-bearing system is to be in constant.

Fig. 2.18 Examples of engineering data of linear guide (by Weck and NSK)
It is furthermore notable that TH (Technische Hochschule) Aachen (Weck,1993) and NSK publicized some valuable data for the linear guide asshown in Fig. 2.18. At present, the linear guide is a core machine element
for the conventional TC and MC, which have prevailed across the whole64

world market. Importantly, the machine tool joint shows, in general,
non-linear characteristic, apart from that under the lower interface pressure.
It is thus very interesting that the linear guide of NSK-brand shows thelinear relation between the applied load and the deflection 3.

Fig. 2.19 Damping tool holder of “Gleichdicke” type called TRIBOS, 2001
(by courtesy of Schunk)
Figure 2.19 shows a tool holder with higher damping capacity by stuffing
plastics within the tool holder without excessing holder’s dimension. The
same idea was also applied to the hollow ball screw by THK around 2005.
In fact, the free decay vibration continues up to 1 sec. in the case of theconventional ball screw, whereas the hollow ball screw stuffed damping
material can decay the vibration within 0.25 sec., while rotating with 3,000
rev/min
3

Professor Weck publicized these data in 1993 in part; however, the detail is not publicized as yet,because of company’s confidentiality in accordance with the message from Dipl.-Ing. Brockmann in 2013.
Refer to the following dissertation. Müni Hakan Ispaylar. (1996) Betriebseigenschaften von ProfilWälzfürungen. Compu TEAM, Würzburg.

65

To this end, we must be aware again that a crucial shortage in theengineering data is the rigidity and damping capacity of the rolling bearing
while rotating more than 2,000 rev/min.

2.3 Elastic Grinding Wheel and Others
Within the context of chatter suppression in metal grinding, we may talk asimilar story to that in metal cutting. Typically, we must reinvestigate thevalidity for using the term “Forced Chatter” caused by, for example, theunbalance of the grinding wheel, which is not a due exciting source
inhabiting within the grinding mechanism.
Obviously, we may have various ideas for the remedy of the regenerative
chatter suppression from the block diagram of chatter loop. For example,
there are two typical remedies: one is to control the regenerative effect, e.g.
variable rotational speed in either work or grinding wheel, and the other isto decrease the contact stiffness of the grinding wheel, e.g. elastic grinding
wheel.
In the former, the variation rate should be at least ± 10% of the grinding
wheel speed in constant, and thus this remedy is only applicable to roughgrinding as reported elsewhere. Regarding the latter, Inasaki et al (2001)
introduced an idea proposed by Bzymek et al in 1994, in which both thesides of wheel body can be facilitated by a considerable number of theholes and grooves with various shapes. Although aiming at the reduction of
the thermal deformation, Naxos Union once developed a grinding wheelwith a considerable number of the grooves placed radially at both the wheelside faces as shown in Fig. 2.20 (beginning of 1980s). In accordance with
the Editor’s experience, such a grinding wheel is very noisy by wind
shearing.
Within the systematic contrivance for the chatter suppression, Folkerts(1993) proposed a notable guide as shown in Fig. 2.21, which can beobtained from the frequency response analysis for the cylindrical grinding66

machine while plunge grinding.

Fig. 2.20 Special-purpose grinding wheel developed by Naxos Union

Fig. 2.21 First-hand view of remedies to increase grinding stability(by Folkerts)
More specifically, Folkerts eyes the compliance and phase shift in thefrequency response and suggests that the unstable conditions can be67determined by the three stability restriction curves, i.e. vertical amplituderestriction, upper and lower phase restriction curves, resulting in the“Stability Field”. Importantly, Folkerts summarizes systematically each
remedy so far known by using the stability field as shown also in Fig. 2.21.
Within those shown in Fig. 2.21, it is very interesting that the chatter can besuppressed by employing the larger work diameter, and importantly, this
remedy implies similar effects to that proposed by Entwistle and Stone(2013). Accidentally, they suggested that the torsional vibration of the work
could result in the improvement of the chatter stability.
Although the idea of stability field is very valuable, the stability fielddepends upon (1) the grinding machine, (2) grinding condition, and (3)grinding wheel to a large extent, and thus obviously it is very difficult toselect a suitable remedy in accordance with each grinding requirement.

Fig. 2.22 A cBN elastic grinding wheel and its vector locus on complex
plane (by Sexton et al)
Having in mind such difficulties, it is recommendable to develop
68furthermore the elastic grinding wheel in consideration of the success story
for the milling cutter of irregular tooth pitch type and with differing helix
angle. Fig. 2.22 reproduces thus a cBN elastic grinding wheel proposed by
Sexton et al. (1982). Of note, this grinding wheel is for cylindrical grindingof hardened CrMo steel (EN 31 as per BS), can be characterized by fixing
the T-section rim with abrasive layer through 40 pads made of neoprene,
and in due course shows fruitful results.
More specifically, the control of the contact stiffness is equivalent to the“Shift of Harmonic Locus” on the complex plane, and as shown also in Fig.
2.22, we can credit such an effect clearly. Conceptually, the designprinciple of this elastic grinding wheel is as same as that of single-point
cutting tool of gooseneck type. Importantly, Sexton et al. (1981) tried
beforehand a cBN elastic grinding wheel for HSS (High Speed Steel) workby the surface grinder, the hub of which was made of Ni porous material
together with realizing about 1.4 × 106 N/m per mm in radial rigidity.

References
Denkena B, de Leon L, Grove T. (2010) Prozessstabilität eines kordelierten
Schaftfräsers. ZwF; 105-1/2: 37- 41.
Entwistle R, Stone B. (2013) Position Paper - Fundamental Issues in
Self-excited Chatter in Grinding. Jour. of Machine Eng.; 13-3: 26-50.
Folkerts W. (1993) Dynamische Prozeßkennwerte des Schleifens und deren
Einfluß auf das Prozeßverhalten. Dr. Thesis, RWTH Aachen.
Inasaki I, et al. (2001) Grinding Chatter - Origin and Suppression. Annalsof CIRP; 50-2: 515-534.
Ito Y. (2008) Chapter 8 Design Guides, Practices, and Firsthand View ofEngineering Developments - Sliding Joints. In: Modular Design for
Machine Tools, McGraw Hill.
Opitz H, et al. (1967) Improvement of the Dynamic Stability of the Milling
Process by Irregular Tooth Pitch. Proc. of 7th Inter MTDR Conf. p.

69

213-227, Pergamon Press.
Sexton J S, Stone B J. (1981) The Development of an Ultrahard Abrasive
Grinding Wheel Which Suppresses Chatter. Annals of CIRP; 30-1:
215-218.
Sexton J S, Howes T D, Stone B J. (1982) The Use of Increased Wheel
Flexibility to Improve Chatter Performance in Grinding. IMechE
Proceedings; 196-25: 291-300.
Sisson T R. (1969) ASTME Technical Paper. MR 69-245.
Slavíček J. (1966) In: Proc. of 6th Inter. MTDR Conf., p. 15, Pergamon.
Stone B J. (1970 a) The Effect on the Chatter Behaviour of Machine Tools
of Cutter with Different Helix Angles on Adjacent Teeth. In: Proc. of 11th
Inter. MTDR Conf., Vol. A; p. 169-180, Pergamon Press.
Stone B J. (1970 b) The effect, on the regenerative chatter behaviour ofmachine tools, of cutters with different approach angles on adjacent teeth.
Research Report No. 34, The MTIRA (Distributed only for members).
Tsutsumi M, Nabeta N, Nishiwaki N. (1980) Damping in Single-row
Rolling Bearings. In: Proc. of the 4th ICPE, p. 374-379, JSPE and JSTP,
Tokyo.
Vanherck P. (1968) In: Proc. of 8th Inter. MTDR Conf., p. 947, Pergamon.
Watanabe K. Sato H. (1988) Development of Nonlinear Building Block
Approach. Trans. of ASME, Jour. of Vibration, Acoustic Stress, andReliability in Design; 110: 36-41.
Weck M. (1993) Trends of Manufacturing Technology Looking Towards
the 21st Century. Industrial Technical Seminar, Kobe, May 19th.
Weinert K, Kersting M. (2005) Konzeptionelle Entwicklung eines
adaptiven Werkzeughalters. ZwF; 100-6:352-354.

70

Chapter 3 Symptoms of Needs to Advanced
Chatter Theory
Although carrying out the uncountable activities, even now the chatter isstill at burning issue in the academia and industries as mentioned in
Chapters 1 and 2. These evidences imply the capability limitation of thechatter theory, which is based on those of Tobias, Tlusty and Merritt, i.e.
“Traditional Chatter Theory”. In contrast, with the advance of the
traditional chatter theory, experimental techniques, chatter suppression
technology and engineering knowledge, we can have a considerable
number of new findings and proposals, and furthermore conduct
challenging trials such as shown in the “Headnotes in Part II”.
In consideration of both the critical issues and the valuable findings so far
obtained, we must pave the way to a new horizon in the chatter theory andconcerns, i.e. “Establishment of Advanced Chatter Theory” and innovativetechnologies for the chatter suppression. In this context, there are the threeleading driving factors, i.e. (1) reinvestigation of essential features of“Forced Chatter Vibration”, (2) improvement for applicability of block
diagram of chatter loop to higher rotational speed of main spindle, and (3)validity verification of self-excited chatter derived from “Drooping
Characteristics in Cutting Force”.
Thus, the Editor proposes first a new classification of the chatter and amodified block diagram of chatter loop in consideration of these driving
factors. In short, the proposed classification can be characterized bydisregarding the forced chatter vibration, which is not caused by the cutting
mechanism itself and by incorporating the new vital factors with theincreasing main spindle speed.
Following these proposals together with some quick notes, the Editor willthen reveal something underpinning these proposals, and discuss duly thecorresponding matters in three basic expressions for the chatter ofregenerative type. Of special note, the Editor does not deal with the71advanced chatter theory for metal grinding, because of not enoughachievements available for such discussion as compared with those formetal cutting (see Section 1.3). Against this context, Professor Lee of
Nihon University will first review in detail the past and present
perspectives for the chatter in metal grinding and then suggest a handful ofissues in Chapter 8. Such issues are related to the contact stiffness anddamping of the grinding wheel and concerns, and necessary to paving theway to the establishment of the advance chatter theory for metal grinding.

3.1 A New Proposal for Classification of Chatter- Reinvestigation into Validity of “Forced Chatter”
In the classification of the chatter having been prevailed, there are twoleading types, i.e. “Forced Chatter” and “Self-excited Chatter”. Importantly,
primary concern is the “Regenerative Type” within the self-excited chatterin practice. This classification is somewhat helpful to conduct R & D(Research and Engineering Development); however, we must now modify
it in consideration of the fundamental definition, some new findings
recently obtained and the changes in machining conditions, e.g. prevalenceof heavy-duty machining per unit time by using higher-cutting speed.
In the modification, we must first recall the following definition of theself-excited chatter, which is based on our long-standing experience. Morespecifically, we know now that primary concern is the chatter ofregenerative type; however, we cannot deny the occurrence of the otherchatter, which is caused by the “Drooping Characteristics in Cutting
Force”. In addition, it is common sense that the forced vibration disappears
completely by eliminating the external and internal excitation sources.
“The machine-attachment-tool-work system vibrates considerably evenwhen not existing any internal and external excitation sources within thesystem”.
Secondly, we must reconsider the validity of using the term, “Forced72

Chatter”, as will be discussed later. For example, it has been said that we
can observe the “Saw Tooth-like Chip” in Ti alloy turning, which affects
the stability limit, because of inducing a vibration with frequency close to
the generated pitch of the saw teeth. Some researchers assert to include
such chatter within the regenerative type and to call it “Material
Property-related Type”. Conceptually, the chip generation seems as one of
the excitation sources; however, we cannot eliminate it in certain cases,
because the chip generation is a part of cutting mechanism itself.
Thirdly, we must investigate the role of some new excitation sources,
which are actualized within the machine-attachment-tool-work system withthe advance of the higher-speed machining. For example, M. Doi et al(1982) reported already the differing behavior in chatter between the works
held by the three-jaw scroll chuck and by the four-jaw independent chuck.
In fact, the work holding stiffness of the three-jaw scroll chuck varies threetimes every one rotation of the main spindle, resulting in the directional
orientation effect of the work stiffness. Consequently, such an influencing
factor could act considerably with increasing the main spindle speed.
Eventually, we must investigate the regenerative type in consideration ofmingling the parametric vibration caused by such a directional orientation(see Section 1.1.3).
In consideration all the controversial points mentioned above, Fig.3.1
shows a new proposal for the classification of the chatter, and as can bereadily seen, this classification may be characterized by regarding theforced excitation source as a disturbance in the self-excited chatter. We willthus discuss its characteristic features in the following, and obviously, thesewill give us some guides to advance and enhance the chatter theory.
With respect to a new classification for the chatter, we must furthermore
continue the meticulous observation for the chatter mark and concerns tounveil the essential features in some uncertain chatter vibrations. In fact,
we face much more unfavorable chatter, the mark of which is microscopic“Blur-like”, in grinding even now. In addition, a machine tool manufacturer
has an experience, in which the blur-like chatter mark or micro-beat-like73chatter mark can be clearly observed while wet-machining Al alloy with
3-flute end mill, i.e. full side cutting mode and cutting speed being morethan 11,000 rev/min. Although we have not verified the validities of thesefindings yet, some researchers already observed similar chatter inhigher-speed machining of the Al alloy, and asserted that this chatter may
be classified into the “Friction-based Excitation Type”. In thefriction-based excitation type, it is said that the chatter frequency is morethan the order of kHz.

Fig. 3.1 A new proposal for classification of chatter
Although the forced chatter in the past sounds “Out-of-place” term, it isinevitable that the cutting mechanism itself includes often the excitation
sources. Obviously, we cannot eliminate such sources, and thus now let us
discuss the detail for the self-excited chatter with forced excitation.
In accordance with the definition so far used, the forced chatter involves
those caused by, for example, the unbalance within the work, rotating tooland chuck, intermittent cutting and the meshing ratio of the driving gearsystem of the machine tool. Correctly, such a forced chatter should be
regarded as one of the forced vibrations, because we can suppress easily it
by eliminating the corresponding excitation source. In contrast, even when74considering that the forced vibration decays rapidly, we must discuss aproblem, where the excitation source inhabits within the cutting mechanism
itself like shock loading in down-cut milling and saw-like chip generationin turning of Ti alloy. For example, the shock load acts frequently at theengagement of cutting tooth to the work in down-cut milling before
decaying the previous forced vibration. Obviously, such excitation
sources cannot eliminate, and thus we may guess its certain influences on
the self-excited chatter.
With the growing importance of the higher-speed cutting for the aircraftcomponent made of Al alloy, we can observe another excitation source, i.e.
run-out (whirling) of end mill, which can be called the “Excitation by
Forced Deviation”. Even when we can set the rotating tool to the main
spindle without run-out in the preparatory work, the rotating tool shows aconsiderable run-out, while running higher-speed. In due course, we face a

Fig. 3.2 Changes in stability chart by run-out of end mill while machining(by courtesy of Insperger, 2015)
new problem, i.e. influence of run-out caused by centrifugal force on
75chatter stability, and thus we will scrutinize it in the following.
In this context, Insperger et al (2008) reported already an interesting
behavior in down-cut milling of Al alloy with two-flute end mill (slot drill).
They show a theoretical stability chart together with experimental values asshown in Fig. 3.2, where ρ1=ρ2 means no run-out, and ρ1=0.9 and ρ2=1.1
mean that the first edge cuts 10 % less than and the second edge cuts 10 %larger than those without run-out, respectively.
Importantly, Insperger et al reported that the stability limit does not depend
upon the run-out, and that the experimental values are not in good
agreement so much with theoretical ones, because of difficulty in correct
estimation of the cutting stiffness. In contrast, we can observe aninteresting behavior at 13,000 and 15,000 rev/min in the rotational speed ofthe main spindle. In short, the unstable state is amplified around theserotational speeds, and thus we may guess the implicit influence of therun-out on the stability limit.

Fig. 3.3 Actual diameter of two-flute end mill while rotating higher-speed(by Rall et al., 1998)
In retrospect, Rall et al (1998) suggested already the change of the actual76diameter of the end mill with the increase of the rotational speed as shownin Fig. 3.3. They conducted a research from viewpoint of the in-process
measurement of the cutting tool. In addition, Opitz et al (1967) reported thediffering effects on the chatter stability while up-cut and down-cut millingby the face mill with irregular pitch tooth type as already shown in Fig.
2.11. Importantly, they suggest that such an interesting behavior is causedby the milling mechanism itself, i.e. time-dependent direction coefficient,which can evaluate the influence of the cutting tooth position on thevibration direction (see Section 2.1).
To this end, it is thus worth suggesting that we must conduct a research into
the influence of forced excitation, which is derived from the cutting
mechanism itself, on the self-excited chatter.

3.2 A Proposal for “Modified Block Diagram of
Chatter Loop”
Along with a proposal for the classification of the chatter, the Editor asserts
the necessities to establish an advanced chatter theory. In fact, we canobserve some further evidences to support such an assertion in addition to
the symptoms, which necessitate the revision of the classification of thechatter.
Thus, Fig. 3.4 shows a new block diagram of chatter loop by modifying the
traditional one. More specifically, the Editor indicates the corresponding
modifications by the red letter, which include the (1) influences of theforced deviation and periodic excitation inhabiting the cutting mechanism,(2) estimation of the real dynamic specific cutting force, (3) reinvestigation
into the dynamic compliance while machining with the higher rotational
speed more than 2,000 rev/min, and (4) influence of the multipleregenerative effect on the stability chart.
To deepen our understanding to such a modification, let us discuss herein
the discrepancy between both the stability charts obtained theoretically andexperimentally in the higher rotational speed as shown in Fig. 3.5. Van Dijk
et al (2008) reported valuable results together with specifying neatly the77

Fig. 3.4 A proposal for modified block diagram of chatter loop

Fig. 3.5 Comparison between computed stability chart and experimental
values (by van Dijk et al)experimental conditions as follows.
78

The theoretical stability chart was computed for the milling machine of
Mikron-brand (Type HSM700) with the two-flute end mill (10 mm indiameter and 57 mm in length), which was held by shrink-fitting. Of note,the necessary data for computing the stability chart were obtained by theimpulse hammer method, and the cutting test was carried out for Al alloy
7075 with mode of slot (full circular) milling.
It is very interesting that the computed stability chart differs considerablyfrom the experimental one at the higher-speed range of the main spindle, i.e.
up to 27,000 rev/min. They suggested that this behavior is caused by the
speed-dependence of the spindle dynamics 1. In addition, they suggested
that the acceleration sensor is preferable to detect the chatter in its earlier
stage of onset as compared with the force and sound sensors, although not
clearly stating how to define the chatter onset (commencement). Thus, it
seems that such a discrepancy is, in part, caused by the uncertainty in thedetermination of the chatter onset.
Importantly, Schmitz et al (2004) publicized already a similar theoreticalresult as shown in Fig. 3.6, where the stability chart for non-rotating main
spindle is compared with that for rotating 2. More specifically, theymeasured the frequency response function by using impact testing, while
the main spindle was either still stand or rotating, and consequently
produced both the stability charts shown in Fig. 3.6. In due course, they
verified the validity of these stability charts by slot cutting for the work
made of 6061-T6 Aluminum with the end mill. As can be also seen, there isconsiderable difference between both the stability charts, and theysuggested the certain influence of gyroscopic and centrifugal effects in boththe front and rear bearings in the main spindle system on the chatterstability. Apparently, we can observe such an interesting behavior when the1

The spindle dynamics depend upon considerably the kinds of main bearings and their arrangement
including the magnitude of pre-load; however, the stiffness and damping capacity of the rolling bearing at
higher rotational speed is not investigated as yet even in the sphere of the machine tool joint. In fact, wehave a few design data as exemplified in Fig. 2.18 for the case of linear guide (see Chapter 7).
2 In end milling, recently the terms “Axial Depth of Cut” and “Radial Depth of Cut” are prevailed, butthese should be used in accordance with the traditional way, i.e. former and latter being “Width of Cut”
and “Depth of Cut”, respectively, apart from the very special cases, so that the analogy to other machining
methods is to be guaranteed.

79Fig. 3.6 Differing phase in stability chart either non-rotating or rotating
main spindle (by courtesy of Schmitz, 2015)
rotating speed is more than 17,000 rev/min. In consequence, we must beaware the dire necessity for investigating the rigidity and damping capacityof the main spindle system while its higher-speed rotating states; however,as already mentioned in Section 2. 2, we have not had any corresponding
engineering data even now (see Figs. 2. 17, 2.18 and Chapter 7).
As exemplified these evidences, the facing problems in the chatter are far
beyond from those observed in the 1960s with the continuous increase of
the spindle speed. On the strength of findings within some challenging
researches, it is thus natural to reconsider the chatter theory having beenrelied on, and we must deploy to new horizons, so that we will be able toseek the essential features of the chatter than ever before.

3.3 Some Representative Issues within Three Basic
Expressions
In accordance with the indications within Fig. 3.4, we will discuss some80

fundamental issues within the three basic expressions for the chatter ofregenerative type. More specifically, we must place the main stress on thefollowing issues.
(1) Although dealing with the vibration problem, the specific cutting force
in the “Cutting Process Equation” has so far been estimated, in nearly
all cases, on the basis of the orthogonal (two-dimensional) static or
quasi-static cutting model.
(2) As same as above, the dynamic characteristics in the “Structural
Equation” have been measured while the machine-attachmenttool-work system is still stand or non-loading condition. Intuitively, thedynamic characteristics in cutting differ from those for still stand andnon-loading.

3.3.1 Issues in Uncut Chip Thickness Equation

The “Uncut Chip Thickness Equation” is of great importance to understandthe characteristic features of the self-excited chatter of regenerative type;however, it is better to discuss something related to this expression together
with the “Cutting Process Equation”, because these two are in mutual
closer relationship.
In retrospect, we used to consider only the regenerative feedback path asalready shown in Fig. 1.1, which corresponds with the wavy surface
generated by the cutting tool at one revolution before. Against to thiscontext, Kondo, Sato et al (1980) were first aware of the noteworthy
regenerative effects of the wavy surfaces generated at two and further
revolutions before as shown in Fig. 3.7.
Reportedly, they call it “Multiple-regenerative Effect”, and skillfully
revealed why the vibrational amplitude is constant after the chatter onset bytaking this effect into consideration. Following that of Kondo et al., Tlusty
and Ismail (1981) applied this idea to milling, and nowadays themultiple-regenerative effect becomes one of the leading factors in the
chatter analysis (Sato, 2013). For example, Jemielniak and Wypysinski(2013) suggested again the importance of the multiple regenerative effects,

81although not showing the detail, and thus it is desirable to quantify thechanging phase in the stability chart in consideration of themultiple-regenerative effect. Against this context, it is said elsewhere thatthe multiple-regenerative effect does not influence on the stability limit,
although such an assertion is not verified yet.

Fig. 3.7 Concept of “Multiple-regenerative Effect” in self-excited chatter(by courtesy of H. Sato)
As a minimum requirement, the uncut chip thickness equation should be
revised in consideration of the “Primary Multiple-regenerative Effect” asfollows.
u(t) = u0(t) – y(t) + μ1y(t-T) + μ2y(t-2T) -------(3.1)
Admitting that, within the uncut chip thickness equation context, anotherissue is to consider the run-out of the rotating cutting tool as shown alreadyin Fig. 3.2, we believe that the uncut chip thickness equation has not further
problems in general concerns. Dare to say, the Editor asserts the necessityof a reinvestigation into what is the “Real Uncut Chip Thickness”
82corresponding with the “Real Dynamic Cutting Component in Cutting
Force” (see Section 3.4).

3.3.2 Issues in Cutting Process Equation

As already shown in Fig. 2.7, Denkena et al verified the excellent
anti-chatter capability of the nicked cutting edge in the end mill, and theysuggested that such a capability is derived from the preferable control ofthe specific cutting force. Obviously, the specific cutting force (cutting
force coefficient, cutting stiffness) is of great importance in the analysis of
the chatter, and thus it is desirable to determine theoretically the specificcutting force on the strength of the cutting mechanism model. As widely
known, we have used often the orthogonal cutting model with single shear
plane without any doubt; however, we must first recall the major
assumptions within this model as follows.
(1) The shear angle can be estimated by various ways depending upon theyield criterion of the engineering material. In fact, we have various
criteria, e.g. hypotheses of maximum main stress, maximum shear
stress and octahedral shear stress; however, even now there areuncertainties in the applicability of each criterion when the engineering
material is specified.
(2) In practice, we can observe the side flow on the rake face of the tool
while cutting; however, the cutting mechanism model does not considerit.
In addition, the specific cutting force varies with the depth of cut and feed
rate. As a result, it is difficult to estimate the specific cutting force by using
such a cutting mechanism model, and in general, we can estimate thespecific cutting force by measuring the cutting force itself.
In this context, we must be aware that in all the research and engineering
reports so far publicized, the specific cutting force is, without exception,
determined by measuring the average value of the cutting force, i.e. static
component in cutting force. As well-known from the past, the static cutting
force is closely related to the dimensional accuracy of the finished83

component, whereas the dynamic cutting force generates, in general, thesurface roughness of the finished component 3.
The chatter is, as literally shown, one of the dynamic phenomena in theengineering problem, and eventually we must consider something
discrepancy in the identification of the specific cutting force so far
employed.
Nevertheless, people in the chatter sphere used to determine the specificcutting force from the measurement of the static cutting force. It seems thatthey have followed the theory and practical technology established in the
1960s without any doubt. In retrospect, we were able to measure only thestatic component of the cutting force when the traditional self-excited
chatter theory was established.
With the advance of the related technologies, of course, we haveendeavored to improve the shortcomings in such traditional ways. Of note,
Altintas et al (2008) publicized some measured results of the dynamic
specific cutting force and process damping while turning with grooving
tool (hereafter to avoid unnecessary confusion, we call it “Nominal
Dynamic Specific Cutting Force”). They employed the test rig consisting of
the tool holder with piezo-actuator-driven fast tool servo and load cell, andfurthermore of laser displacement sensor. The former is for control thevibration frequency and amplitude of the cutting tool, and the latter canfacilitate the measurement of cutting tool displacement. Following that of
Altintas et al., Sellmeier et al (2009) conducted a similar investigation into
the measurement of the process damping by using the milling machine oftraveling spindle type. In this case, the spindle is supported by electromagnetic guide and driven by the linear motor. In the experiment, the spindle
was under the sinusoidal excitation with amplitude control, and the cutting
force was measured by the dynamometer of Kistler-brand while end
milling.
In fact, these are very challenging trials; however, the Editor asserts that
3

Some researchers conduct machining the work with wavy surface and measure the average value of the
cutting force. After then, they determine the specific cutting force, and duly call it “Dynamic SpecificCutting Force”. Thus, there are confusions to certain extent.

84

those of Altintas et al and Sellmeier et al measured the specific cutting
force for fluctuation component, i.e. nominal value, but not real dynamic
one, as will be discussed in Section 3.4. Consequently, we have some doubt
regarding whether the measured values are reliable or not, and in addition,
both the test rigs appear as to be not suitable for the measurement of thereal dynamic specific cutting force. In short, these are for the investigation
into ultrasonic machining with phase control itself, and supposedly we may
not guarantee the one-to-one relation between the measured result by thesetest rigs and that in the chatter in practice. Conceptually, the dynamic
waviness should be, at least, estimated from the chatter mark left on thefinished surface, and then such a signal should be input to the oscillator.
Admitting that we have benefited by applying the nominal dynamic
specific cutting force so far used to some extent as will be stated later, on
the contrary, we must reconsider the applicability limit of such a nominal
dynamic specific cutting force. This is because we can now measure thereal dynamic specific cutting force as will be stated in Section 3.4.
In the case of composite material, Kecik et al (2012) determined thenominal specific cutting force and suggested an interesting behavior.
Importantly, they clarified that the nominal specific cutting force changes
depending upon the fiber orientation as shown in Fig. 3.8 (a), even when
the fiber direction is constant. This results from the relative position changeof the fiber with the rotational cutting edge. Eventually, they can formulate
both the nominal specific cutting force for tangential and radial
components by using the regression model as follows 4.
𝐾𝐾", % = 𝑎𝑎'" (𝑎𝑎( + 𝑎𝑎) θ + 𝑎𝑎* 𝜃𝜃 * + 𝑎𝑎, 𝜃𝜃 , + 𝑎𝑎- 𝜃𝜃 - + 𝑎𝑎. 𝜃𝜃 . ) ------(3.2)

where, θ is in radians.
4

In accordance with the information from Kecik, Fig. 3.8 (a) is based on that of Sheikh-Ahmad J Y.
(Machining of Polymer Composite. Springer-Verlag, 2009).

85

(a) Variation of specific cutting force with fiber orientationand tooth position (ψ= 45 deg.)(b) Comparison of stability charts for constant Kn and Kt
with those variable
Fig. 3.8 Stability charts for composite material (by courtesy of Kecik)
86

As can be readily seen, the stability chart shows some differing features asshown in Fig. 3.8 (b), when the nominal specific cutting force varies withthe rotational angle of the end mill. The stability charts were produced formilling with one-flute PCD (Poly Crystalline Diamond) end mill of 12 mm
in diameter, 6 mm in (radial) depth of cut, and 0.03 mm/tooth in feed,
where the chip thickness coefficient κ is 0.9 4. In addition, the dynamic
characteristics of CNC (Computerized Numerical Control) milling machine
were obtained by the modal analysis, i.e. m = 0.8 kg, c = 57.3 Ns/m and k =2.22×101 N/m.
To this end, it is worth suggesting that the cutting process equation should
be modified as follows, when we will estimate the deterioration magnitudeof the chatter stability by the inevitable disturbance like the impact force
while milling.
F(t) = kcu(t) + f0sinωt (forced vibration) , t = 2π/ω ---(3.3)

3.3.3 Issues in Structural Equation

Within the “Structural Equation” context, there are three leading issues, i.e.
(1) measurement of dynamic characteristics, m, c and k shown already in
Fig. 1.1, (2) clarification for distribution diagram with respect to therigidity and damping coefficient within a whole system, and (3) effect or
influence of non-linearity in structural body component on chatter.
Of these, the item (1) is at burning issue and vital, and thus we will
discuss it in the following. For the item (2), we have not enoughengineering data at present; however, even in the discussion aboutprocess damping, it is better to refer to the “Damping Distribution
Diagram” as well as the “Deflection (Rigidity) Distribution Diagram” ofthe machine-attachment-tool-work system as a whole, so that we canclarify quantitatively the validity of process damping in consideration of
its relative damping capacity within a whole system.
The idea of the distribution diagram is one of the auxiliary design
methodologies having been used since 1960s, and a deflection distribution

87diagram for the planomiller of portal type is reproduced in Fig. 3.9 (Geiger,
1965). Regarding the item (3), we must be aware of the very importance offitting tolerance design in both the structural body components. In fact, theanti-chatter capability can be improved by designing a suitable fitting
tolerance; however, such an outstanding structural design is not fruition yet(see Section 2.2).

Fig. 3.9 Deflection distribution diagram for planomiller of portal type(by Geiger)
Conceptually, the machine-attachment-tool-work system is, in principle, ofmultiple-degree of freedom, and thus we employ the equivalent mass,
damping coefficient and spring constant in the utmost simplified structural
equation for one-degree of freedom system. We can however suggest thefollowing problems even in the simplified structural equation.
(1) With the advance of FEM (Finite Element Method) analysis for the
structure, we can compute the dynamic behavior of the machineattachment-tool-work system to some extent; however, the availabilityof the computed values is limited because of including uncertainties, e.g.
88

damping capacity at the joint.
(2) In the case of thin-walled work, the dynamic behavior changes
continuously with machining, and thus we need to get the structural
equation with real-time adaptability.
(3) We used to measure experimentally these coefficients, for example, by
the impact hammer method; however, it is difficult to measure them inconsideration of all the directional orientations in the dynamic behavior
correctly.
(4) In general, the measurement of the coefficients is carried out in the still
stand of the system; however, we need those in the machining condition.
More specifically, the dynamic characteristics have been measured
while the machine-attachment-too-work system is in the still stand or
non-loading operation so far. As can be easily imagined, such measured
values could be far from those in actual machining states at present, and
it is thus natural to discuss the availability of the stability chart.
(5) Extremely regarding the damping coefficient, its reliable measuringmethod is not established yet, because of the great difficulty in thequantified measurement of the damping capacity as already stated in
Chapter 2, and also revealed in the sphere of machine tool joints (Ito,2008).
Now let us discuss furthermore the measurement of dynamic characteristics
while machining. In this context, we must also reinvestigate, up to howmuch spindle speed, the dynamic characteristics, i.e. m, c and k, measured
in still stand are available. In the 1960s, there could be non-influential
deviations between those for 2,000 rev/min in rotational speed and still
stand of the spindle state, and thus the measured characteristics were withinallowable values when applying them in practice.
Figure 3.10 reproduces some measured values for m, c, k and due
frequency by the impulse excitation, while end milling (Matsumura andHaibara, 2002). Although not stating the differences between those for still89

Fig. 3.10 Measured values for m, c, and k while end milling in 2002(by courtesy of Matsumura, 2014)
stand and in milling, as can be readily seen, the measured values varyconsiderably depending upon the rotational angle of the cutting edge.
Conceptually, the milling process is of vibrational phenomena withmultiple-degree of freedom and, considering it, that of Matusmura andHaibara may suggest the necessity of identifying the dynamic
characteristics in the structural equation while actual cutting, but not being
in still stand.
Of special note, Nicolescu and Archenti (2013) investigated the dynamic
characteristics while turning the cylindrical work supported at both theends with centers. More specifically, they detected first both the vibrationalsignals, i.e. those for stable turning and with chatter, by the microphone.
Then, they conducted the simulation for the “Motion of Equation” excitedby the white noise, and then by comparing the simulated result to the
output signal of the microphone, the damping coefficient and spring
constant are identified.
90

Fig. 3.11 Test rig for measurement of dynamic characteristics while end
milling (by courtesy of Sims, University of Sheffield, 2015)
Following these, the Dynamic Research Group of University of Sheffield
has launched out an investigation to measure the dynamic characteristics
while the aerospace component is finished with higher-speed milling. Fig.
3.11 reproduces the test rig, which is being used under the aegis of EPSRC(GR/S49858/01). Importantly, the frequency response of the milling cutter
and concerns shows the non-linearity, which could be caused by the joint
between the tool holder and main spindle. In this case, we must be
furthermore aware of the counter-effect of the electromagnetic exciter,
which plays in turn a role of damper so far suggested elsewhere, although
the Group seems not pay any special attention to it.
For the ease of further discussion, the logarithmic damping decrement isapproximately calculated from the dynamic characteristics so far measured
as shown in Tables 3.1 (a) and (b). In this case, we assume that themachine-attachment-too-work system is one-degree of freedom with decay
vibration, and that the logarithmic damping decrement δ can be written as

where, m: Mass, k: Spring constant, c: Damping coefficient
cc: Critical damping coefficient91

(a) While cutting(b)(b) While in still-stand or non-loading operation
Table 3.1 Conversion of measured m, c and k into equivalent logarithmic
damping decrement
From the magnitudes of the equivalent logarithmic damping decrement
shown in Table 3.1, we may understand first the difficulty in themeasurement of the damping capacity, because of their widely scatteredvalues. Then, we must scrutinize the reasons regarding why we observe theincredible magnitudes as exemplified by those of Ahmadi and Sellmeier. In
case of that of Sellmeier, the equivalent logarithmic damping decrement islarger at the first natural frequency of the system than that for the highest
natural frequency. Apparently, we may expect the effect of the92

“Lower-speed Stability”; however, the equivalent logarithmic damping
decrement itself is too large as compared with that in the mechanicalvibration system. In addition, we may find certain uncertainties in themeasured values regarding whether the damping coefficient is of
directional-orientation and frequency-dependent characteristics or not by
nature.
Importantly, it has been said that the total damping capacity of a machine
tool as a whole is less than 0.3 in logarithmic damping decrement, and thatthe machine-attachment-too-work system does not occur any chattervibration when its logarithmic damping decrement is more than 0.6. More
importantly, the machine tool joint can facilitate the sources of the larger
damping capacity, provided that the joint is not located at the node in thevibration mode.
To this end, it is also worth suggesting that the structural equation should
be further generalized by incorporating, for example the “Directional
Orientation in Chucking Stiffness”, and “Non-linearity in Stiffness ofStructural Body Component”. In the case of work holding by the three-jaw
chuck, the structural equation yields toF(t) cosβ = m(d2/dt2)y(t) + c(d/dt)y(t) + (k + kw sin3ωt)y(t) ----(3.5)
where, kw = Average stiffness of chuck-work system
In addition, the Editor emphasizes furthermore that we have so far
discussed the necessity of the measurement of dynamic characteristics, m, c
and k, for example, in the main spindle branch of MC only. In contrast, wehave not paid any attentions to those belonging to the table-fixture-worksystem, i.e. table branch. This is because we assume that the table branch
is dynamically stiffer than that of the main spindle branch; however, incertain cases, we must combine the dynamic characteristics of both thebranches, although such a research is about to launch out (see Chapter 5).
93

3.4 Drooping Characteristics in Real Dynamic Componentof Cutting Force and Self-excited ChatterIn the measurement of the cutting force, Kistler AG contrived an innovative
force sensor of piezoelectric type in the late 1960s, and in consequence, we
can measure accurately the cutting force. Importantly, we can discriminate
three components, i.e. static, fluctuation and dynamic components, as
shown schematically in Fig. 3.12. In this case, the fluctuation component iscaused by the change of the number of cutting teeth in simultaneously
engaging to the work.

Fig. 3.12 Static, fluctuation and dynamic components in cutting force
More importantly, Opitz et al (1970) and Langhammer (1972) revealed,
especially in Dr. Thesis of Langhammer, the characteristic features of the
dynamic component in cutting force, as shown, for example, in Fig. 3.13.
As can be readily seen, there are apparent differences in the behavior of the
dynamic component from static one together with their magnitudes. It may
thus worth suggesting that we must eye a new direction in the94determination of the “Real Dynamic Specific Cutting Force” as follows.
(1) The nominal dynamic specific cutting force has been so far determinedby measuring the average value of the cutting force, i.e. static
component in Fig. 3.12. Is such a nominal dynamic specific cutting
force reliable and applicable to the analysis of the chatter vibration?
(2) What is the role of the dynamic component in the cutting force unveiled
by Langhammer in the analysis of the chatter? Must we determine thereal dynamic specific cutting force by using the dynamic component?
Supposedly, there have been no publications related to these controversial
issues, and thus for the sake of further research activities, some quick notes
will be given in the following.

Fig. 3.13 Static and dynamic components of cutting force in carbon steel
turning ( by Langhammer )As can be readily seen from Fig. 3.13, the static component in the cutting95

force is in nearly constant with increase of the cutting speed, whereas the
dynamic component shows the typical “Drooping Characteristics”, i.e.
dynamic component reducing with cutting speed, in turning of the work
made of carbon steel. In fact, Langhammer obtained a similar result whenturning the work made of alloy steel (DIN 16MnCr5G).
As already suggested by Arnold in the past and again resurged recently by
Entwistle and Stone (2013), the drooping characteristics in the cutting force
may induce the self-excited chatter without any regenerative effects(Arnold, 1946), provided that the dynamic component is larger enough.
Thus, Fig. 3.14 illustrates the mechanism of such a chatter type.

Fig. 3.14 Theoretical model for self-excited chatter vibration derived from“Drooping Characteristics” in cutting force
If x(t) represents the displacement of the tool in the direction of the force,the principal component of the cutting force is given byThus, the equation of motion for one degree-of-freedom yields to96

where, b = Cutting width, u0 = Depth of cut, R0 and α = Positive coefficient
determined by work material, cutting conditions, shape and size of cuttingedge and so on.
When the coefficient for velocity is negative, the system becomes unstable,
and thus the chatter onset can be regulated by

Intuitively, another facing issue is whether the nominal dynamic cutting
force proposed by Tobias is identical to the real dynamic cutting force
measured by Langhammer, and thus we must, at least, conduct thefollowing research.
(1) Measurement of the dynamic plowing force in both the stable and theself-excited chatter, which can be regarded as an evaluation factor ofprocess damping when verifying the hypothesis suggested by Tobias.
The plowing force can be obtained by applying the extrapolation
method to the “Cutting Force - Feed Rate or Depth of Cut” chart. The
force corresponding to the nil in depth of cut or feed rate axis is theplowing force. For example, from Fig. 3.13, we can obtain the static
plowing force as to be 70 kgf in the case of the principal component of
300 kgf in full range.
(2) To unveil the correlation of the dynamic cutting force with thecorresponding instantaneous uncut chip thickness.
Langhammer publicized furthermore another interesting results, whichwere obtained when turning the work made of heat resistant steel. As
reported elsewhere, the heat resistant steel is prone to generate the saw-likechip, and thus Fig. 3.15 shows also such information. In fact, the dynamic
component is larger at the cutting speed, where the saw-like chip can beobserved. On the strength of those shown by Langhammer, we could
expect that the dynamic component of the cutting force is a clue forunveiling the essential features of the self-excited chatter.
97

Fig. 3.15 Static and dynamic components in cutting force when machining
heat-resistant steel (by Langhammer)
Summarizing, it is necessary to investigate to what extent the dynamic
component contributes to the self-excited chatter. In such an investigation,
we must be aware that the cutting stiffness has been evaluated by using the
average value (static component) of the measured cutting force, and thus anewly arisen problem is to verify the validity of the determination method
for the cutting stiffness so far used.

References
Altintas Y, Eynian M, Onozuka H. (2008) Identification of dynamic cutting
force coefficients and chatter stability with process damping. CIRP Annals- Manufacturing Technology; 57: 371-374.
Arnold R N. (1946) The Mechanism of Tool Vibration in the Cutting ofSteel, Cutting Tools Research: Report of Subcommittee on Carbide Tools.
98Proc. I Mech E: 261-284.
Doi M, Masuko M, Ito Y. (1982) Re-observation of the Chatter Vibration in
Chuck Works - Effects of Jaw Number on the Appearance of a Scale-like
Chatter Mark. In: Proc. of 10th NAMRI of SME, 409-416.
Entwistle R, Stone B. (2013) Position Paper - Fundamental Issues in
Self-excited Chatter in Grinding. Jour. of Machine Eng.; 13-3: 26-50.
Geiger H G. (1965) Statische und dynamische Untersuchungen an
Schwerwerkzeugmaschinen. Dissertation, TH Aachen, 1965.
Insperger T. et al. (2008) On the chatter frequencies of milling processes
with runout. Inter. Jour. of Machine Tools & Manufacture; 48: 1081-1089.
Ito Y. (2008) Modular Design for Machine Tools. McGraw-Hill.
Jemielniak K, Wypysinski R. (2013) Review of Potential Advantages andPitfalls of Numerical Simulation of Self-excited Vibrations. Jour. of
Machine Engineering; 13-1: 77-90.
Kecik K, et al. (2012) Chatter control in the milling process of composite
materials. Jour. of Physica: Conference Series 382: 012012.
Langhammer K. (1972) Die Zerspankraftkomponenten als Kenngroßen zur
Verschleißbestimmung an Hartmetall-Drehwerkzeugen. Dissertation der
RWTH Aachen.
Kondo Y. et al. (1980) Behaviour of self-excited chatter due to multiple
regenerative effect. Trans. of JSME(C); 46-409:1024-1032.
Matsumura T, Haibara T. (2002) Simulation of Dynamic Cutting Process inMilling Operation. Preprint of Assembly of Manufacturing & Machine
Tool Division, JSME.
Nicolescu M, Archenti A. (2013) Dynamic Parameter Identification in
Nonlinear Machining Systems. Jour. of Machine Engineering; 13-3:
91-116.
Opitz H, Dregger E U, Röse H. (1967) Improvement of the Dynamic
Stability of the Milling Process by Irregular Tooth Pitch. In: Proc. of 7thInter. MTDR Conf., Pergamon, p. 213 - 227.
Opitz H. et al. (1970) Statische und dynamische Schnittkräfte beim Drehenund ihre Bedeutung für den Bearbeitungsprozess. Forschungsberichte des99

lands Nordrhein-westfalen, Nr. 2144, Westdeutscher Verlag.
Rall K et al. (1998) Vermessung rotierender Werkzeuge in HSCFräsemaschinen. ZwF; 93-4: 127.
Sato H. (2013) Multiple Regenerative Effect Governing Chatter Behaviour
After Onset. Jour. of Machine Engineering; 13-3: 51-76.
Schmitz T L, Ziegert J C, Stainslaus C. (2004) A Method for Predicting
Chatter Stability for System with Speed-dependent Spindle Dynamics.
Trans. North American Manufacturing Research Institution of SME; 32:
17-24.
Sellmeier V, Hackeloeer B, Denkena B. (2009) Process Damping in Milling- Measurement of Process Damping Forces for Chamfered Tools by Meansof an Electromagnetically Guided Spindle. 12th CIRP Conf. on Modellingof Machine Operations, San Sebastian.
Tlusty J, Ismail F. (1981) Basic non-linearity in machining chatter. CIRP
Annals; 30-1: 299-304.
Van Dijk N J M. et al. (Oct. 2008) Real-Time Detection and Control of
Machine Tool Chatter in High Speed Milling. 2nd Inter. Conf. “Innovative
Cutting Processes & Smart Machining”, Cluny.

100PART IILEADING-EDGE RESEARCH AND ENGINEERINGDEVELOPMENTWITHIN ADVANCED CHATTER VIBRATION THEORY

Headnotes
In Part I, we describe first the fundamental knowledge about the chatter,and in due course note quickly the leading remedies for the chatter
suppression like the milling cutter with irregular pitch and differing helix
angle in cutting teeth. Importantly, on the strength of the achievements
obtained from a handful of forerunning and challenging researches, wesuggest a considerable number of the issues in Chapter 3, which cannot
regulate by the chatter theory at present. Such issues may give us certain
clues to advance the theory and suppression remedies of the chatter, i.e.
further research guides into the chatter.
Following Part I, we display quickly a first-hand view for the research andengineering development conducted in the year 2005 and beyond herein.
Along with the first-hand view, we display the interesting observation in
the chatter while higher-speed machining the aircraft component at present
in Chapter 4. Obviously, these ascertain the dire necessity for conducting
the research into the issues suggested in Chapter 3.
Figure II.1 illustrates such a first-hand view for the research objectives byclassifying them into the following three categories.
(1) The long-standing traditional subjects like “Chatter Suppression” and“Lower-speed Stability and Process Damping”, which are based on
those of Tobias, Tlusty and Merritt. Obviously, these are dominant evennow.
(2) Trial and challenging subjects like “Influence of Run-out in End Mill on
Stability Chart” and “Identification of Coefficients in Structural
Equation While Cutting”, which intend to modify the regenerative
chatter theory to be compatible with the higher rotational speed of themain spindle at present.
(3) Interesting subjects like “Chatter in Consideration of Joint Stiffness of101

(a) Traditional research area(b) Challenging and innovative research to unveil essentials

102

(c) Interesting research although including uncertainties
and controversial points
Fig. II.1 Present perspectives for research into chatter between
2005 and 2014 (Country Codes as per ISO 3166)
Wheel Flange While Cylindrical Grinding”, although including
something uncertainty and controversial points.
In short, Fig. II.1 is produced by the literature survey and concerns, where
the objectives are around 40 papers including the Dr. thesis, to unveil some
symptoms for the new research direction within the chatter. In this context,
the list of references in Chapters 5 and 6 is somewhat valuable tounderstand the past and present perspectives for the research into thechatter.
With growing importance of milling to machine the aircraft component andmolding die, it is thus natural that nearly all researches aim at milling.
Milling is, in principle, complicated and difficult to clarify its mechanismby the analytical method. This is because the cutter path trajectory can be
represented by the Trochoidal curve, and the chip thickness is given byboth the curves. In contrast, we can apply the advanced simulation software103

and dynamic information processing to overcoming such difficulties, andthese tools can facilitate the ease of work to a large extent.
We can observe similar difficulties in drilling and grinding; however, thecomputational simulation can facilitate considerably to solve suchcomplicated mechanisms at present. In contrast, a considerable number ofthe research papers deal with the computation only and don’t pay any
attentions to the essential feature in the facing problem.
As will be clear from Part I, importantly one of the crucial issues at present
is to discriminate the chatter pattern because we can observe various
complex chatters including unknown ones having not been experienced so
far with increasing the machining speed. Eventually, such various chatters
are caused by the self-excitation of regenerative type; however, it is veryrare case to observe the pure regenerative type at present. In contrast, eachchatter pattern includes more or less the influences of forced excitation
derived from the cutting mechanism itself and also the non-linearity in themachine-attachment-tool-work system. Thus, a trouble shooting method isfurthermore proposed in Chapter 4 after investigating the chatter mark so
far observed. In the proposed method, the chatter pattern may be identified
from the sample (reference) chatter mark together with suggesting somerecommendable tools to suppress the corresponding chatter.
In due course, in Part II we publicize the two contribution papers from thepotential researchers, which deal with some leading issues mentioned
above, to deepen the understanding for the essential feature in the chatter(see Section 3.3.3 and Fig. 3.6). As widely known, we used to detectvarious signals related to the chatter, e.g. sound, vibration acceleration,
cutting force and so on, with various transducers; however, we have usedsuch signals to identify only the chatter onset so far. Against this context,
these contributions suggest further importance of signal processing in detailby exemplifying concretely that signals at chatter onset and immediately
before onset involves the valuable information to unveil the essential
features of the chatter.
More specifically, Professors Archenti and Nicolescu of KTH Royal104

Institute of Technology introduced the concepts of statistical dynamics forthe identification of the dynamic parameters of the machining system, in
actual machining operations, by using both the non-parametric and theparametric computational models in Chapter 5. Importantly, these proposed
procedures can be characterized by real time estimation of the operational
dynamic parameters of the machining system, represented by theinteraction of the two subsystems, machine tool elastic structure andcutting process. More importantly, they suggest the importance ofchemistry between the real-time identification of dynamics in themachining system with the statistical signal processing as exemplified by
the application of “Markov Process Representation”. Chapter 5 can playfurthermore the role of a “Position Paper” with respect to the output signal
processing when discriminating the properties of the chatter.
Professor Schmitz of UNC Charlotte conducts a numerical andexperimental investigation in Chapter 6 to accurately predict both stableand unstable milling conditions. He uses the well-known “Poincaré Map”
(a phase map between displacement and velocity) to study the self-excited
vibrations due to the regenerative effect (called “Secondary Hopf”) as wellas instabilities that result in behavior that repeats every second, third,
fourth, and so on tooth period (called “Period-n bifurcations”), rather thaneach tooth period (“Forced vibrations”).
In consideration that a crucial issue at present is to improve theapplicability of the stability chart while carrying out higher-speed
machining, furthermore, a quick note is given for the present and near
future perspectives with respect to the dynamic behavior of the main
spindle in Chapter 7. In short, it is notable that the dynamic behavior of themain spindle is not clarified as yet while rotating more than 2,000 rev/min,and this is one of the major obstacles to establish the stability chart, whichis available for the machining condition with higher-cutting speed. Dare to
say, people in chatter concerns are not familiar to the characteristics of themain spindle in detail and its influences on the chatter, although they areinterested in such subjects. Thus, the Editor suggests some leading research

105

subjects to be investigated hereafter. Paraphrasing, we cannot apply thestability chart to the machining condition with the higher-cutting speed,
where we face certain difficulties to determine the parameters of the“Structural Expression” in the chatter theory. As can be easily imagined,
one of the determinants in this case is the rigidity and damping capacity ofthe main spindle.
In accordance with the aims and purpose of this book, Chapter 8 provides
us with the though-provoking quick notes for the research guide to thechatter in metal grinding. As well and widely known, the research into thechatter in metal grinding has been inactive since 1970s, and thus the Editor
suggest roughly that we have no systematic remedies for the facing
problems, which could be derived from the contact stiffness, grinding
stiffness, grinding damping and so on. In Chapter 8, these influencing
factors to be in current issues are detailed following the quick notes in
Section 1.3. It is extremely noteworthy to clarify the influence of the“Unbalance of Grain Distribution” in the grinding wheel on the chatter.

106

Chapter 4 Observation for Various Chatter Marksand a Proposal of Trouble Shooting for
Identification of Chatter Patterns
With continuing requirements for the product innovation, we need toenhance machining requirements for the part of the product, and thus themachine tool user is often suffered from the chatter. In due course, themachine tool manufacturer must solve such trouble in accordance with theuser’s claims. Obviously, such claims are very valuable to understand thefacing chatter problems; however, because of fierce competition among
machine tool manufacturers, in general, we cannot obtain their details. In
this context, we must be aware that the chatter mark is one of the dominant
clues to solve such chatter problems.
As reported elsewhere, we can observe at present the chatter with more
complicated pattern (type) ever than before with prevailing thehigher-speed machining, and often we face certain difficulty to identify its
causality. In such cases, the chatter mark may also provide us with useful
information.
In this chapter, thus, we discuss first the characteristic features in the
chatter mark accidentally obtained from some case studies on machining by
MC at present, and then in consideration of the discussion in Chapter 3, wesuggest a troubleshooting method to identify the chatter pattern from thechatter mark. In addition, we suggest how to seek the recommendable
cutting tool when we can identify the chatter pattern.

4.1 Case Studies on Observation of Chatter Marks
As can be readily understood from Part I, we can observe various chatterpatterns at present with prevailing the higher-speed machining. In certain
cases, we can observe a chatter pattern, which is unfamiliar and far beyond
our knowledge resulting in the difficulty of identifying its causality.
107

As already mentioned in the headnote, the chatter mark gives us with someclues to identify the chatter pattern and in the simplest case, the chatter ofregenerative type generates the micro-inclined ridge-like mark along thefeed direction of the cutter as shown in the “Title Page”. In grinding, at theburning issue is to finish the surface without any “Blur-like” texture, and in
this case, we can identify, whether it is derived from the “Forced Vibration,e.g. that caused by the unbalance of the grinding wheel” or “Regenerative
Effect”, by polishing the surface with the oilstone (Arkansas stone). Morespecifically, the former seems the micro-ridge with “Beat wave-like”,
which is perpendicular to feed direction of the grinding wheel, whereas thelatter is an inclined micro-ridge to feed direction of the grinding wheel, i.e.
that of typical regenerative type.
Admitting the effectiveness of the chatter mark in identifying the chatterpattern, however, we face often certain difficulties to obtain the obvious
and reliable information for the chatter mark, because of the company’s

Fig. 4.1 Representative chatter mark for regenerative typeconfidentiality. Against this context, the Editor can obtain fortunately some108

examples of the chatter marks from the publicized material as follows,
although being not enough information, but rough ones to discuss
something necessary.
Figure 4.1 shows a typical chatter mark caused by the regenerative effectwhen the spindle speed is 12,000 rev/min. As can be readily seen from thestability chart shown in the bottom right of Fig. 4.1, we can suppress it byshifting the spindle speed from 12,000 rev/min to 20,000 rev/min (see leftof Fig. 4.1).

Fig. 4.2 Regenerative chatter mark mingling with beat-like cutter markand its suppression
Figure 4.2 shows an interesting chatter mark, which may be regarded as aregenerative type mingling with the beat-like mark at certain interval in
feed direction. Could be, such a chatter is derived from the influence of theexcitation factor within the cutting mechanism, e.g. run-out of end mill, as
implied by cutting tooth chipping. This suggestion may be supported by thefact: we can suppress such a chatter mark by changing only the cutting
tool-tool holder system as shown also in Fig. 4.2.
109

Figure 4.3 shows a chatter mark observed in thread milling and its
suppression by changing the cutting tool. More specifically, in the left ofFig. 4.3, we can observe the chatter mark, which is generated by a synergyof the torsional and bending vibrations, and in which the crown-pleat-likemark may correspond with the torsional chatter as similar as that generated
by the cutting edge while twist-drilling the pocket hole (Roukema and
Altintas, 2006). In contrast, in the right of Fig. 4.3, we can observe thesuppression of the torsional chatter, but still the aftermath of the bending
chatter, although reducing the vibration amplitude.

Fig. 4.3 Chatter mark generated by synergy of torsionaland bending vibration
Admitting that we can identify, in certain cases, the chatter pattern from thechatter mark, we can observe, in contrast, unfamiliar chatter mark withprevailing the higher-speed machining and also growing demands formachining of difficult-to-machine material like heat-resistance steel. Fig.
4.4 shows such a chatter mark, which consists of blur-like and beat-likemarks along with large burr at disengagement of the end mill and the110beat-like mark appears at certain interval as indirectly proven by thefluctuation of the cutting force. Importantly, there are no apparent chattermarks indicating that related to the regenerative type.

Fig. 4.4 Effect of special function for chatter suppression while end millingMore importantly, the chatter can be suppressed by a special function
integrated within a machine tool itself (no information because ofcompany’s confidentiality): however, it seems that there remains thedisturbance in the cutter mark and the small fluctuation of the cutting force.
Thus, it is necessary to investigate what is the essential feature of thefluctuation of the cutting force, e.g. either dynamic component of the
cutting force in stable milling (see Fig. 3.12) or cutting force corresponding
with the blur-like chatter mark.
111

4.2 A Proposal for Troubleshooting to Identify
Chatter Patterns
Figure 4.5 demonstrates one of the typical evidences in the chatter
suppression remedy at the factory floor. As can be seen from it, we try, in
general, the chatter suppression by relying only the qualitative and simple
data, e.g. vibration amplitude, and also employing a simple remedy withthe cheaper cost, e.g. change of the cutting tool. In fact, the chatter can besuppressed by only changing the slot drill of two-flute type to that ofthree-flute type, and the engineer and technician in concern don’t conduct
any further activities.

Fig. 4.5 A suppression remedy by changing cutting tool
It is however recommendable to pursuit the causality of the chatter together
with identifying its pattern. On such an occasion, it is very useful to use thetrouble shooting method by referring the chatter mark being observed tothe model pattern (reference chatter mark).
In this context, it is furthermore desirable that we will be able to indicate acorrelation between the chatter mark and the cutting tool preferably112facilitating the chatter suppression in practice; however, nearly all the
cutting tool manufacturers advertise, in general, only the effectiveness of
their products to suppress the chatter without any guidance for thecorresponding chatter pattern. Of course, in the case of the milling cutter ofirregular tooth pitch type and with differing helix angle, we can identify
such a correlation without any problems.
Now let us discuss this issue by taking some sample cutting tools
demonstrated at JIMTOF (Japan International Machine Tool Fair) 2018. In
the discussion, it is better to use a linkage diagram, which can delineate theclose tie among the cutting tool, its characteristic features, and chatterpattern possible to suppress together with the suppression principle tocertain extent.

Fig. 4.6 Linkage diagram for visualizing close tie among cutting tool, its
characteristic features and chatter pattern possible to suppress (1)Figure 4.6 shows a representative case study on the end mill and slot drill,
and as can be seen from Fig. 4.6, these cutters can facilitate the chatter
suppression in accordance with the self-excited chatter theory of113regenerative type. More specifically, the former controls the regenerative
effect together with rigidity reinforcement, whereas the latter controls the
specific cutting force by the wave-like cutting edge as similar as that shownin Fig. 2.7. It is furthermore notable that we may expect the reduction of
the run-out in the end mill by the tapered web while rotating higher-speed,
although the manufacturer does not pay any attention to it.

Fig. 4.7 Linkage diagram for visualizing close tie among cutting tool, its
characteristic features and chatter pattern possible to suppress (2)In contrast, Fig. 4.7 shows two cutting tools, which are for generalized
remedies to chatter, and thus it is difficult to identify their utmost effective
chatter pattern. More specifically, the long arbor can increase its rigidity byintegrating the core made of sintered carbide and shaping the thick flangeroot, and thus is expectable to suppress the vibration in general. At present,
the long arbor is for the milling cutters of 50 ~ 100 mm in diameter. In
retrospect, Giddings & Lewis employed once the boring bar inserted partly
the sintered carbide as shown in Fig. 2.1 in the 1960s, and obtained thefruitful result, although such a boring bar was very expensive on that114

occasion.
As similar as the long arbor of MST-brand, the deep grooving tool withsemi-active damper is one of the very popular remedies for the chatter
suppression. In this case, it is very interesting that we can adjust thedamping capacity on-site by controlling the pre-load between the O-ringand the “Damping plates (mass for tuning)” by the calibration screw (adjust
screw) as can be seen from the bottom left of Fig. 4.7. In deep grooving
tool, furthermore, we must be aware of its extreme effectiveness forsuppressing the vibration with higher frequency as suggested elsewhere;
however, the manufacturer does not care about it nevertheless one of thevaluable information for advertisement.
Admitting that the tool manufacturer has often contrived the innovative
tool for the chatter suppression, as exemplified above, the manufacturer
does not delve into the potential function and performance of the developed
tool and may lose the obtainable benefits. For example, Walter has
contrived a new drill made of sintered carbide (Type Advance DC 160), in
which the main stresses are placed on (1) a new thinning method, i.e. could
be a variant of XR type, (2) the double-margin for better guiding, (3) the
innovative two-layered coating by TiSiAlCrN and AlTiNi and (4) 140
degree in point angle. As a result, the cutting force reduces considerably
and the drill guiding accuracy is improved to some extent, resulting in the
better hole-location accuracy.
Intuitively, we may, as will be clear from above, expect the reduction of thespecific cutting force duly resulting in the suppression of the torsional
chatter, which was shown, for example, in Fig. 4.3, together with lessbending chatter. We must thus examine whether such suppression isfunctioned or not.
Now let us discuss what is a desirable trouble shooting method. As will beclear from Figs. 4.1 ~ 4.4, a method enables first the chatter pattern to be
identified from the chatter mark, and then as shown for example in Fig. 4.6,
one of the recommendable cutting tools should be selected by using thereverse flow in Fig. 4.6.

115

Table 4.1 Various chatter marks possible to observe in 2010sFig. 4.8 A proposal for trouble shooting of chatter in metal cuttingMore specifically, we may classify the chatter pattern on the strength of thefactory floor experience as shown in Table 4.1, which can be also deduced
from Part I, and simultaneously we can propose a flow of trouble shooting
as shown in Fig. 4.8, although it is far from completion. Paraphrasing, atburning and crucial issue at present is to establish such a trouble shooting
116

method by accumulating further knowledge about the correlation between
the chatter mark and its causality.
To this end, the Editor suggest an attractive research into the observation of
the chatter mark generated around the chatter onset while face milling withfull cutter width. As can be readily seen from Fig. 2.11, it is very interesting
to observe what is differing feature in the chatter mark while up-cut anddown-cut modes within one revolution of the cutter.

References
Roukema J C, Altintas Y. (2006) Time domain simulation of torsional-axial
vibrations in drilling. Inter. Jour. of machine Tools and Manufacture; 46:
2073-2085.

117

Chapter 5 Statistical Dynamics - Analysis of
Machining Systems Operational Conditions
In this chapter the subject of statistical dynamics are discussed and nonparametric and parametric models for machining system identification are
derived. The common characteristic for all discussed models is that they
may be used for computing the operational dynamic parameters (ODP) of
the closed loop machining system. Though the input to these models
originates from the machining operations, not all models can be
implemented for real-time identification. Generally, non-parametric models
may be used solely for off-line identification, i.e., first recording thevibration signal from machining operations and then analysing the signal
and identifying the nature of the excitation. Parametric models
implemented in recursive algorithms are used for real-time identification ofmachining systems dynamic characteristics.
The main objectives of the chapter are: (i) the development of parametricand non-parametric models based on identification techniques with the
purpose of integrating into a single step within the estimation of dynamic
parameters characterising the machining system, (ii) in non-parametricidentification, implementing techniques for ODPs and random excitation
estimation, (iii) in parametric identification, the development of therecursive computational model of the machining system based on the dataobtained during the actual operational regime. Through these contributions,a step is taken beyond the classical approach to analyse the dynamics of amachining system by separately identifying the structural and processparameters. In the proposed process, the two substructures, tool/toolholder
and workpiece/fixture, are coupled, in addition to the open loop (elastic
structure of the machine tool), by a feedback loop closing the energy loop,
through the thermoplastic chip formation mechanism. The machining
118

system can only be completely analysed only in closed loop i.e. in
operational conditions since specially designed off-line experiments with
controlled input, such as modal testing, give the response from only theopen loop.

5.1 Introduction
In the modern manufacturing environment, high productivity and quality
are key requirements for competitive production. As the metal removal
rates and the quality of machined surfaces are impaired by vibrations, acritical issue in the art and science of machining technology is to control
the machining system dynamics.
Even if the technology of virtual manufacturing is steadily growing in the
production environment, experimental investigations of dynamic systems
still play a key role because they provide valuable information to
production engineers concerning the influence of process modification on
the system’s performance, the prediction of failures in the system andsupport for the maintenance of the production systems.
Generally, a machining system represents a closed-loop system with themachine tool’s mechanical structure in the main loop and the machining
process dynamics in the feedback loop. From dynamics point of view, themachining system can be characterized by the modal characteristics of thejoint system. We denote these modal parameters by operational dynamic
parameters (ODPs), as they represent the parameters of the joint machine
tool structure and cutting process, respectively (Archenti and Nicolescu,
2009 and Archenti, 2011. The modal parameters in this respect are theoperational frequency and damping and corresponding terms of operational
mass and stiffness.
The most important scientific issue considered in machining dynamics isprobably the analysis of self-excited vibrations or chatter. This topic has
119attracted the scholars’ attention for over 100 years. Taylor stated in 1907
that chatter is the “most obscure and delicate of all problems facing themachinist” (F.W. Taylor, 1907). Even though the machining process should
always be run free of chatter, the analysis of chatter as a non-linear
phenomenon is considered essential in the theory of machining system
dynamics. The interest resides especially in studying the intricate
interaction of the internal parameters of the machining system and acomprehensive understanding of their motion regulation is beneficial for
optimising the design of machining systems. Many analytical, numericaland experimental methods were developed in this field (e.g. Tlusty [1954],
Tobias [1965], Tlusty [1983], Rahman and Ito [1986], Altintas and Budak[1995], Insperger and Stepan [2002], Budak and Tunc, [2010], Quintana[2011]).
The theory of self-excited vibrations, and consequently of machining
system chatter, has experienced significant progress in the last decades. H.
Poincare founded in the 19th century the nonlinear dynamics theory thathas greatly advanced with the rapid development of the calculation
technique and the important contributions of many scientists as the Hopf
bifurcation theory that is extensively used to analyse various self-excited
vibrations (Wenjing Ding, 2010).
Chatter in machining has detrimental effects on the quality of the parts andon the safety of the operations; parts incused even slightly by chatter marks
have to be discarded. In other fields of physics, though undesirable, selfexcited vibrations are tolerated within certain limits as for instance inaeromechanical systems (flutter, aircraft flight dynamics) andaerothermodynamics (flame instability), in fluid dynamics, – known as
Kármán vortices, in dry friction in many mechanical situations, and so on.
The core part of the theory of chatter is concerned with methods ofpredicting the stability of equilibrium and deducing stability criteria.
The motivation of the study presented in this chapter is the development ofmethods for real time analysis of the dynamics of machining systems, i.e.
the computation of the operational dynamic parameters. The focus is on the120

theoretical and practical aspects of the parameter identification of theclosed loop machining systems. Generally, the parameter identification of
dynamic systems belongs to the field of statistical dynamics. As we will
later demonstrate, the benefit of parameter identification is twofold: (i) it
gives an efficient method to represent, by a synthetic model, the machining
system in operational conditions and (ii) it offers a robust stability criterion
that is inferred from the equivalent physical model. By this approach,cumbersome estimation of system stability by independently studying themachining system’s constituent parts is avoided. In addition, the parameter
identification method provides means to separate the sources of vibration,
which supports implementation of robust technologies for vibration
suppression. The identification of the machining system considered in thischapter is based on both non-parametric and parametric models(autoregressive-moving average, ARMA). Therefore, a central part in thischapter is the relation of this Markovian representation of a stationary
stochastic process to the autoregressive moving average (ARMA) models.
The chapter is organised in three parts. In the first section, in the view ofstatistical dynamics, a linear Langevin equation for a stationary stochastic
process is derived under the assumption of the Gaussian Markov processwith the whiteness of random force excitation. The transition probability
density function is demonstrated to be a solution of the Fokker-PlanckKolmogorov (FKP) equation. The transition probability density function isused to construct a criterion for the discrimination between forced and selfexcited vibrations, i.e. detection of the stability border. This forms thenonparametric representation of the stability criterion. In the second section,
for parametric identification, the stochastic process of the system isconstructed and described by an ARMA (autoregressive moving average)
model. Practical implementation of the discrimination algorithm inmechatronic systems for machining operation milling and turning are
exemplified through this chapter. The chapter presents a unified theory
originating from the Markov processes for both nonparametric andparametric approaches. The structure of the chapter is illustrated in Fig. 5.1.
121

Fig. 5.1 Structure of the chapter

5.2 Statistical Dynamics
The excitations that occur in many dynamic systems are of a random natureand cannot be described adequately by the deterministic function of time
(Nicolescu, 1996). An adequate description of excitation, and therefore of
the response of the system to such excitation has to be developed within theframework of the statistical dynamics theory. Statistical methods arewidely used for the analysis of the dynamic phenomena. Statistical
dynamics as a general discipline examines various random phenomena indynamic systems. The use of statistical models opens the possibility toextend the analysis to the real response of the dynamic systems. There arethree main groups of problems where statistical dynamics find applications(Svetlitsky, 2003):
i. The first group consist of problems in which the scope is to estimate
the response level of a system exposed to random excitation, and to
study the dependence of the response level on different parameters.

122

ii.

The second group of problems belongs to the reliability theory with atypical example of “the first-passage time”, which deals withprediction of the probability that during a given time interval(response of a dynamic systems) will stay within a given admissible
domain.
iii. The third field of applications of statistical dynamics is concernedwith the identification of dynamic systems. In machining systems,
the parameters of the system cannot be precisely estimated from offline experiments. In the case of a machine tool structure interactingwith the cutting process, the correct model of the system can solely
be based on the data from real machining tests.
When applying identification methods for modelling dynamic machining
systems, the input itself is not measurable (observable) and only the
response to a random input can be measured. The field of application of
statistical dynamics that is the focus in this chapter is related to twoconcepts: (i) identification and (ii) classification of the response of dynamic
systems.
The term identification refers to the formulation of a mathematical modelof the dynamic system based upon on-line signal measurements of system
response, and belongs to a class of inverse dynamic problems encounteredin various technological fields (Ljung, 2006). The classification term aims
to determine, from the model, a function to characterise the nature of thesystem’s response, i.e., whether x(t) represents a forced/free vibration
response or if it is a self-excited vibration. The problem of identificationand classification of a system’s response to find the border between the
random forced vibration and self-excited vibration is the key issue for theevaluation of the stability boundary in operational conditions. The responseof the system to a broadband random force will be a nonzero steady-statesignal both in stable and in unstable state.
Special problems of so-called qualitatively and semi-qualitative
identification are of special importance in studying dynamic stability ofmachining systems. The qualitatively identification implies that the123discrimination between different types of dynamic systems is based on thestatistical analysis of measured response. The solution to this problem
should precede the estimation of system parameters, i.e. quantitative
identification. The results of a qualitative identification may be of direct
practical importance, for instance for developing of methods for the
analysis and control of self-excited vibrations (e.g. chatter in machining,
flutter in aerodynamics and Kármán vortex in fluid dynamics). The term
semi qualitative identification is used for those problems in which only few
key parameters are sought. The results from qualitative identification canbe used to detect chatter vibrations of machining systems from its measured
random response. Algorithms for chatter detection may be used forselection of machining parameters to avoid instabilities of chatter type
(Archenti, 2011).
From the dynamics of a machining system point of view, there are twomain objectives, (i) the identification of the system’s response and (ii) the
prediction of the stability limit. With respect to the first issue, the lack ofqualitative criteria for evaluation of a machining system´s dynamical
behaviour led to extensive use of Fast Fourier Transform (FFT) analysis fora relative comparison of signals recorded during machining. Quantitative
analyses are then performed to detect whether or not chatter occurs. As
these analyses are based on the signal’s amplitude measurement at certain
characteristic frequencies, they are relative in their nature and consequently
lead to subjective decisions. The problem of discrimination between
random forced vibrations and self-excited vibrations is a key issue for theevaluation of the system’s stability boundary in operational conditions. The
response of a dynamic system to a broadband random excitation will be anonzero steady-state signal both in stable and in unstable state. In case of
self-excited vibrations, the excitation persists even in the absence of therandom excitation (Blankenship, 1978). This is an important issue incomputing the border separating the self-excited and forced vibration in
operational conditions because this determines the control strategy that may
124be implemented to reduce or cancel the vibration. Following thismethodology, control algorithms can be directly implemented inmechatronic devices for identification and control of system stability.
Regarding the second problem, the main techniques used today are related
to a judicial (limits are not absolute) use of “stability lobes” to findrelatively stable operating regimes (Rivin, 2002). The traditional evaluationof machining system’s dynamic behaviour has invariably been approachedin the following steps (Brecher, 2009 and see Fig. 5.2):

(a)

(b)

(c)

Fig. 5.2 Analysis of the dynamic behaviour of the machining system: (a)Extraction of dynamic parameters of the elastic structure, (b) Stability
charts, (c) Chatter marks as a result of dynamically unstable machining
1. Identification of the dynamic properties of the elastic structure ofmachine tools. Generally, this step is done experimentally often
using experimental modal analysis (EMA).
2. Identification of the characteristics of the cutting process, i.e. thedynamic parameters describing the transfer function of thesubsystem represented by cutting process dynamics
3. Evaluation of the stability lobe diagram of the machining systemfrom step 1 and step 2.
Because the evaluation of dynamic stability and of modal parameters,
carried out following the above approach, does not take into consideration
the actual operational conditions, the results have limited applicability.
125Take for instance a 5-axis milling machine, in which the tool follows acomplex 3D trajectory, and where the modal direction of the machine tool
depends upon the relative orientation of the cutting tool and the workpiece.
This situation is difficult to adequately capture with traditional EMA
techniques, as the receptance is customarily measured in two orthogonal
directions. Furthermore, the classical methodology described above makes
use of external forces that are of a different nature than cutting forces. The
separation into the two subsystems does not take into consideration themutual interaction between the two subsystems in real operations. In orderto do this, it is necessary to monitor and analyse the real system in fulloperation. Design and measured data could then be put together to make arealistic digital model of a physical machine system individual, that could
be used as input in machining simulation software to find the root causes ofstability problems. Also, in the classical methodology, straight paths of the
cutting tool have to be adopted since the stability limit is detected by
gradually increasing the depth of cut. This makes it in many situations
inapplicable in the case of workpieces with complex shapes.
From a dynamics point of view, a machining system (MS) is formed by
two interacting subsystems; the mechanical structure of the machine tooland the cutting process. In the absence of the process, the MS represents
the structural open loop between two substructures, tool/toolholder andworkpiece/fixture, through the machine tool structure, e.g. columns, bed,
movable and fixed parts, see Fig. 5.3 (a). With the process considered, thetwo substructures, tool/toolholder and workpiece/fixture, are coupled, in
addition to the open loop, by a feedback loop closing the energy loop,
through the plastic chip formation mechanism, see Fig. 5.3 (b). As theoperational behaviour of the system is described by the interaction between
process and machine, the system can be complete analysed only in closedloop since specially designed off-line experiments with controlled input,
such as modal testing, give the response from only open-loop.
126Open-loop

Closed-loop

ColumnSpindleWorkpiece

Spring anddampers

Table

Base

(a)
(b)
Fig. 5.3 Machine tool represented as an open-loop system (a), and closedloop system (b)5.3 Random Processes
Problems are only considered in this chapter where the systems response toa random excitation can be observed. This may be a case of the machine
tool structure interacting with the cutting process. In this case the input
excitation is not measurable. A force sensor does not measure the input
excitation but some complex response that is transmitted through thestructural path from the tool – workpiece interface to the sensor location.
This signal is partly attenuated on this path and partly amplified by the
local structural resonances resulting in signal behaviour of a random nature(Archenti, 2010). In machining, there are many mechanisms that contributeto the random excitation, among others the dynamic variation of the contact
workpiece – cutting edge and the intermittent engagement of the multitooth cutter of impulse nature. This, in combination with the dynamic
variations of the material flow on the cutting edge, leads to a broadband
frequency excitation of the machine tool structure.
127

Real dynamic systems that possess some inertia may respond only to alimited range of excitation frequencies, with very high input frequencies
having practically no influence on the response. Therefore, if the excitationis broadband, with its spectral density close to a constant value within the
bandwidth of the system (within the range of those excitation frequencies
to which the system may respond significantly), then the response of thesystem will be the same as to white-noise excitation.
An adequate description of random excitation, and therefore of the
response of the machining system to such excitation, must be developed
within the framework of the random process theory (Dimentberg, 2004).
The adaptation of such a probabilistic point of view implies that somestatistical characteristics of both the random excitation and the system’s
response to this excitation are considered.
The problem of identifiability of an autoregressive moving average process
is demonstrated to be a complete solution of the Markovian representationof the process.

5.4 Markov Process
A function or process X(t) is called random if its values are randomvariable (Chua, 2005). The argument t is considered to be deterministic and
it is interpreted as time. It is well known that a basic characteristic of arandom variable X is its cumulative probability function
F(x) = P(x < X)

(5.1)

where P(x<X) denotes the probability of the event (x>X) or the probability
density functionp ( x) =

dF
dx

(5.2)

For a random function or process such a representation will not becomplete in general. The complete description of a random process X(t) is
128

available through the infinite-dimensional probability density p(x1, x2, ...,xn), where n tends to infinite. Since this complete representation is notpossible it is important to develop various models for the random process,which permit the expression of the infinite-dimensional joint probability
density (or its corresponding joint cumulative function) in terms of a finite
series of functions. One such model that is of practical importance to ourapproach is the Markov random process.
Among other methods such as Statistical linearization (Kazakov [1954,
1955], Sawaragi [1962], Nogouchi [1985], Harrison and Hammond [1984]),
Equivalent non-linear equation (Caughey [1986], Cai and Lin [1988]),
Perturbation and functional series (Crandall [1963], Manning [1975], Roy
and Spanos [1990]), it has been demonstrated that the response of dynamic
system to broadband random excitation can be accurately represented interms of Markov processes. The state transition probability density function
for such processes is governed by the Fokker-Planck-Kolmogorov (FPK)
equation known as the forward diffusion equation (Mohanty [1986],
Gardner [1994]). A continuous Markov process is defined in terms ofconditional probability density functionsp( xn , tn x n-1, tn-1...x1,t1 ) = px ( xn ,.tn xn-1, tn-1 ); tn > tn-1 > ... > t1

(5.3)

Assuming that t2 > t > t1 and x(t) is a stationary Markov process, then, thetransition probability density function satisfies the Chapman-KolmogorovSmoluchowski equation (Sun Jian-Qiao, 2006; Stochastic Dynamics andControl, Elsevier)p X ( x 2 , t 2 x 1 , t1 ) = ò p X ( x 2 , t 2 x, t ) p X ( x, t x 1 , t1 )dx

(5.4)

R

For a random Markov stationary process which is both wide-sense
Markovian and weakly stationary the correlation coefficient functionr XX (t - s) = r XX (t - u)r XX (u - s); t £ u £ s

(5.5)

or

rXX (t ) = r XX (j ) r XX (t - j )

(5.6)
129

Taking derivatives with respect to τ

r! XX (t ) = r XX (j)r! XX (t - j)

(5.7)

and upon setting τ = ϕ

r! XX (t ) = r XX (t )r! XX (0) = -br XX (t )(5.8)

which, taking into account that ρXX(0) = 1, has the unique solutionr XX (t ) = exp(-bt ); b > 0(5.9)

From expression (5.9) the auto-correlation function of the Markov process
can be evaluated
RXX (t ) = s X2 exp(-b t )(5.10)The Fourier transform of Eq. (5.10) gives the power spectra density withthe variance σ2 and the time constant β-1.
S XX (w ) =bs X2
p (b 2 + w 2 )(5.11)A Markov process, x(t), may be generated by the following stochastic
differential equation (Polidori, 2000).
dx(t) = A(x(t))dt + B(x(t))L(t)(5.12)In the above equation, L(t) is assumed to be a Gaussian random force withzero mean and with δ-correlated noiseL(t ) = 0;

L(t ) L(t ') = qd (t - t ')(5.13)

where q is the intensity factor of the excitation and the angle brackets
denote averaging. Eq. (5.12) is a first order Langevin equation(Mohanty,1986) and for B(x(t)) constant the equation is called Langevinequation with additive noise forcedx(t) = A(x(t))dt + σL(t)(5.14)In the context of Brownian motion, x(t) is interpreted as the velocity of aparticle moving in one dimension, and the Ornstein-Uhlenbeck, Beck
(1987) theory introduces the damping force A = -αx(t), α >0130Because L(t) is a stochastic quantity, it follows that x(t) is a stochastic
quantity too. Then, the essential problem is to find the probability
distribution p(x). One way to describe the evolution of the probability
distribution is by means of the Fokker-Planck-Kolmogorov (FPK) equation
that, for one variable, can be represented in the following form (Risken,
1996).
é ¶ù
¶¶2¶p( x, t ) = ê - D(1) ( x(t )) + 2 D(2) ( x(t )) - ... n D ( n) ( x(t )) ú p( x, t )
¶t
¶x
¶x
ë ¶x
û

(5.15)

where D(1) and D(2) are the drift and diffusion coefficients, respectively. The
drift and diffusion coefficients can be calculated using Kramers-Moyal
expansion (Van Kampen [1981], Ghasemi [2007]).
D ( n ) ( x, t ) =11nlim [ x(t - t ) - x(t )]n ! t ®0 t

(5.16)

When the process x(t) satisfies an Langevin differential equation driven bya white noise excitation, the right hand side of Eq. (5.15) retains only the
first two terms. In this case, the state transition probability densitypX(x,t;y,s), s < t, of a Markov process satisfies the FKP equation¶¶¶ 2 é B( x, t )ùpX ( x, t y , s) = - [ A( x, t ) p X ] + 2 ê
pX ú
¶t
¶x
¶x ë 2!
û(5.17)

The first term containing A(x, t) on the right hand side of Eq. (5.17) isagain the drift term associated with the deterministic behaviour of thesystem, while the second term containing B(x, t) is the diffusion term dueto the stochasticity of the excitation. Stationary solutions are obtained when
the time derivative of the probability density vanishes. The solution to the
FPK equation must satisfy the initial condition
lim p( x, t; y, s) = d ( x - y)(5.18)t ®¥where δ is the Dirac delta-function. It is possible to relate the drift anddiffusion coefficients in the FPK equation directly to the parameters in, forinstance, the stochastic dynamic equation that generates the Markov
process131

dX(t) = -αX(t)dt + σdL(t)(5.19)

where α and σ are constants and L(t) is a unit Brownian motion describedby Eq. (5.13). In this case, the coefficients of the FPK equation for thesolution of the Markovian Process X(t) are evaluated as1
E é DX X = x ùû = -a x
Dt ë1
B = lim E éë DX 2 X = x ùû =Dt ®0 Dt
1
= lim E éëa 2 x 2 Dt 2 - 2as xDt DL(t ) + s 2 DL2 (t ) ùû = s 2
Dt ®0 Dt
A = limDt ®0(5.20)The FPK equation is represented by¶¶s 2 ¶ 2 px
pX ( x, t y , s) = - [ -a xp X ] +¶t¶x
2 ¶x 2(5.21)The solution of the Langevin equation (Eq. 5.14) represents a Markov
process, which implies that the joint probability density function of its
components is described by an initial density function at time t0 and thetransition probability density function defining the conditional probability
density function at time t given the values at time t0. The solution to Eq.
(5.19) expresses the Markov process X(t)t

X (t ) = x0e-a t + ò e-a (t - s ) L( s)dt

(5.22)

0

5.5 Non-parametric Models
Let us now to extend the analysis to a second order equation of motion fora SDOF non-linear damping model!!
x(t ) + D( E ) x! + g ( x) = W (t )(5.23)
where D is the non-linear damping function with D(0,0) = 0, g(x) is a nonlinear elastic force and W(t) is a white noise process with spectral density
S0,132

W ( t )W (t + t ) = 2p S0d ( t )(5.24)

where the angle brackets denote expectation. This expression shows thatthe white noise is uncorrelated for any small finite time interval. Thismeans that the future state of variables is independent of the past state
being dependent only on the present state. This defines a Markov process.
x

If U x = ò g (x )dx is the potential energy function and D(E) is a function of0

the total energy

1
E ( x, x! ) = x! 2 + U x
2(5.25)The expression in Eq. (5.23) represents the force balance. Multiplying thisequation by x! (t ) the energy balance equation is obtained¶E= W (t ) x! - D( E ) x! 2 (t )¶t(5.26)

where the first term on the right hand represents the energy introduced in
the system, the second term corresponds to the energy dissipated.
The FPK equation for the Eq. (5.23) may be formulated according to

¶p
¶p ¶
¶2 p
+ x! + éë( - g ( x) - D( E ) x! ) p ùû - p S0 2 = 0¶t¶x ¶x!
¶x!

(5.27)The solution p( x, x! ) of the joint stationary process [ x(t ) x!(t )]has been derived
by Caughey (1964)

æ 1 E
ö
p( x, x! ) = C exp ç D
(
x
)
d
x
÷
ò
p
S
0 0
è
ø

(5.28)

where C is a normalizing constant. The integralE

d1 ( E ) = ò D(x )dx(5.29)

0133

designates the damping potential. In the special case of linear damping,
D(E) = c, a constant, the solution of the FPK equation isæ cE ö
p( x, x! ) = C exp ç ÷è p S0 ø(5.30)

From the general case represented by Eq. (5.23) there are some response
distributions that can be developed. Let us now introduce two newvariables (Roberts [1986], Iourtchenko, [2002]).
x = X sin(ω0t + φ) and x! = w0 X cos(w0t + j )(5.31)

It can be seen that(X = x 2 + x! 2 / w02)

1/2

j = arctan(w0 x / x! ) - w0t
In this transformation of variables, X is the amplitude envelope sequenceand φ is the phase sequence. Then, Eq. (5.23) can be transformed into thefollowing pair of equations
1
X! =[ - D( X sin q , w0 X cos q )] cos q + W (t ) cos q ](5.32a)1[ D( X sin q , w0 X cos q )] sin q - W (t ) sin q ]X w0(5.32b)w0

j! =Where θ = ω0t + φ. The corresponding Langevin stochastic differential
equations for the amplitude X is

W (t )
1
X! = - d1 ( X ) + 1 + W2 (t )w0X

(5.33)

where d1(X) is averaged nonlinear damping function
2p

d1 ( X ) =1
D( X sin q , w0 X cos q ) cos q dq
2p ò0

and

134

(5.34)

W1 (t ) =p S ww (w0 )2w02(5.35)

Sww is the spectral density. When the excitation is white noise then Sww(ω0)
is S0. W2(t) is white noise with intensity factor 2W1(t).
It is obvious from Eq. (5.32) that the amplitude X(t) is uncoupled from thephase φ, i.e., the process X(t) is a one-dimensional Markov process withthe transition density function p(X, t|X0, t0) governed by the following FPK
equationü¶p¶ ì¶ 2 p( X , t )ïé d1 ( X ) W1 (t ) ù
ï
=
p
(
X
,
t
)
+
W
(
t
)
íê
ý
ú
2
¶t ¶X ï
X û
¶X 2
ï
îë w0
þ

(5.36)

If the random excitation is stationary, then the response will approachstationarity

lim p( X , t | X 0 , t 0 ) = p( X )(5.37)t ®¥and the stationary probability density w(X) of the amplitude process X(t) isgoverned by FPK equationüï¶ ïìé d1 ( X ) W1 (t ) ù¶ 2 p( X )
+
p
(
X
)
=
W
(
t
)
íê
ý
ú
2
¶X îïë w0
X û
¶X 2
þï

(5.38)

which has the solution(5.39)

p(X) = 2CX exp(-F(X))

where C is a normalizing constant andX

F(X ) =1d (x )dz
w0W1 (t ) ò0 1

(5.40)In the remaining part of this section, the non-parametric estimation of thesystem damping from the stochastic response based on the method describe
above will be illustrated. In case of light damping, narrow band response tobroad band excitation will result. This condition is primarily assumed when135

analysing the dynamic system described by Eq. (5.23). The slow variationof the energy that results following this condition forms the basis for the
method of stochastic averaging. In the first step, a variable substitution is
performed. The pair (X, φ) as described by Eq. (5.32) approaches a joint
Markov process. The transition probability density function (p.d.f.) of thejoint Markov process (X, φ) is governed by the FPK equation. It can benoticed that the amplitude X is uncoupled from the phase φ and thetransition probability function of the amplitude X is governed by the FPK
Eq. (5.36). By an accurate estimate of the p.d.f. p(X) from the response
signal of the system excited by the broad band excitation, the function F(X)
is calculated from Eq. (5.40).
F(X) = ln(w(X)) – 2ln(C)(5.41)and damping energy function d1(X) from Eq. (5.34)

d (X )d
G( X ) = 1
dX
w0W1 (t )(5.42)

Finally, the damping in Eq. (5.23) is estimated from

H(X ) = ò2p0[h(v sin q ) + vh¢( X sin q )]dq

as a solution to the Schlomilch's integral equation (Nayfeh, 1985).
The numerical simulation of Eq. (5.23) has been performed by using the! , theRunge-Kunta method. For a linear viscous damping, D = 0.010 x
response and the amplitude square variation in time are illustrated in Fig.
5.4.
The estimation of the p.d.f p(X2) from the system response is represented asa normalised logarithmic function and the corresponding linear
interpolation function is shown as the function of amplitude square in Fig.
5.5. The fitted coefficients and goodness of fit statistics are shown in thecaption below.

136

X2

X

time (sec)time (sec)

(a)

(b)

Fig. 5.4 (a) Time response x(t) and (b) the square amplitude X^2(t)ln(p(X2)

Coefficients with 95% Conf.
limit p1 = 0.0211 (0.02093,
0.02127)
Goodness of fit: SSE: 11.23
R-square: 0.9876
RMSE: 0.1201

2

X

Fig. 5.5 Logarithmic p.d.f. of amplitude square and the linear fitting
The calculation of the damping value follows the above described steps
From F(X) = 0.0211X2 and d1(X)=0.0211X
2and the damping is estimated from the integral
H(X ) = ò2p

00.02011v sin q dq = 0.01055v
2

Similarly, the estimation of the damping characteristic for the simulation of
Eq. (5.23) with damping 0.04 is illustrated in Fig. 5.6.
A summary over three linear damping estimates by fitting the logarithmic
function of p.d.f of amplitude square and the corresponding interpolation
function is shown in Fig. 5.7.

137

The procedure for identification of the damping characteristic frommeasured stationary response produces a very accurate estimate of damping
characteristic for a linear case. In the next example the damping
characteristic in Eq. (5.23) is represented by the dry friction damping D(E)
= R sign( x! ). For three R values, 0.05; 0.1; and 0.2 the estimated results arepresented in Fig. 5.8.

2ln(p(X )Coefficients (with 95% confidence limits):
p1 = 0.04104 (0.04072, 0.04136)
p2 = 3.183 (3.16, 3.205)
Goodness of fit: SSE: 17.9
R-square: 0.988

2X

Fig. 5.6 Estimation of logarithmic p.d.f. of amplitude square for asimulation case with linear damping 0.04
zth

0.01

0.02

0.04

zest.

0.01055
0.01040.0106
0.98760.020510.02093,0.02127
0.988

0.041040.04072,
0.04136
0.988

Conf.

2

ln(p(X )Corr. Sq.

X

2

Fig. 5.7 Linear interpolation for damping 0.01, 0.02, 0.04
138

(a)

(b)
Theo. Val.
EstimatedConf.
interval

0.05
0.051020.004467,
0.005737

0.1
0.10110.008864,
0.011350.2
0.23930.2149,0.2636(c)Fig. 5.8 (a) simulated system’s time response; (b) amplitude squarevariation in time; (c) interpolation function of the logarithmic p.d.f. ofsquare amplitude
From both the above examples the association between the logarithmic
p.d.f. of the square amplitude and the damping estimate is apparent. Even
for a dry friction damping case a good estimate of the damping

139

characteristic can be noticed. Also, it can be observed that the higher thedamping, the higher the fitting coefficient.
The operational damping identification is demonstrated from a signal
recorded from a turning operation. The machining system’s response
during both stable and unstable processes is recorded and presented in Fig.
5.9 (a). The corresponding power density functions are illustrated in Fig.
5.9 (b).

stableunstablestable(a)(b)
Fig. 5.9 (a) Time record of the machining system’s response for stable andunstable processes; (b) PDF for stable and unstable machining processes,
respectively
Process
Est.
damp
Conf. Int.
R-squareSSE

stable0.128

unstable
0.00790.106,
0.150
0.653
545

0.0078,0.81060.970587.52

Fig. 5.10 Estimation of the damping characteristics from the interpolation
function of the logarithmic p.d.f. of the square amplitude140

It is important to recognise that in case of machining the input force is notmeasurable and therefore the variance of the excitation is not known. As aconsequence, the damping estimates are not accurate. Only the relative
ratio between them is reliable.

Also, the accuracy of the estimation

procedure is dependent on how well the input force excites the system’snatural frequencies, in other words how well the white-noise excitation isapproximated by the broad band excitation. In the stable process case, as isalso revealed by the spectra in Fig. 5.9 (b), the response has a broad bandof frequencies and therefore the estimation is poor. This compared to theunstable process where the response during self-excited vibration isperiodic with the frequency close to one of the system’s natural frequencies.
Also, in this case, the damping is light, which helps to concentrate theenergy in a narrow band.
The procedure based on the stochastic averaging method relates thedamping characteristic to the measured probability density of the amplitude.
The approach is of a non-parametric nature, which makes it convenient for
testing hypotheses of damping mechanisms from measured random
vibration data. However, as practical applications demonstrate, without
knowledge about the input excitation variance it is impossible to correctly
estimate the damping characteristic. Therefore, a much simpler
classification criterion can be formulated.
At this point it is worth to consider the significance of the quantity d1(X).
As represented in Eq. (5.40), the d1(X) term is the non-linear damping
function averaged over one period. If d1(X) > 0 for X > 0, then the system’s
response x(t) given by Eq. (5.23) is purely forced response to a random
excitation; it is not possible for self-excited oscillations to occur. In theabsence of random excitation, the steady-state response x(t) will vanish. If
we assume that d1(X) has at least one nonzero root, then there exists aninterval (X1, X2) such that d1(X) < 0. This means that in the absence ofrandom excitation, W(t) = 0 in Eq. (5.23), the self-excited oscillations are141possible. It is now possible to derive a classification criterion for the
response of Eq. (5.23) in terms of the statistical characteristics of the
response (Dimentberg, 1988). For this purpose, it is convenient to replace
X in Eq. (5.39) by
X = X2

Then the stationary solution to the FPK equation given by Eq. (5.39)
becomes(5.43)w( X ) = C exp( F ( X ))From Eq. (5.43) we can further obtaind ( X)dln( w( X )) = - 1
dX
2w0 X W1

(5.44)

From Eq. (5.44) it can be seen that if, and only if, d1(X) > 0 then w( X ) ismonotonically decreasing for every X > 0. Then Eq. (5.44) provides thenecessary and sufficient criteria for identification and classification of the
response x(t) of the system; if an observed response signal x(t) has amonotonically decreasing w( X ) then it implies that x(t) represents theforced vibration response of the system to the broadband excitation. If on
the other hand w( X ) is increasing then the system may be regarding as aself-excited type.
To verify this criterion for detection of self-excited systems, the response torandom excitation of two different types of systems was simulated; onestable system with damping ratio, ξ1 and natural frequency ω0, and theother a Van der Pol oscillator with the same damping ratio and natural
frequency. The damping function of the Van der Pol oscillator isD( x, x!) = x ( x2 - a ) x!; x , a > 0(5.45)

The constant α was selected to obtain oscillations of amplitude X0 in theabsence of input excitation. The stationary probability density function for
the Van der Pol oscillator is given by (Zhang, 1994)æörp( x, x! ) = C exp ç a r (w02 x 2 + x! 2 )2 - 2 (w02 x 2 + x! 2 )2 ÷
8w0
è
ø
142

(5.46)

where r = ξ/S0, S0 is the spectral density of the white noise excitation. Fig.
5.11 illustrates a typical representation of the probability density functionof a Van der Pol oscillator.

Fig. 5.11 P.d.f. of a Van der Pol oscillator
In the absence of random excitation all trajectories are "attracted" to thestable limit cycle (attractor). With zero mean white noise disturbance, thesteady-state trajectories will fluctuate around the "attractor" (Zhang, 1994).
The result of application of the classification criterion (5.44) in case of astable system is presented in Fig. 5.12. It shows that in case of pure random
excitation the probability of amplitude square monotonically decreases tozero.

Fig. 5.12 Identification of the system’s response for a stable case143

Next, detection of self-excite oscillation is demonstrated in Fig. 5.13,
where the application of the classification criterion (5.44) results in a
deviation from monotonicity of the p.d.f. of square amplitude.

Fig. 5.13 Detection of self-exited oscillations
As illustrated in Fig. 5.13, the function ln( w( X ) ) is monotonicallyincreasing and thus correctly detecting a self-excited dynamic system.
More importantly, in the case of machining system (Fig. 5.14) the criterion(5.44) accurately identifies the stable process.

Fig. 5.14 Identification of stable machining system

5.6 Nonparametric Models Based on Energy Method
The problem of discrimination between random forced vibration and self144

excited oscillations is a key issue in the evaluation of the system stability
boundary during operational conditions. This is the boundary between
forced vibration and self-excited vibration (chatter). The response of a
dynamic system to a broadband random excitation will be a nonzero
steady-state signal both in stable and in unstable states. In case of chattervibration, the excitation persists even in the absence of the random
excitation. The discrimination between self-excited and forced vibrations ofa dynamical system in operational conditions is also important for selecting
the type and strategy of control that may be implemented to reduce orcancel the vibration.
The nonparametric identification method presented in this section follows
the approach developed by Roberts (1986), Krenk (1999) and Rüding
(2001). The approach is implemented in three stages for generation of the
response of a machining system using numerical simulation. At the firststage, a Gaussian white noise process is created. At second stage, theequation of motion is integrated with a suitably chosen time step, using a
Runge-Kutta algorithm of 4th order that enables accurate response histories
to be obtained. Finally, the response data are processed appropriately toestimate the system’s parameters.
In Eq. (5.23) if it is assumed that the state space variables X and X! enter theequation only through the total energy function E, then the damping is afunction of the energy only. The total energy is given byE=

1 !2
X +U (X )2(5.47)whereX

U ( X ) = ò u (x )dx0

In Eq. (5.47) the right-hand side represents the sum of the kinetic energy(first term) and the potential energy (second term). With a change ofvariables, X1 = X and X2 = X! , Eq. (5.23) is transformed into a set of two
145

stochastic differential equations. Under the assumption of Gaussian white
noise excitation, the state space vector (X1, X2) represents a Markov
process and the probability density is the solution of FPK equation, Eq.
(5.27) where p(x1, x2; t) is the joint probability distribution of the state
space vector (x1, x2). The initial conditions are given in the form: p(x1, x2;t0) = δ(x1 - x10) δ(x2 - x20) for x1(t0) = 0 and x2(t0) = x20. Different boundary
conditions are possible such as reflective boundary, absorbing boundaryand periodic boundary as discussed by Risken (1995).
The solution to Eq. (5.27) is given in the form described by Eq. (5.28).
Equation (5.29) shows that the damping function D(E) can be obtained
from the derivative of the damping potential d1(E). The damping potential
is computed from Eq. (5.27) after the join probability p(x1, x2) is evaluated.
Because the probability density distribution of the energy envelope process,
E(t), can be estimated from the system’s response, it is required to relate
the distribution p(E) to the distribution p(x1, x2).

5.6.1 Stiffness Estimation
The estimation of system stiffness from the stochastic response obtained
from Eq. (5.23) follows the Rüding approach (2001). The response X is
calculated at zero level where the energy is kinetic energy and at extremes
where the energy is potential energy. A nonlinear system with linear-cubic
stiffness was used to generate a stochastic response. The damping ratio in
the system is low, ξ = 0.1%. The results of stiffness identification are
illustrated in Fig. 5.15. Each blue dot represents a sample (1/2 x! 2 , X2). The
values of the potential energy U(x), the target values, calculated from Eq.
(5.46) are represented by the red line and the potential energy U(x) for anequivalent linear system is represented by the green dashed line. There are
two important observations: (i) the points are centred on the target line,
which represents the analytical expression, and (ii) the departure fromlinearity is apparent.
146

Fig. 5.15 Kinetic 1/2 and potential U(x) energy for a light damped system,
0.1% and linear-cubic stiffness
By increasing the damping to 1% the level of scatter around the target
value increases as illustrated in Fig. 5.16. This is due to the variation of theenergy from one period to another. As damping tends to zero, the energy
will also vary infinitely slow, and the points will be virtually located on thetarget line. Therefore, the distance between a point and the line represents
the energy variation during half of a period, i.e. the time between samplinga kinetic energy value and a potential energy value (or between a zero andan extreme value).
By examining the plots in Figs. 5.15 and 5.16, it is reasonable to accept thatan estimate of U(x) can be extracted from the dot cloud. The procedure isbased on dividing the energy plane in zones of equal energy. This approachis illustrated in Fig. 5.17. As the potential function U(x) is unknown at the
beginning of the procedure, the potential energy is approximated to thelinear value. After that, the samples (1/2 x! 2 , x2) are averaged in each zone
and one average value is calculated for each zone, as illustrated by thegreen dots in Figure 5.17. The estimating procedure shows excellent results,
but increasing deviations from the target is apparent at higher energy levels.
147

This is because of the smaller amount of samples at a higher energy level.
Eliminating the samples above level 2 will considerably improve theestimation. The next step is to fit a polynomial to the estimates to computethe potential energy U(x). Then from Eq. (5.47) the stiffness function may
be calculated.

Fig. 5.16 Kinetic 1/2 and potential U(x) energy for a heavier damped system, 1%

Fig. 5.17 Estimation of U(x) in zones of equal energy. The dots represent
the average U(x) in each energy zone

5.6.2 Damping Estimation
Damping estimation is treated in two steps. The first step starts from thedamping potential. The equation of motion is represented by Eqs. (5.23)
and (5.27), and can be rewritten as148æ p ! ( x, x! ö
d1 ( E )= - ln ç x , x
÷
p S0
C
è
ø

(5.48)

where d1(E) is the damping potential as described by Eq. (5.27).
By estimating the probability density function from the system’s response,the damping potential can be calculated. This function, for a linear system,
is represented in Fig. 5.18. By fitting a linear polynomial to theexperimental data, the ratio between damping potential and excitation
intensity, S0 can be calculated.

(a)

(b)

Fig. 5.18 Estimation of damping potential (a) linear system and linear
polynomial fitting, (b) nonlinear damping system, dry friction, and cubic
polynomial fitting
149

In the second step, the equivalent damping function, heq, in Eq. (5.27) isestimated from the covariance function at various energy levels according
to

heq =

æ X ö
4
ln ç - 1 ÷c( E )T ( E ) è X 2 ø(5.49)

where c(E) is the participation factor in the covariance function, T(E) thenatural period and X1 and X2 are the extreme values of the covariancefunction in the first period.
¥

Rx! (t | E ) = E exp ( - heqt ) å c j ( E ) 2 cos( jwt )1
2(5.50)

j =1

For a linear system the covariance functions at different energy levels areshown in Fig. 5.19. Having calculated d1(E)/S0 and heq the excitation
intensity S0 can be estimated. For each energy level the extreme values(shown in Fig. 5.19 at X1 for potential energy, and at X2 for kinetic energy)
are extracted from the corresponding covariance functions and theequivalent damping computed according to Eq. (5.49). The calculated
values are represented by dots in Fig. 5.20. As the system is linear thedamping is independent of energy level and therefore constant. It can also
be noticed in Fig. 5.19 that the natural period, and therefore the natural
frequency, are constant for different energy levels. Each curve in Fig. 5.19
represents an energy level corresponding to the dots in Fig. 5.20.

Fig. 5.19 Covariance function estimated at different energy levels
150Fig. 5.20 Estimation of the equivalent damping function, heq, frommeasured autocovariance functions
After plotting the equivalent damping values, a line is fitted to represent theconstant damping. As the energy level is increased, a larger deviation from theanalytical value is apparent.

5.6.3 Identification of Machining System ODPs
The procedure developed above will be applied for the identification of thecharacteristics of the machining system. Steel bars (C < 0.20%) with a lengthof 1200 mm and an initial diameter of 42 mm were machined in longitudinal
turning between tailstock and chuck at a cutting speed of vc= 180 m/min, a
feed rate of fr = 0.3mm/rev and a variable depth of cut (0.5 – 3mm). Cemented
carbide inserts with 1.2mm nose radius were used in all experiments. The
vibration signals were measured by a pre-polarized microphone and sample at
12.8 kHz sampling rate. In Fig. 5.21 an example of stable machining ispresented, while Fig. 5.22 illustrates an unstable machining process.

Fig.5.21 Stable machining (D = 38 mm)
151

Fig. 5.22 Unstable machining, chatter

Fig.5.23 Unstable process, as the tool approaches the centre of the bar
heavy vibration sets on (measured sound pressure)Coming down to diameters below 34 mm, as the tool approaches the centreof the bar, chatter vibration is generated. Fig. 5.23 illustrates the time signal
showing first a stable process, then the chatter in the middle of the bar, thenas the tool approaches the chuck, the system recovers stability. In Fig. 5.24,
the unstable and stable signals are represented in frequency domain.

152

Fig. 5.24 The stable and unstable signals in frequency domain

5.6.4 Operational Stiffness Identification
Following the procedure described above, the overall stiffness of themachining system is estimated. For a stable process, Fig. 5.25 illustrates thepotential function U(x). The function shows a soft nonlinear characteristic.
After fitting to a quadratic polynomial function, the stiffness function canbe extracted from Eq. (5.47). The blue line represents the theoretical linear
system at the system’s natural frequency.

Fig. 5.25 Stiffness estimation - stable machining
153

The estimation of the overall stiffness of the machining system entering
unstable behaviour is presented in Fig. 5.26. The potential function U(x)and consequently the stiffness function show a hard nonlinear behaviour
which is characteristic for self-excited vibrations. The deviation from anonlinear behaviour is apparent by comparing this to the linear
characteristic shown by the blue dotted line. As these results show, if thestiffness of a machining system enters the inelastic range of the material, or
the process becomes nonlinear, or a combination of both, and the degreeof-nonlinearity is large, using a linearization method for the nonlinear
stiffness may yield large errors in response estimation.

Fig. 5.26 Stiffness estimation - unstable machining

5.6.5 Operational Damping Identification
Following the approach described above, the damping potential andequivalent damping are estimated from the system’s response both in stable154

and unstable conditions. However, the results are presented only for theunstable machining system.

Fig. 5.27 Estimation and fitting of damping potential for unstablemachining

Fig. 5.28 Covariance function calculated for various energy levels
The estimation procedure starts with computation of the damping potential
d1(E) from the experimental computation of the probability density
function at different levels of energy. From Eq. (5.48), the function d1(E) is
computed and then fitted to a polynomial. As in the case of stiffness, the155damping in an unstable machining system shows a nonlinear behaviour anda cubic polynomial is then employed. In the second step, the covariancefunction is calculated for gradually increasing levels of energy. Some ofthese functions are represented in Fig. 5.28. It is worth noting that, as thestiffness is nonlinear, the natural period depends on the energy level. The
same can be stated for the equivalent damping. A modified form of Eq.
(5.49) is used to take into account the changes in the natural period. The
total energy is calculated as a sum of the kinetic and potential energies. The
kinetic energy is extracted from the derivative of the response while
potential energy from an iterative procedure applied to the identified
potential function U(x).

Fig. 5.29 Computation of equivalent damping function

5.7 Parametric Models for Machining System
Identification
In section 5.5, non-parametric identification procedures have been used toestimate the ODPs from the response of the machining system in stable andunstable conditions. The main benefit of this approach is that the estimateddynamic parameters are extracted in operational conditions directly from156

the interaction between the elastic structure and the cutting process.
Knowledge of ODP opens new opportunities for optimising the machining
system. Strategies for improving, if necessary, the system may bestraightforwardly implemented by comparing different solutions. Nonparametric identification has, however, the limitation that it requires longsample sequences. In addition, as the non-parametric models developed
herein are based on SDOF system, special treatment of data is required.
Another limitation is that this approach requires knowledge of the response
probability distribution.
In this section, the identification technique based on parametric models ispresented. Parametric models can be applied to any numbers of DOF and intheir recursive implementation can take into account the nonlinear nature of
the system. A parametric model is a special class of representation of asystem, where the input in the model is driven by white noise processes andthe model is described by rational system functions, including
autoregressive (AR) (Burg, least square, Yule Walker, geometric lattice,
instrumental variable), ARX (autoregressive with eXogeneous variables,iv4), moving average (MA), autoregressive-moving average (ARMA), Box
Jenkins, Output Error models (Box [1970], Pamdit [1983], and Söderström[2001]). The process output of this class of models has a power spectral
density (PSD) that is entirely described in terms of model parameters andthe variance of the white noise process.
Parametric models described in this section have their origin in the Markov
representation of stochastic processes. The problem of identifiability of aparametric autoregressive moving average (ARMA) process is discussed in
the opening of this section and a complete solution is obtained by using theMarkovian representation of the process.

5.7.1 Gauss-Markov Sequence
A sequence N(k) is called a purely random sequence if(5.51)

157

for all k, where f[N(k)] is the probability density of the sequence N(k),k=0,1,2, ..., n. Let X(k) be a random sequence generated by a linear
difference equation(5.52)
where a(k) is a known deterministic sequence and N(k) is a random
sequence. The conditional density of the random sequence(5.53)
In other words, the future state of the process X(k+1) depends only on thepresent state of the process X(k) and the random sequence N(k). Hence,
X(k) is a Markov sequence. However, a sequence X(k+1) generated by asecond order difference equation depends on present and past. For example,
consider a second-order difference equation(5.54)

where a1(k) and a2(k) are known deterministic functions and N(k) is arandom sequence.
Let(5.55)The second difference equation can now be expressed as a vector matrix
equation(5.56)

158

X(k) is of dimension 2 and is a Markov vector sequence. In this way, a nthorder difference equation can be reduced to a vector matrix equation ofdimension n. Therefore, a sequence generated by a nth order linear
difference equation is a Markov sequence. The ARMA model will generatea Markov vector sequence. It can be noticed that while the vector sequenceis Markov, the components of the vector are not Markov. A Markov
sequence is called a Gauss-Markov sequence if p[X(k)] and p[X(k+1)|X(k)]are Gaussian probability density functions for all k.
Let’s now consider a general representation. When a discrete-time system
is time-invariant, the state space representation of a system is given as(Akaike, 1974)

vn +1 = Avn + Bun +1(5.57)

yn = Cvn

Where n denotes the time and un is a q x 1 vector of the input to the system,
yn is a r x 1 vector of the output and is vn a r x 1 vector of the state. A, B
and C are and C are respectively p x p, p x q and p x r matrices.
The Markovian representation is a stochastic analogue of (5.57) and isgiven byvn+1 = Avn + Bzn +1(5.58)

yn = Cvn + wn

where wn is uncorrelated with vn. In Eq. (5.58) zn+1 is the innovation of the
input un at time n+1.
If the observations {yt} are from a stationary scalar ARMA(p, q) process
then
p

q

j =1

j =1

yt = åf j yt - j + zt + åq j zt - j(5.59)

Arranging the AR coefficients as a companion matrix (φj = 0 for j > p)159

æ f1 f2
ç
ç1 0
F =ç0 1
ç
ç!
ç0 #
è

... fds -1 fds ö
÷
...
0 0 ÷
...
0 0 ÷
÷
" ! ! ÷0
1 0 ÷ø

(5.60)

Then we can write (θj = 0 for j > q)

X t = F X t -1 + V zt , V = ( w0 , w1 ,..., wq ) 'yt = (1q1 ...q ds -1 ) X t(5.61)

wi's are the impulse response functions of the time-invariant linear systemdefined by (5.59) with the input zn and the output yn. This result shows that
an ARMA process always has a Markovian representation.

5.7.2 Parametric ARMA Models
A serious limitation in practical validation of stability diagrams originatesfrom the lack of a rigorous criterion to unambiguously detect the stability
limit, i.e., impending chatter. In other words, formulation of adiscrimination criterion to distinguish between forced and chattervibrations will be much more useful from both a theoretical and, especially,a practical point of view. Many of the methods used to analyse, control andoptimise machining systems are based on off-line procedures or test
environments that do not represent the actual machining conditions.
The objective of the parametric modelling is to infer a model of the processfrom the recorded data. In other words, parametric modelling represents
mapping from the data space to the model parametric space. The procedureby which a model is inferred from measured data is called identification.
More precisely, parametric identification attempts to find the parameters ina linear model that will give the best fit to the data. The identification of astationary time series as the output of a dynamic system whose input iswhite noise a(t), can be carried out in several ways. One way is to use the160parsimonious parameterization, which is employing ARMA(p, q)
representation.
Given a time series of data yt, the ARMA model is a tool for analysing andpredicting future values in this series. The model consists of two parts, anautoregressive (AR) part and a moving average (MA) part. The combinedmodel is usually then referred to as the ARMA(p,q) model were p is theorder of the autoregressive part and q is the order of the moving average
part.
Sensors:- Microphone- Accelerometer
Spindle idle
AR-filter
ARMA
models

Machining

Structure model

Natural frequencies fn

Coefficients

Damping ratio ξnFig. 5.30 ARMA model identification scheme (Archenti, 2008)
The model for ARMA process can be expressed as (Brockwell, 2002),
Y ( z ) = H ( z )U ( z )(5.62)

where Y(z), U(z) and H(z) are the z-transforms (the z-transform is thediscrete-time counterpart to the Laplace transform for continuous-timesystems) of the output sequence, input sequence and the system impulse
response (transfer function), respectively, and161

H (z ) =c 0 + c1z -1 + c 2 z -2 + ! + c q z -q

(5.63)

1 - a1 - a1z -1 - a2 z - 2 !a p z - p

where the bi and ai coefficients of the polynomials of the MA part and AR
part, respectively.
For a second order system with impulse response function given byequationh(t ) = Ae -at sin(bt + f )(5.64)
the ci and ai parameters can be related to physical parameters, A(amplitude), α (damping), β (frequency)
H (z ) =A sin f + Ae -aDt sin(bDt - f )z -1

1 - 2e - aDt cos(bDt )z -1 + e - 2aDt z - 2(5.65)

where Δt is the sampling interval in seconds.

5.7.3 Power Spectral Density of ARMA Models
Estimation of the power spectral density (PSD) of sampled data containing
stochastic components is traditionally performed with the help of FFT.
There are however, a number of problems related to spectral analysis basedon the Fourier approach.
The major limitation of FFT spectral analysis is the lack of ability todiscriminate the spectral components of two signals with the frequency
close to each other. This becomes a big problem when attempting toanalyse short time series because the frequency resolution is approximately
the inverse of the available data time length. Another drawback is the needfor curve-fitting for modal parameter extraction.
The discrete-time counterpart of the Fourier transform is the discrete
Fourier transform (DFT). The classical PSD based on DFT is given by
Cooley (1965)
1 N -1f yy =
y n e - j 2pmn / N
N n =02

å(5.66)

162

where N is the number of samples. The power spectrum for a processdescribed by Eq. (5.66) is obtained by evaluating the impulse response
function around the unit circle,z^-1 = exp(-j2πfΔt)
Another problem with spectral analysis based on the Fourier approach isspectral leakage (Marple, 1987). The frequencies represented by lines inthe spectrum, are harmonics of the fundamental frequency. Spectral
leakage causes energy from distinct spectral features to leak into nearby
frequency channels. This gives rise to false components in the frequency
spectrum of the signal.
The power spectrum for a process described by Eq. (5.66) is obtained byevaluating the impulse response function around the unit circle, z-1 = exp(j2πfΔt). Using only AR parameters, the PSD function is determined asfollows, equations 2 Dtf AR =1+m

å am e2(5.67)- j 2pfnDt

n =1

The PSD can therefore be determined solely from a knowledge of thecoefficient a1, a2, …, ap, and the variance, σ2.
In the on-line identification procedure, the estimation of physicalparameters, chatter frequency, ωch and damping ratio ξ can be used forcontrol of dynamic stability. It is important to stress that in the context of
stochastic modelling, the estimated physical parameters are meaningful
only from a statistical point of view, i.e. they are properly significant withina certain confidence interval.
The transmutation process from the ARMA parameter domain to the ωch - ξdomain gives the advantage of robust chatter identification criteria.
Theoretically, dynamic stability can be defined in terms of negative
damping. A system is dynamically stable if the damping is positive andunstable when damping becomes negative. In machining, as we areinterested in avoiding instabilities like chatter, when damping starts to163

decrease towards zero it is a proof that the system is approaching stability
threshold. Therefore, monitoring damping in an on-line identification
scheme can give a good indication of the dynamic state of the system.
The motion of an n degree-of-freedom system can be represented by asecond-order differential system of equations
Mx!!(t ) + Cx! (t ) + Kx (t ) = f (t )(5.68)

If the model of an ARMA process is represented by Eq. (5.68) then the ARparameters are related to M, C and K through the characteristic equation of
the form,2m

å

a i µ 2m - i =

t =0

m

æ

ö

Õ çè µ - expìíîx j wnj Dt + iwnj 1 - x 2j Dt üýþ ÷ø ´ !
i =1

m

æ
ì
2 üö
!´
ç µ - expí- x j w nj Dt + iw nj 1 - x j Dt ý ÷îþøi =1 è

(5.69)

Õ

where x jw nj and ±iwnj i - x j2 , j =1, … n are the eigenvalues of the system ofEq. (5.68).
The structural parameters ξi and ωnj, j = 1, … 2n, may be determined fromthe AR parameters estimates equation(5.70)

5.7.4 Operational Dynamic Parameters (ODP)
In machining, real-time systems for detecting self-excited vibrations are
based on vibration amplitude monitoring. However, the vibration amplitude
criterion is not consistent with the nature of the vibration source. A highamplitude vibration can be the result of a stable system working close toresonance, or in condition of tougher cutting parameters. Likewise, selfexcited vibrations need a very low energy level to be excited and are
therefore difficult to detect before they are fully developed. A non164

conservative mechanical system with positive damping is said to be
dynamically stable, whereas one with negative damping is said to beunstable. This gives a robust criterion for discrimination between forced
and self-excited vibrations, which is not related to the vibration amplitude
criteria.
The concept of ODP is introduced due to the fact that the machining system
is a closed-loop system between the cutting process dynamics and themachine tool elastic structure. The structure damping of the machine toolelastic structure and the dynamic cutting process damping cannot be
separated from a response measure point of view. ODPs include thecontribution of the structural vibration modes and process vibration modes
generated during a machining system operation. The ODPs are theoperational damping ratio (ODR) and operational frequency (OF). The
ODR, ξop, is the overall damping of the machine tool structure ξmod (modal
damping) and the dynamic cutting process ξcp (process damping)

xop = xmod + xcp(5.71)

ξop is used in this study as the index of the chatter stability boundary.
In Eq. (5.68), M, C and K represent n x n mass, damping and stiffness
matrices respectively; f(t) is a vector of external excitation. Matrices C andK contain both structural and process damping and stiffness respectively.
x(t ), x! (t ), !x!(t ) are n ϕx 1 vectors of displacement, velocity and acceleration of
the n degrees of freedom. The eigenvalue problem associated with Eq.
(5.68) is given by( l M + l C + K ) Φ = 0, i = 1, 2,..., 2n2ii(5.72)iwhere λi and ϕi are the eigenvalues and eigenvectors respectively.
Let xj(kΔT), k = 0, 1,2 … be the discrete samples of the displacement of
the j-mass. ΔT is the sampling interval. Then the observations xj(kΔT) canbe represented by an ARMA modelpq

å a x(t - i) =å c e(t - i), a = 1
i =0

i

i =0

i

(5.73)

0165

Model parameter estimation for a given model structure seeks theparameter vector θ(t) [a1, a2 … ap, c1, c2, … cq] and the residual varianceσe2(t) at every time t = 1, 2, … n.
The characteristic equation of (5.74) can be writtenpnå a x(t - i) =Õ (µ - µ )(µ - µ )
i =0

y

i =1

j

*
j

(5.74)

whereµ j1,2 = exp(-(wop ) j (xop ) j DT ± i(wop ) j 1 - x j2 DT )(5.75)

From equation (5.75), the ODP, operational damping, (ξop)j and frequency(ωop)j are calculated at each time t and for each eigenvalue.
(xop ) j =ln(µ j µ *j )éæ µ - µ *j öù÷ln(µ j µ ) - 4 êtan-1 ç j
ç µ + µ * ÷ú
êë
j
j
è
øúû

2

(5.76)* 2jé -1 æ µ j - µ *j öù1* 2÷(wop ) j = ln(µ j µ j ) - 4 êtan çç µ + µ * ÷ú
2DT
ú
j øû
è j
ëê

2

(5.77)

5.7.5 Stability Analysis
A series of experiments were carried out with the scope of demonstrating theidentification procedure based on a parametric model approach. Both off-line
and on-line identification methods were used to find modal parametersdescribing the static and dynamic stability of the system.
Figure 5.31 displays the experimental set-up of the test and data acquisition
system. The experiments were carried out in a conventional turning machine
equipped with a chuck, revolver and tailstock. To acquire data for machining
analysis, the acoustic signal from the machining process was recorded.
Experimental modal analysis was used for analysing the interaction between
machine tool elastic structure and workpiece. For further investigation of thechuck-workpice-tailstock system, an FEM model was developed. Natural
frequencies and mode shapes were computed and used for analysis.
166

A long slender workpiece of steel was used (length 585 mm), and preparedwith a series of tapered segments (diameter 43 – 49 mm). For robustness andsensitivity analysis purpose, three different workpieces all with the samematerial properties and geometries were used, Table 5.1.

(a)

(b)

(c)

Fig. 5.31 a) Experimental set-up (b) unstable state only the workpiecefrequency mode is excited (c). In stable state, several modes shapes are excited
Workpiece

Tool, Insert Sandvik

Feed[mm/rev]

Steel 42CrMoS4, C = 0.38%
Length 585, Ø43-49 [mm]CNMG120412-PM
Grade 4025

0.25

Cutting
speed[m/min]
150
155
161

Table 5.1 Experimental conditions
From Experimental Modal Analysis (EMA), the following modalparameters were obtained167

Natural frequency [Hz]
26.35
59.76
114.35198.21
311.14

Damping ratio [%]
0.15
0.21
0.23
0.26
0.11

5.7.6 ARMA Model-based Off-line Identification
The suitable order for the models was chosen after a number of trials, both on
stable and unstable processes, in order to find the optimum order for each of thetime series. The reliability of the identification approach is reflected by either
plotting the residual curves for each model or by evaluating the standard
deviation for each parameter. The ARMA parameters, estimated from theappropriate models along with their standard deviations are calculated and theODPs are then extracted in Table 5.2.
Stable region

Unstable region

Roots

0.1148±0.8124i

0.2875±0.8856i

Frequency [Hz]

256

270

Damping ratio [%]

5.1

0.011

Table 5.2 ODPs for stable and unstable process, respectively

5.7.7 Parametric Modelling for a Milling Operation
The tests were performed in full immersion (slotting) with an end tool
holder, see Fig. 5.32 (b). A tool holder type R390-020A20-11M with three
inserts and diameter D = 20mm was selected. The overhang of the tollholder was l = 60 mm, i.e., three times the diameter. At this ratio l/D therisk of tool holder chatter is reduced. The tool holder is designed for three
inserts. Inserts for steel type R390- 11 T3 12E-PM 1030 were employed.
The nose radius was 1.2 mm. The tool diameter selected was larger enoughnot to influence the process dynamics by generating chatter.
Workpieces with dimensions 70 x 70 x 70 mm were adapted to the palletsystem size and pre-prepared for machining, see Fig. 5.32 (a). Three slots
were machined at every height level on the workpiece.
168

Chemical Composition Element of
workpiece material
C
Mn
P
S
Cr
Cu
N

(a)

min/max<=0.20(1.00 - 1.60) 773)<=0.050<=0.050<=0.30<=0.40<=0.009(b)(c)
Fig. 5.32 Workpiece clamped on a pallet system (a), full immersion(slotting) (b), comparison between vibration measured at the top of theworkpiece and at the bottom of the workpiece. Red curve: vibration
amplitude at the top, Green curve: vibration amplitude at the bottom (c)169

Fig. 5.33 Increasing the axial depth of cut until reaching the stability
threshold. Left: time records with various
axial depth of cut; blue 1mm, green – 1.5 mm; red - 2 mm; Right: Frequency domain autospectrum; Red curve – 2mm and Blue curve - 2.5 mm
The first mode of vibration of the pallet system is a bending mode, whichmeans that for the same magnitude of the dynamic force, the vibration
amplitude at the top of the workpiece will be higher than at the bottom. For
the second mode of vibration, which is a torsion mode, the vibration
amplitude at the extremities of the workpiece is larger than at the centre. In
Fig. 5.32 (c), the vibration amplitude was measured during machining at
the top of the workpiece (red curve) and during machining at the bottom of
the workpiece (green curve).

5.7.8 Evaluation of the Operational Modal Parameters
The EMA was used to determine the modal parameters in off – operation
conditions. This means that the machine tool mechanical structure wasseparated from the cutting process. In order to assess the interaction
between the machine tool structure and the cutting process, the stochastic
parametric modelling was employed. The structure is assumed to be excitedby a random force and the output signal measured is identified and170modelled by parametric models, so called ARMA models. As the interest in
this report is to evaluate and compare pallet systems, the analysis will refer
only to the interaction between pallet system’s structure and cutting process.
At higher excitation levels, not only is the pallet system brought into
vibration but even the cutting tool holder (see Fig. 5.34). Also, in thewaterfall diagram, the tooth passing frequency can be identified.

Tooth passing
frequency

Cutting tool
natural frequency

Fig. 5.34 Waterfall diagram at feed rate fr = 0.12 mm/tooth and axial depthof cut docax = 2.5 mm
The procedure for evaluation of the ODP is:
1. Identify the output signals: acoustic noise, acceleration in x-, y- andz-directions
2. Build ARMA synthetic models
3. Transform ARMA models to physical models
4. Estimate frequency and damping ratios from physical models.
An example is illustrated below. The time record from machining with
Macro pallet system with depth of cut, docax = 2.8mm, cutting speed, vc =270 m/min and feed rate, fr = 0.14 mm/tooth is represented in Fig. 5.35.
171

Fig. 5.35 Time record of the machining on pallet system
The acoustic signal represented in Fig. 5.34 is then identified by an ARMA
model. An ARMA model is a parametric model consisting in an AR partand an MA part:
Discrete-time ARMA model: A(z)y(t) = C(z)e(t)AR partOutput signal

MA partInput noise

Parameterization:
Polynomial orders: AR na=12 MA nc=11
A(z) = 1 - 0.8087 (+/- 0.2946) z-1 - 0.1551 (+/- 0.444) z-2 + 0.01815 (+/- 0.2205) z-3 + 0.1578 (+/0.07877) z-4 + 0.01117 (+/- 0.09527) z-5 + 0.2581 (+/- 0.07606) z-6 - 0.06554 (+/- 0.08284) z-7 0.461 (+/- 0.07969) z-8 + 0.4674 (+/- 0.1457) z-9 - 0.1881 (+/- 0.2604) z-10 - 0.04093 (+/- 0.2113)
z-11 + 0.1853 (+/- 0.08648) z-12

C(z) = 1 - 0.4828 (+/- 0.2952) z-1 - 0.1279 (+/- 0.3529) z-2 + 0.1373 (+/- 0.1566) z-3- 0.04195 (+/- 0.0483) z-4 + 0.01585 (+/- 0.05173) z-5 + 0.1642 (+/- 0.05134) z^-6 + 0.1326 (+/- 0.04548) z-7 - 0.4016 (+/- 0.05148) z-8 + 0.3727 (+/0.1616) z^-9 +0.07662 (+/- 0.2277) z^-10 - 0.07114 (+/- 0.1135) z-11

Fit to estimation data: 23.97% (prediction focus)
FPE: 0.1948, MSE: 0.191

172

In the present case, the model was identified by an AR of order 12 and an
MA of order 11. The ARMA power spectrum is illustrated in Fig. 5.36.

Tooth passing frequency

1st frequency mode of the pallet system

1st frequency mode of the tool holdernd

2 frequency mode of the pallet system

Fig. 5.36 ARMA Power spectrum of the output signal
From the ARMA parameters, the physical model can be built. The
operational frequencies and damping are then estimated. The damping andfrequency values are statistical estimates of the interaction between
mechanical structure and process. The intermittent cutting force exciting
the structure at tooth passing frequency generate a forced vibration. The
closer the tooth passing frequency to the structure frequency modes, thehigher are the amplitude of vibration. Also, as the excitation is the cutting
force, it is proportional to the depth of cut and the chip thickness. The
higher one of these parameters is, the higher the amplitude of vibration.
In a machining process on the pallet system with docax = 1 to 3 mm, feed
rate fr = 0.14 mm/tooth, cutting speed vc = 270 m/min, the dominant
dynamic source of excitation is the tooth passing frequency. This is
illustrated in Fig. 5.37, where the tooth passing frequency is indicated. The
ODPs are presented in Table 5.3.

173

Fig. 5.37 ARMA power spectrum for a machining with docax = 3 mm, fr =0.14 mm/tooth, vc = 270 m/min (left) and waterfall plot (right)
Frequency (Hz) Damping (%)
Mode 1 (tooth passing)
380
NA
nd
Mode 2 (2 pallet system) 2224
0.9
rdMode 3 (3 pallet system) 3676
1.6
Mode 4 (tool holder)
4916
0.72
Table 5.3 ODPs in stable machining (non-filtered signal)
Since the tooth passing frequency is outside the first mode range, the forced
vibration caused by the excitation at this frequency is controlled by thestatic stiffness of the structural system. In addition, as the modal damping
is defined only in the region around the resonance, this damping cannot becalculated for the excitation force at tooth passing frequency. In stablemachining on pallet system, the only excitation that exists is at the tooth
passing frequency.
In order to evaluate the operational modal parameters of the pallet system,
the signal is filtered to eliminate the tooth passing frequency. The signal
processing leads to the estimation of a new spectral function represented inFig. 5.38. The diagram now reveals the first frequency mode of the palletsystem. In Table 5.4 the modal parameters are calculated. Parameters were
calculated for a machining operation on pallet system with docax = 3 mm

174

and the feed rate fr = 0.14 mm/tooth. The cutting speed is vc = 270 m/min.
In this case the material removal took place on the top of the workpiece.

Fig. 5.38 ARMA spectrum of the signal after removing thetooth passing frequency
Frequency (Hz)Damping (%)

Mode 1 (1st pallet system mode)

1079

4.78

Mode 2 (tool holder mode)

4778

1.77

Table 5.4 ODPs for machining at top of the workpiece
By continuing the material removing process, as the tool cuts down on theworkpiece, the vibration at the first mode is decreasing. However, thevibration at the tool frequency increases mainly due to tool wear. Thissituation is shown in Fig. 5.39 where the ARMA power spectrum recorded
from machining at top level is compared to the machining at the bottom
level of the workpiece. While the damping at the first mode of the pallet istwice at the bottom than at the top of the workpiece, the damping at the toolmode is reducing drastically. One of the reasons is machining at a relatively
high cutting speed, 270 m/min, which caused thermal damage of the cutting
teeth. The tool wear was measured after 14min. machining.
175Rake face

Plastic deformation of the cutting edge

Fig. 5.39 Left - Comparison between ARMA power spectrum at top of theworkpiece (red line) and that at the bottom level (blue line). Right – plastic
deformation on the cutting edge

Frequency (Hz)Damping (%)

Mode 1 (1st pallet system mode)

1072

8.67

Mode 2 (tool holder mode)

4985

0.54

Table 5.5 ODP parameters at the bottom of workpiece after 14 minmachining
When increasing the depth of cut above the stability threshold, chatteroccurs at the pallet system first mode. Fig. 5.40 illustrates the ARMA
power spectrum of the signal recorded during machining with feed rate, fr= 0.14 mm/tooth, cutting sped, vc = 286 m/min and depth of cut, docax =3.5 mm.
The depth of cut exceeds the value corresponding to the stability thresholdand high vibration is generated in the form of chatter. This can be clearlyobserved in Fig. 5.40 and in Table 5.6 where the instability at the firstfrequency mode is proved by the very low damping. It can be noticed fromthe auto-spectrum that a very large energy is concentrated at the first mode.
The energy is about three orders of magnitude higher than in stablemachining.
176

st

1 frequency modeof the tool holder
1st frequency modeof the pallet system

Fig. 5.40 Auto-spectrum for unstable cutting conditions
As illustrated in Fig. 5.40 the dominant frequency modes present are thefirst mode of the pallet system and the frequency mode of the tool holder.
In Table 5.7 the operational damping ration is very close to zero, which
describes a genuine unstable process.
Frequency (Hz) Damping (%)
Mode 1 (pallet system)429
0.01
Mode 2 (tool holder)
5078
0.03
Table 5.6 Operational dynamic parameters for unstable cutting conditions

Right

Tool
Left

Feed direction

Fig. 5.41 Chatter generation in slot milling; full immersion operation,
chatter marks
177

5.7.9 Real-time Identification of Machining Systems
By real time identification we define procedures for on-line monitoring of
the process and to infer a model at the same time as the data is collected.
The model is updated at each time instant new data becomes available.
A batch method gives only an average behaviour and cannot monitor timevarying systems. If the model, however, is needed to constantly monitor
and control a system during operational conditions, as for instance a
machining system under cutting action, it is necessary to identify the modelat the same time as new data is acquired (Goethals, 2004). This way themodel is recursively updated at each time instant the new data sample
becomes available. By ‘recursive’ term it is understood that at time t thealgorithm produces a model by processing the data acquired at time t in a
fixed (t-independent) and finite number of arithmetic operations making
use of prior information condensed into a fixed (t-independent) and finite
memory space.
The reason for using real time identification is that the dynamics ofmachining systems change in time. In order to detect the onset of chatterand transition from forced vibration to self-excited vibration, close trackingof the system is required.
Recursive identification methods have the following general features(Ljung, 1983):− they are the main part of adaptive systems where action is taken
based on the most recent model;− they do not occupy much memory space, since not all data are
stored;− they can track time-varying parameters; and− they can be used as a first step in a fault detection algorithm to findout if the system characteristics have changed significantly.
There are two main drawbacks to recursive identification:− the model structure has to be selected before starting the recursive
identification; and178

− the accuracy of recursive methods is not as good as that of off-line
methods.
Some well-known recursive methods are (Ljung, 1981):− recursive prediction-error methods (RPEM);− recursive pseudolinear regression methods (RPLR); and− recursive instrumental-variable methods (RIV).
The model-based identification method used in this chapter is based on therecursive prediction-error method (RPEM) (Söderström, 2001). The model
structure is based on a parametric process where the input to the model isdriven by white noise processes and the model is described by a rational
system function and represented by the recursive autoregressive moving
average (RARMA) model structure. The process output of this model has
the power spectral density (PSD) that is entirely described in terms ofmodel parameters and the variance of the white noise process.

Fig. 5.42 Recursive model-based identification. The algorithm is used toestimate the mathematical model that represents the real operational
machining systemLet, as before, yj(kΔT), k = 0, 1, 2 …n be the discrete samples of themeasurement response of the displacement of the j-mass for a dynamic
system represented by Eq. (5.68). ΔT is the sampling interval. Then the179

observations yj(kΔT) can be represented by an ARMA model described by
Eq. (5.62). The measurement equation is then of the form
Yi = H(kΔt, Xj; Θ) + ek
(5.78)The RARMA method is designed to minimize the criterion
V(θ) = E f(t, θ, ε(t,θ))
where ε(t,θ) is the prediction error associated with the parameter q, and E
is the expectation operator.
!

e (t ,q ) = y (t ) - y (t , q )where y! (t ,q ) is the prediction of y(t) and1 N
Ef (t )
åN ®¥ Nt =1

Ef (t ) = lim

This is done by recursively adjusting the estimates in the negative gradientdirection modified by multiplication with a strictly positive define matrix
R-1. The purpose of RARMA is to recursively identify the parameters Θ in
Eq. (5.79) from the response measurements in the time domain. The
recursive algorithm follows the steps!
!
Q k = Q k -1 + µk Rk-1y k ;Qk -1 e k ;Qk -1!
e k ; Q = yk - yk ; Q(5.79)

Rk = Rk -1 + µk éëy k ;Qk -1y kT;Qk -1 - Rk -1 ùû

where Ψk is the gradient of y.
Té dù
y (t ,q ) = ê - e (t ,q ú which is a column vector, and γ is the gain.
ë dqû

5.7.10 Model Structure Selection and Model Validation
The selection of an appropriate model (from a set of candidate models) torepresent an observed system involves traditionally a series of iterative
steps. The two major steps in this series are (Hannan, 1979):− model order selection; and− checking the selected order validity.

180

This procedure is normally referred to as the model order selection andvalidation procedure and the primary objective, both regarding batch
ARMA modelling or recursive ARMA modelling, is the choice andvalidation of AR(p)-order and MA(q)-order.

5.7.11 Model Structure Selection
Once a set of data has been acquired and the removal of DC level, linear
trend (corrupting the AR spectra estimates) and seasonal components has
been done (assuming stationary properties), appropriate values for the orderp and q have to be chosen (Burnham, 2002). Two common and well-known
criteria for choosing model order are proven by Akaike, Burnham (2011).
The first criterion, the final prediction error (FPE), selects the model orderso that the average error variance for on-step prediction is minimized(Akaike, 1969). The second criterion, Akaike (1973) information criterion(AIC), uses the maximum likelihood approach. A model with a lower valueof FPE and AIC is considered to be a better model. By increasing themodel order, the sum of squares of the residual will decrease, and a better
fit is achieved. In many practical situations, however, it is not advantageousfrom an identification point of view to choose large p and q random. A highmodel order will take longer to compute than a model with low order. This
time constraint is not crucial for batch identification as the modelling isnormally performed off-line.
In real-time recursive schemes for chatter detection, where time-varying
processes are tracked, the computation time becomes crucial. If for instancea vibratory process is tracked in time, the choice of model order is a tradeoff between increased resolution and decreased variance. In such a system,
filtering of data might be helpful when signals vary around a large signal
level, improving computational accuracy by removing frequency bands of
no interest. This leads to a reduction of spectral components and models of
lower order can be used for identification. A high order model will
generally result in a small estimated noise variance. However, the meansquared error of the forecast will depend not only on the noise variance of181

the fitted model but also on estimation errors of the model parameters.
These will be larger for high-order models.
Further, as the vibratory process is assumed to be underdamped, complex
conjugate pairs of roots are expected to be obtained from the response
characteristic equation. Since the characteristic equation is determined by
the AR coefficients, an even order p=2n, of the AR components should beassumed in order to not force real roots. The choice of model order for MA
component should be done q=2n-1 (Pandit, 1983).

5.7.12 Model Validation
Once a model has been chosen as a candidate to represent a system it must
be checked against a validation criterion. There is no standard method to do
this, however, one way is to confront the candidate model with as muchinformation about the true system as possible. This includes a priori
knowledge, experimental data and experience of using the model (Bohlin,1991). From this perspective, some useful techniques that can be used for
discarding models are discussed in Ljung (2006).

5.7.13 On-line Identification – Turning Case
A turning experiment demonstrates the recursive identification of ODPs.
The longitudinal turning operation was carried out in a conventional
turning machine with cutting conditions specified in Fig. 5.43. To acquire
data for machining analysis, the sound from the machining process wasrecorded. A long slender workpiece of steel was used. The suitable order ofthe models was chosen after a number of trials (based on AIC), both on
stable and unstable processes, in order to find the optimum order for eachof the time series. The ARMA(3,2) model parameters, estimated from theappropriate models along with their standard deviations are depicted forstable and unstable machining, respectively, in Table 5.7. As can be noticed,
the RARMA-ODP algorithm identifies two dominant operationalfrequencies and related damping ratios.

182

Material
Workpiece
InsertFeed
Cutting speed

Cutting conditions
Steel grade 0.35-0.40% C, HBS 262.
Length: 585mm
Diameter: 45mm
Sandvik CNMG 120412
0.25 mm/rev
150, 155, 160 m/min

Fig. 5.43 Experimental set up for identification of chatter in turning
AR parameters
MA parameters
Stable machining
a1
a2
a3
b1
b2
0.2472-0.5619-0.7519
0.09649‐0.5029±0.0794±0.0645±0.0127±0.0121±0.0107
Unstable machining
a1
a2
a3
b1
b2
-0.39470.4593-0.9344-0.54640.4476±0.0273±0.0181±0.0262±0.0720±0.0716
Table 5.7 ODP parameters in stable and unstable machining
Figure 5.44 illustrates the results from on‐line modelling of the first andsecond identified modes during the machining of the slender bar. As the
cutting tool approaches the middle section of the bar, the system’s stiffnessand damping reach very low values. The machining system becomes
unstable. As the tool approaches the chuck, the system is gradually
recovering its stable conditions.
183

(a)

(b)

(c)

Fig. 5.44 In‐process identification and modelling of damping and natural
frequency in turning. Sound pressure level measured by a microphone (a),Identified first natural frequency and damping ratio (b), Identified second
natural frequency and damping ratio (c)The sound pressure level was acquired by a microphone. When chatter iscompletely developed, its frequency stabilizes at f1 = 290 Hz for the firstmode, Fig. 5.44 (b). With increased chatter intensity a significant drop in
the damping ratio is apparent, see Table 5.6 and Fig. 5.44 (b). In Fig. 5.44(c), the ODPs for the second mode are tracked.

5.7.14 Discussions and Conclusions
In this chapter the subject of statistical dynamics has been discussed for the
analysis of stability of machining systems. Markov process representation184

in continuous and discrete representation is used in this study for systemparameter identification. In continuous form the Markov process leads to a
FKP equation, which is described by a partial differential equation whose
solution has been demonstrated can be used as a criterion for stability
analysis of systems of random ex citation. In the discrete case the Markov
process has been showed to be generated by an ARMA process. An ARMA
process can also be used to formulate a parameter based criteria for
dynamic stability of machining processes. In sections 5.5 and 5.6 thiscriterion was used for recursive and batch identification.
The parameter estimation through statistical methods is subjected toestimation errors. In estimating of unknown parameters, errors are likely toarise and it may be appropriate to estimate the interval in which theparameter will lie. Some basic techniques for interval estimation can befound. The major limitation with time-domain approaches is the lack of areliable model order determination mechanism.
Comparing batch and recursive estimation methods, recursive methods
usually have lower accuracy due to limited time to compute models.
Another issue concerning recursive estimation methods accuracy is the factthat model order and model structure has to be chosen before identifying
the system characteristics. This selection is traditionally done by using
some well-known criterion. Choosing model order according to such
criterion can in some cases lead to high model orders (depending on
degrees of freedom etc.). This can result in unnecessarily large data
effecting the computation time. Unfortunately, in industrial applications, noprior knowledge of the number of freedom of the system to perform theanalysis is known. In such situations the choice of the sampling frequency
becomes critical.
Furthermore, to reduce bias a more flexible model structure (manyparameters) is needed whilst reduced variance requires decreasing thenumber of parameters. Thus, the selection of a model order is determinedby two conditions: flexibility; and parsimony (a parameterization isparsimonious if it has as few parameters as possible).
185

To summarise, there are several benefits arising from employing stochastic
models for analysis of dynamics of machining systems:− RARMA models are characterised by high frequency resolution andobjective assessment of component significance. This is in contrast
to non-parametric models, such as FFT technique, where thefrequency resolution is approximately the inverse of the available
data time length.
− Estimation of ARMA model parameters can be implemented inrecursive schemes and used for real-time identification of the
dynamics of the join system: machine tool elastic structure – cutting
process.
− Parsimony – relative short data lengths and few parameters in themodel can describe the system behaviour and consequently may be
implemented in fast algorithms.

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191

Chapter 6 Milling Dynamics: Forced Vibration,
Self-excited Vibration, and Period-n Bifurcations
6.1 Introduction
This chapter investigates the three possible milling response types: forced
vibration, self-excited vibration, and period-n bifurcations [1-45]. In
milling, a rotating tool with defined cutting edges is moved relative to aworkpiece in order to remove material and obtain the desired workpiecegeometry and dimensions. The tool is typically mounted in a holder which
is attached to the spindle. The spindle provides the tool’s rotational speed,
torque, and power. Multiple axes are then used to manipulate thetool-holder-spindle relative to the workpiece. As the tool rotates relative tothe workpiece, the instantaneous chip thickness varies periodically. This
gives a periodic forcing function which excites the structural dynamics ofthe tool-holder-spindle-machine and workpiece. The forcing function
exhibits, in general, content at the tooth passing frequency (i.e., the productof the spindle speed and number of cutter teeth) and its integer multiples(or harmonics).
The cutting force causes deflections of the tool and/or workpiece. These
vibrations are imprinted on the workpiece surface as a wavy profile andproduce time-delayed surface regeneration, which occurs from tooth totooth. This provides a feedback mechanism because the instantaneous chip
thickness depends not only on the process geometry, but also on both thecurrent vibration and the surface left by the previous tooth (one tooth
period, or time from one tooth to the next, earlier). The variable chip
thickness governs the cutting force which, in turn, affects subsequent tool
vibrations. The result is the possibility for the three response types.
Forced vibration, or stable machining conditions, is the desired response
because the force level, vibration level, and surface roughness are lower
than for the other two options. When stable machining conditions are192selected, an additional consideration for high quality part manufacture issurface location error, or part geometric errors that occur due to forced
vibrations. Surface location error has been modeled and predicted for stable
milling conditions by several authors [46-61]. In these publications, thedifference between the machined surface location and the commanded
location is measured and/or predicted to determine the influence of (stable)
machining conditions on the error.
Due to the time-delayed surface regeneration, various bifurcations (i.e., theappearance of a qualitatively different solution as a control parameter isvaried) are exhibited. These bifurcations include: 1) secondary Hopf
instability (traditional chatter described by self-excited vibration); and 2)
period-n motions (e.g., a period-2, flip, or period doubling bifurcation).
Operating under secondary Hopf bifurcation conditions is avoided due tothe large tool and/or workpiece motions and cutting forces and,
subsequently, poor workpiece surface finish, reduced part accuracy, andpotential damage to the workpiece-tool-spindle system. Period-nbifurcations are not as common in practice, but are also generally avoided
due to increased surface roughness.
The outline for this chapter follows. To begin, a milling time domain
simulation is described. The simulation is then used for a selected dynamic
system to generate a stability map that separates stable and unstable spindle
speed-axial depth of cut combinations by applying a periodic
sampling-based stability metric. The metric is then extended to incorporate
subharmonic sampling and the stability map is updated to individually
identify stable, secondary Hopf, and period-n bifurcation zones. To
augment the stability map, bifurcation diagrams are then presented todisplay the specific stability behavior versus spindle speed for a fixed axial
depth of cut. To add insight to the bifurcation diagrams, Poincaré maps areincluded, as well as the supporting time dependent displacements. Finally,
surface location error results are presented for both stable and period-2
behaviors. Experimental results are included to validate the time domain
simulation predictions for both the Poincaré maps and surface location
error analysis.

193

Fig. 6.1 Cutting force geometry. The normal and tangential direction
cutting forces, Fn and Ft, are displayed. The fixed 𝑥𝑥 (feed) and 𝑦𝑦
directions, as well as the rotating normal direction, n, are also shown. The
angle f defines the tooth angle. The tool feed is to the right for theclockwise tool rotation and the axial depth is in the z direction.

6.2 Time Domain Simulation
Time domain simulation enables the numerical solution of the time-delay
equations of motion for milling in small time steps [59]. It is well-suited toincorporating the inherent complexities of milling dynamics, including thenonlinearity that occurs if the tooth leaves the cut due to large amplitude
vibrations and complicated tool geometries (incorporating runout, ordifferent radii, of the cutter teeth, non-proportional teeth spacing, andvariable helix). The simulation is based on the regenerative force, dynamic
deflection model described by Smith and Tlusty [26]. As opposed toanalytical stability maps that provide a global picture of the stability
behavior, time domain simulation provides information regarding the local
cutting force and vibration behavior for the selected cutting conditions. The
simulation used in this study proceeds as follows (see Fig. 6.1):
1. the instantaneous chip thickness, h(t), is determined using thevibration of the current and previous teeth at the selected tooth angle

194

2. the cutting force components in the tangential (𝑡𝑡) and normal (𝑛𝑛)directions are calculated using:𝐹𝐹& (𝑡𝑡) = 𝑘𝑘&+ 𝑏𝑏ℎ(𝑡𝑡) + 𝑘𝑘&/ 𝑏𝑏,
𝐹𝐹1 (𝑡𝑡) = 𝑘𝑘1+ 𝑏𝑏ℎ(𝑡𝑡) + 𝑘𝑘1/ 𝑏𝑏, ------(6.1)

where 𝑏𝑏 is the axial depth of cut and the cutting force coefficientsare identified by the subscripts 𝑡𝑡 or 𝑛𝑛 for direction and 𝑐𝑐 or 𝑒𝑒for cutting or edge effect.

3. the force components are used to find the new displacements bynumerical solution of the differential equations of motion in the 𝑥𝑥
and 𝑦𝑦 directions:𝑚𝑚5 𝑥𝑥̈ + 𝑐𝑐5 𝑥𝑥̇ + 𝑘𝑘5 𝑥𝑥 = 𝐹𝐹& (𝑡𝑡) cos f + 𝐹𝐹1 (𝑡𝑡 ) sin f,
𝑚𝑚= 𝑦𝑦̈ + 𝑐𝑐= 𝑦𝑦̇ + 𝑘𝑘= 𝑦𝑦 = 𝐹𝐹& (𝑡𝑡) sin f − 𝐹𝐹1 (𝑡𝑡 ) cos f, ------(6.2)

where 𝑚𝑚 is the modal mass, 𝑐𝑐 is the modal viscous damping
coefficient, and 𝑘𝑘 and the modal stiffness. The subscripts identify
the direction and multiple degrees of freedom in each direction canbe accommodated.
4. the tool rotation angle is incremented and the process is repeated.
The instantaneous chip thickness depends on the nominal, toothangle-dependent chip thickness, the current vibration in the direction
normal to the surface, and the vibration of previous teeth at the same angle.
The chip thickness can be expressed using the circular tooth pathapproximation as:ℎ(𝑡𝑡 ) = 𝑓𝑓& sin f + 𝑛𝑛(𝑡𝑡 − 𝜏𝜏) − 𝑛𝑛(𝑡𝑡), ------(6.3)

where ft is the commanded feed per tooth, f is the tooth angle, n is thenormal direction (see Fig. 6.1), and t is the tooth period and represents thetime delay. The tooth period is defined as:
195

t =

60
WN t

(sec), ------(6.4)

where W is the spindle speed in rpm and Nt is the number of teeth. The
vibration in the direction of the surface normal for the current toothdepends on the x and y vibrations as well as the tooth angle according to:𝑛𝑛 = 𝑥𝑥 sin f − 𝑦𝑦 cos f ------(6.5)

For the simulation, the strategy is to divide the angle of the cut into a
discrete number of steps. At each small time step, D𝑡𝑡, the cutter angle isincremented by the corresponding small angle, Df. This approach enables
convenient computation of the chip thickness for each simulation step
because: 1) the possible teeth orientations are predefined; and 2) the surface
created by the previous teeth at each angle may be stored. The cutter
rotation:ABC

Df = DE (deg) ------(6.6)
depends on the selection of the number of steps per revolution, 𝑆𝑆𝑆𝑆. The
corresponding time step is:BC

D𝑡𝑡 = DEW (sec). ------(6.7)
A vector of angles is defined to represent the potential orientations of theteeth as the cutter is rotated through one revolution of the circular tool path,f = [0, Df , 2Df , 3Df , … , (𝑆𝑆𝑆𝑆 − 1)Df ]. The locations of the teeth
within the cut are then defined by referencing entries in this vector.

196

In order to accommodate the helix angle for the tool’s cutting edges, thetool may be sectioned into a number of axial slices. Each slice is treated as
an individual straight tooth endmill with radius r, where the thickness ofeach slice is a small fraction, D𝑏𝑏, of the axial depth of cut, 𝑏𝑏. Each slice
incorporates a distance delay:
𝑟𝑟c = D𝑏𝑏 tan g ------(6.8)

relative to the prior slice (nearer the cutter free end), which becomes theangular delay between slices:c=

DN OPQ g
R=

SDN OPQ g
T(rad) ------(6.9)

for the rotating endmill, where 𝑑𝑑 is the endmill diameter and g is the helix
angle. In order to ensure that the angles for each axial slice match thepredefined tooth angles, the delay angle between slices is:

c = Df. ------(6.10)

This places a constraint on the D𝑏𝑏 value. By substituting Df for c andrearranging, the required slice width is:T(Df)D𝑏𝑏 = SOPQ g . ------(6.11)
Once the x and y direction displacements are determined (Eq. 6.2), the final
spatial trajectory for each tooth is determined by summing thesevibration-induced displacements with the nominal cycloidal motion of theteeth due to the combined translation and rotation. This final spatial
trajectory is finally used to define the machined surface and, subsequently,
to predict the SLE and surface roughness. The nominal cycloidal motion
197

components in the x and y directions are defined in Eqs. 6.12 and 6.13,
where i is the time step index and Df is the linear feed per time step (see Eq.
6.14).
𝑥𝑥1VW = 𝑟𝑟 sin f + 𝑖𝑖D𝑓𝑓 ------(6.12)
𝑦𝑦1VW = 𝑟𝑟 cos f ------(6.13)
Y[Z Z
D𝑓𝑓 = DE------(6.14)

This simple description can be extended to include:
1. multiple tool modes – the 𝑥𝑥 and 𝑦𝑦 forces are used to calculate theacceleration, velocity, and displacement for each tool mode

(represented by the modal parameters) and the results are summed ineach direction

2. flexible workpiece – the 𝑥𝑥 and 𝑦𝑦 forces are also used to determine
the workpiece deflections, again by numerical integration, and therelativetool-workpiece

vibration

is

used

to

calculatetheinstantaneous chip thickness
3. runout of the cutter teeth – the chip thickness is updated by therunout of the current tooth
4. unequal teeth spacing – the tooth angle vector is modified to accountfor the actual tooth pitch.
Using this time domain simulation approach [59], the forces, displacements,and velocities are calculated. These results are then used to generatestability maps, bifurcation diagrams, and Poincaré maps.

6.3 Stability Maps
Stability maps, or stability lobe diagrams, identify the limiting axial depthof cut (vertical axis) as a function of spindle speed (horizontal axis).
198

Traditionally, this limit is represented as a single contour which separates
stable (forced vibration only) from unstable (secondary Hopf or period-nbifurcations) parameter combinations. In milling, the forcing frequency isdefined by the tooth period, which depends on the spindle speed andnumber of teeth; see Eq. 6.4. Therefore, to evaluate the process
performance, the output signals (force and vibration) from the time domain
simulation were sampled at the tooth period. This provided once-per-tooth
sampling. If the once-per-tooth sampled points repeat, the response isperiodic with the forcing function and the process is stable. If they do notrepeat, the response exhibits either secondary Hopf or period-n bifurcations
and is considered unstable.
To construct a stability map using the time domain simulation results, aseparate simulation was completed at each position in the selected grid of
spindle speed and axial depth values. To automatically establish thestability limit using the predicted time domain signals, the following
stability metric was implemented [62]:

𝑀𝑀1 =

∑d
efg|5_` (a)b5_` (abc)|, ------(6.15)[

where 𝑥𝑥hc is the vector of once-per-tooth sampled 𝑥𝑥 displacements and𝑁𝑁 is the length of the 𝑥𝑥hc vector. Other variables, such as 𝑦𝑦

displacement or cutting force could be selected as well. With this stability

metric, the absolute value of the differences in successive sampled points issummed and then normalized. The sampled points repeat for a stable cut,so the 𝑀𝑀1 value is ideally zero. For unstable cuts, on the other hand,

𝑀𝑀1 > 0 and grows over time, due to the asynchronous motion insecondary Hopf instability and jumps from one fixed point to the next inperiod-n bifurcations.

199

Fig. 6.2 Flexure-based experimental setup with laser vibrometer (LV), laser
tachometer (LT), and capacitance probe (CP). The feed direction and theflexible direction for the single degree of freedom flexure are also
identified. The setup was located on a Haas TM-1 CNC milling machine.
The flexure-based setup displayed in Fig. 6.2 was used to define a physicalsystem for simulation and testing [63-64]. The setup included aparallelogram leaf-type flexure with an aluminum workpiece mounted on
top. The in-process vibration data was collected using a Polytec OFV-5000
laser vibrometer (velocity) and Lion Precision DMT20 capacitance probe(displacement). Both were aligned with the flexible direction for the single
degree of freedom flexure. Note that the feed direction was perpendicular
to

this

flexible

direction.

Once-per-tooth

samplingforstability

identification was accomplished using a laser tachometer and reflective
target attached to the rotating tool holder. The flexure dynamics wereidentified by modal testing: 125.8 Hz natural frequency, 0.0136 viscousdamping ratio, and 1.75×106 N/m stiffness in the flexible direction. The
dynamics for the 19.1 mm diameter, 0 deg helix angle tool (one insert)200

were symmetric: 1188 Hz natural frequency, 0.095 viscous damping ratio,
and 4.24×107 N/m stiffness. The 6061-T6 aluminum alloy cutting force
coefficients were: 𝑘𝑘&+ = 770×106 N/m2, 𝑘𝑘1+ = 368×106 N/m2, 𝑘𝑘&/ =22×103 N/m, and 𝑘𝑘1/ = 22×103 N/m. The experimental up milling cutting

conditions were: 5 mm axial depth, 2 mm radial depth, 0.35 mm/tooth, andvariable spindle speed. Spindle speed values were selected to span fromforced vibration, self-excited vibration, and period-n conditions while
holding all other parameters constant.

Fig. 6.3 Simulated stability map for selected dynamic system with a 2 mmradial depth and 0.35 mm/tooth feed per tooth (𝑀𝑀1 = 1 µm contour).

The stability map for up milling with a 2 mm radial depth and 0.35
mm/tooth feed per tooth is displayed in Fig. 6.3. To construct the map, theEq. 6.15 stability metric was applied to the time domain simulation x
direction workpiece displacements; this was the flexible direction for the
flexure (the feed was in the y direction, which corresponded to the flexure’s
stiff direction). The resolution for the time domain simulations was 5 rpm
in spindle speed and 0.1 mm in axial depth. The initial transients were
removed and the 𝑀𝑀1 value for each simulation was calculated from the201

final 25% of each time record. An arbitrarily small value of M1 = 1 µm wasselectedtodifferentiate

betweenstableandunstableparameter

combinations; this contour is shown in Fig. 6.3 and identifies the stability
limit. Both stable (no color) and unstable (shaded) zones are observed. As
noted, the stable area identifies the spindle speed-axial depth combinations
where only forced vibration is observed. The unstable are may be either
secondary Hopf or period-n bifurcations.
The 𝑀𝑀1 stability metric defined in Eq. 6.15 and applied in Fig. 6.3 does
not separately identify secondary Hopf and period-n bifurcations. However,using subharmonic sampling at nt (n = 2, 3, 4, …), the corresponding

period-n bifurcations can be established [65]. For example, when samplingat 2t, the stability metric becomes “blind” to period-2 bifurcations, or
behavior that repeats every other tooth period rather than each period as inforced vibration. By sampling at every other tooth passage (2t), theperiod-2 behavior appears as synchronous motion (stable). The same is true
for period-3 bifurcations if the sampling interval is 3t, and so on. Thissubharmonic sampling can be applied to add detail to the Fig. 6.3 stability
map. To do so, the M1 metric is augmented to enable sampling at otherinteger tooth periods. Generically, the metric can be expressed as shown in
Eq. 16, where the integer n = 1, 2, 3, … defines the sampling period (i.e.,nt).

𝑀𝑀𝑀𝑀 =∑defg|5_l (a)b5_l (abc)|[------(6.16)

The subharmonic sampling approach was implemented to construct astability map that individually identifies each bifurcation type. The metrics
𝑀𝑀1 through 𝑀𝑀7, which represent t through 7t integer sampling periods,

are used to isolate the stable zone as well as the different bifurcation types:
period-2, -3, -4, -5, -6, -7, and secondary Hopf.
202

Fig. 6.4 New stability map. Period-2 (circle), period-3 (triangle), period-5(+), period-7 (´), and secondary Hopf (open space) bifurcations are
identified. The dots indicate stable (forced vibration) behavior.
The logic used to construct the stability map follows:
if 𝑀𝑀1 £ 1 µm

plot a dot (stable)

elseif 𝑀𝑀2 £ 1 µm

plot a circle (period-2)

elseif 𝑀𝑀3 £ 1 µm

plot a triangle (period-3)

elseif 𝑀𝑀4 £ 1 µm and 𝑀𝑀2 > 1 µm

plot a square (period-4, excludes period-2)

elseif 𝑀𝑀5 £ 1 µm

plot a + (period-5)

elseif 𝑀𝑀6 £ 1 µm and 𝑀𝑀2 > 1 µm and 𝑀𝑀3 > 1 µm

plot a diamond (period-6, excludes period-2 and period-3)

elseif 𝑀𝑀7 £ 1 µm

plot an ´ (period-7)

else

do nothing (secondary Hopf or high order period-n)
end203

The result is displayed in Fig. 6.4. The stable zone is the dark area filledwith dots and the various instabilities are indicated by the symbols. The
open space represents secondary Hopf bifurcation. The period-n bifurcation
zones are also identified by the numerals at the top of the figure (i.e.,
periods-2, 3, 5, and 7).

6.4 Bifurcation Diagrams
In the bifurcation diagrams developed for this study, the once-per-tooth
sampled displacement, xs (vertical axis), was plotted against the spindle
speed (horizontal axis). The transition in behavior from stable to period-nor secondary Hopf bifurcations is then directly observed. A stable cut
appears as a single point (i.e., the sampled points repeat when only forced
vibration is present). A period-2 bifurcation, on the other hand, appears as apair of points offset from each other in the vertical direction. This
represents the two solutions, or fixed points, that appear due to the behavior
that repeats every other tooth period. A secondary Hopf bifurcation is seen
as a vertical distribution of points; this represents the range ofonce-per-tooth sampled displacements that occur due to the quasi-periodic,
asynchronous behavior.
Returning to Fig. 6.4, it is seen that the stability behavior changes withspindle speed at an axial depth of b = 10 mm. Starting at the left end of thestability map, stable conditions are observed at 2500 rpm (dots). Thischanges to secondary Hopf instability (open area) at 2753 rpm. Zones of
period-7, 5, and 3 occur prior to a return to stable conditions at 2858 rpm.
The corresponding bifurcation diagram for this axial depth over the spindle
speed range from 2500 rpm to 2900 rpm is provided in Fig. 6.5, where thespindle speed resolution is 1 rpm.

204

Fig. 6.5 Bifurcation diagram for a 10 mm axial depth of cut and spindle
speeds from 2500 rpm to 2900 rpm. The vertical axis is the once-per-tooth
sampled workpiece displacements and the horizontal axis is the spindle
speed. The inset shows the period-5 bifurcation range between 2761 rpm
and 2765 rpm identified in Fig. 6.4.
A second bifurcation diagram is displayed in Fig. 6.6. The axial depth isagain 10 mm, but the spindle speed range is 2900 rpm to 3400 rpm. This
new spindle speed range spans the period-2 zone shown in Fig. 6.4. It isseen that stable behavior persists from 2900 rpm to 2998 rpm (a single
point at each spindle speed). Period-2 instability then occurs between 2998
rpm and 3047 rpm (two points). Combined period-2 and secondary Hopf
conditions are observed between 3047 rpm and 3104 rpm (two separatelines of points). The spindle speed range between 3104 rpm and 3145 rpm
is characterized by secondary Hopf instability before transitioning back to
the combined behavior between 3145 and 3200 rpm. Period-2 bifurcations
occur from the range from 3200 rpm to 3282 rpm. Stable cutting thenpersists between 3282 rpm and the right hand side of the diagram at 3400
rpm.

205

3047

3200

3282

2998
3104

3145

Fig. 6.6 Bifurcation diagram for a 10 mm axial depth of cut and spindle
speeds from 2900 rpm to 3400 rpm. Stable, period-2, combined period-2
and secondary Hopf, and secondary Hopf behavior is observed.

6.5 Poincaré Maps
Poincaré maps were constructed using both experiments and simulations.
For the experiments, the displacement and velocity of the flexible
workpiece (Fig. 6.2) were recorded using a capacitance probe and laser
vibrometer and then sampled once-per-tooth period using a laser
tachometer. In simulation, the displacement and velocity were predicted,
but the same sampling strategy was applied. By plotting the displacement
versus velocity, the phase space trajectory could be observed in both cases.
The once-per-tooth period samples were then superimposed and used tointerrogate the milling process behavior.
For stable cuts, the motion is periodic with the tooth period, so the sampled
points repeat and a single grouping of points is observed. When secondary
Hopf instability occurs, the motion is quasi-periodic with tool rotation
because the chatter frequency is (generally) incommensurate with the tooth
passing frequency. In this case, the once-per-tooth sampled points do notrepeat and they form an elliptical distribution. For a period-2 bifurcation,
the motion repeats only once every other cycle (i.e., it is a sub-harmonic of
206

the forcing frequency). In this case, the once-per-tooth sampled points
alternate between two solutions. For period-n bifurcations, the sampled
points appear at n distinct locations in the Poincaré map.
The time domain simulation result for an up milling operation with an axial
depth of 5 mm, radial depth of 2 mm, 0.35 mm/tooth feed, and spindle
speed of 3180 rpm is displayed in Fig. 6.7. Both the x (flexible direction)
displacement and the once-per-tooth sampled points are included. It is seen
that, after the initial transients attenuate, two solutions occur that alternate
from one tooth passage to the next. This represents a period-2 bifurcation.
Note this spindle speed-axial depth combination appears within the
period-2 zone in Fig. 6.4.

Fig. 6.7 Simulated x displacement for 3180 rpm, 5 mm axial depth, 2 mmradial depth, and 0.35 mm feed per tooth. The continuous signal (line) andonce-per-tooth sampled points (circles) show period-2 behavior.
The simulated and experimental Poincaré maps for {W, b} = {3180 rpm, 5mm} are provided in Fig. 6.8. Two fixed points appear for the period-2
207

100

100

50

50

dx/dt (mm/s)

dx/dt (mm/s)

bifurcation condition. The simulation (left) and experiment (right) agree.

0

0

-50

-50

-100
-200

-100

0

x (µm)

100

200

-100
-200

-100

0
x (µm)

100

Fig. 6.8 Predicted (left) and measured (right) Poincaré maps for 3180 rpm.
Period-2 behavior is seen. Note that x indicates the flexible direction for the
flexure. The feed direction was y for the experiments.

Fig. 6.9 Simulated x displacement for 3300 rpm, 5 mm axial depth, 2 mmradial depth, and 0.35 mm feed per tooth. The cut is stable.
The spindle speed was next increased to 3300 rpm with all otherparameters remaining constant. The time domain simulation results are
displayed in Fig. 6.9 and the corresponding Poincaré maps in Fig. 6.10. For
this spindle speed, stable behavior is obtained (see Fig. 6.4). The simulation
208

200

100

100

50

50

dx/dt (mm/s)

dx/dt (mm/s)

and experiment again agree.

0

-50

-100
-200

0

-50

-100

0

x (µm)

100

200

-100
-200

-100

0
x (µm)

100

Fig. 6.10 Predicted (left) and measured (right) Poincaré maps for 3300 rpm.
Stable behavior is seen.

Fig. 6.11 Simulated x displacement for 3600 rpm, 5 mm axial depth, 2 mmradial depth, and 0.35 mm feed per tooth. The cut is stable.
The final spindle speed was 3600 rpm. The time domain simulation results
are displayed in Fig. 6.11 and the corresponding Poincaré maps in Fig. 6.12.
For this spindle speed, stable behavior is again obtained (see Fig. 6.4), but
the forced vibration response is larger than Figs. 6.9 and 6.10 because thefirst harmonic of the tooth passing frequency (3600/60×2 = 120 Hz) is close
to the flexure’s natural frequency (125.8 Hz). The simulation andexperiment again agree.
209

200

100

50

50

dx/dt (mm/s)

dx/dt (mm/s)

100

0

-50

-100
-200

0

-50

-100

0

x (µm)

100

200

-100
-200

-100

0
x (µm)

100

200

Fig. 6.12 Predicted (left) and measured (right) Poincaré maps for 3600 rpm.
Stable behavior is seen with increased amplitude relative to 3300 rpm (Fig.
6.10).

6.6 Surface Location Errors
The process dynamics can impose significant limitations on milling
efficiency due to secondary Hopf bifurcations that lead to large forces,
displacements, and poor surface quality. However, productivity can also be
limited by forced vibrations which cause surface location error (SLE), or
workpiece geometric inaccuracies that result from dynamic displacementsof the tool during stable milling. In order to explore the variation in SLEwith spindle speed, the workpiece geometry shown in Fig. 6.13 wasselected. The initial ribs were machined directly on the flexure so it was
ensured that the part was aligned with the machine axes. Low axial andradial depths were selected to minimize vibration levels and the same
conditions were used to machine each rib. Prior to beginning the SLE
experiments, a test workpiece was machined and the four ribs were
measured on a coordinate measuring machine, or CMM, to evaluate therepeatability of the starting rib dimensions (Zeiss Prismo). The mean value
was 9.82 mm with a standard deviation of 2.8 µm.

210

Fig. 6.13 The SLE workpiece included four ribs that were initiallymachined to the same dimensions. The 5 mm axial depth, 2 mm radial
depth cuts were then performed on one edge at a different spindle speed foreach rib. The SLE was calculated as the difference between thecommanded and measured, M, rib widths. The flexible direction for the
flexure (Fig. 6.2) is identified.
Based on Figs. 6.3 and 6.4, the 11 spindle speeds in Table 6.1 were used tomachine 11 ribs (three total workpieces). As shown, this enabled SLE for
both period-2 and stable behavior to be evaluated. Other than spindle speed,
all other machining conditions were identical between the tests (up milling,
5 mm axial depth, 2 mm radial depth, and 0.35 mm/tooth feed).
Spindle speed (rpm)Behavior

3180

Period-2

3190

Period-23200

Period-2

3210

Period-2

3270

Stable3300

Stable3330

Stable3360

Stable3400

Stable3500

Stable3600

StableTable 6.1 Spindle speeds and bifurcation behavior for experiments.
211

A comparison between the simulated and experimental SLE results ispresented in Fig. 6.14 and Table 6.2. Four tests were completed underperiod-2 conditions and seven were performed under stable conditions.
Good agreement is observed between prediction and measurement. The
average error between prediction and measurement is 0.5 µm for all tests.

Fig. 6.14 SLE prediction from simulation (line) and experimental results(circles). The four period-2 bifurcation tests are identified.
Spindle speed (rpm)Behavior

Measured SLE(µm)

Predicted SLE(µm)

Error (µm)

3180

Period-2

-4

0

-4

3190

Period-2

-8

-6

-2

3200

Period-2

-13

-15

+2

3210

Period-2

-19

-21

+2

3270

Stable

-29

-30

+1

3300

Stable

-32

-32

0

3330

Stable

-33

-35

+2

3360

Stable

-38

-38

0

3400

Stable

-44

-42

-2

3500

Stable

-55

-58

+3

3600

Stable

-90

-85

-5

Table 6.2 Comparison of measured and predicted SLE results for rib
cutting tests.
212

9.52

y (mm)

9.5
9.48
9.46
9.44
9.42

24

25

26
x (mm)

27

28

Fig. 6.15 Commanded surface (dashed line), CMM scan (solid line), andsimulation result (circles) for 3180 rpm (period-2). These resultscorrespond to Figs. 6.7 and 6.8.
Figures 6.15-6.17 provide a direct comparison between the time domain
simulation and the CMM surface points obtained by continuous scanning

Fig. 6.16 Commanded surface (dashed line), CMM scan (solid line), andsimulation result (circles) for 3300 rpm (stable). These results correspondto Figs. 6.9 and 6.10.
213

Fig. 6.17 Commanded surface (dashed line), CMM scan (solid line), andsimulation result (circles) for 3600 rpm (stable). These results correspondto Figs. 6.11 and 6.12.
along the machined surface. In these figures the commanded surface is
identified by the dashed line, the solid line is the CMM data, and the circles
are the simulation results. The SLE is the difference between thecommanded and actual surface and, again, good agreement is observed
between simulation and measurement.

Fig. 6.18 Combined stability and SLE map for rib cutting process dynamics.
The secondary Hopf instability is represented by the dark zone, theperiod-2 behavior is identified by the dotted zone, and the SLE is given by
the contours (lines of constant SLE) with numerical labels (SLE in µm).
214

The stability and SLE information is combined in Fig. 6.18. In this figure,the dark area represents secondary Hopf instability, the dotted area
identifies the period-2 bifurcations, and the contour lines give the SLE as afunction of spindle speed (horizontal axis) and axial depth of cut (verticalaxis). Zero SLE is seen near the traditional best speed of 3774 rpm(125.8×60/2). However, a steep gradient for small changes in spindle speed
is also seen near this speed (i.e., the zero SLE contour is vertical). A zero
SLE contour is also observed within the period-2 zone. Interestingly, theSLE gradient is not as steep within the period-2 zone as it is near the best
speed at 3774 rpm.

6.7 Conclusions
This chapter explored the three possible milling response types: forced
vibration, self-excited vibration, and period-n bifurcations, using both time
domain simulation and experiment. Stability maps, bifurcation diagrams,
and Poincaré maps were used to visualize the milling behavior.
Experiments were used to validate the predictions. It was shown that all
three behavior types are possible with changes in spindle speed and axial
depth of cut. It was also demonstrated that, even for stable conditions,surface location error can limit milling productivity due to part geometricerrors introduced by the forced vibration response.

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221

Chapter 7 Quick Notes for Rigidity and Damping
Capacity of Main Spindle Supported
by Rolling Bearing
7.1 Introduction
With the increase of challenging research into the applicability of thestability chart to the higher-speed range, we can observe the obvious
disagreement of the theoretical stability chart with that obtained by theexperiment as already shown in Figs. 3.5 and 3.6. In short, it was suggested
that such behavior are caused by the speed-dependence of the spindle
dynamics, and furthermore gyroscopic and centrifugal effects in both thefront and rear bearings in the main spindle system. It is however regrettablethat the validity of these suggestions is not verified as yet.
Paraphrasing, such a disagreement is derived considerably from theinadequate determination of the parameters in the “Structural Equation” inthe chatter theory and not acceptable to contrive the remedy for the chatter
suppression. Of note, people in chatter concerns are not familiar to theknowledge so far obtained in the sphere of the “Machine Tool Joint”,
although the main spindle system is one of the leading machine tool joints.
In general, the main spindle system can be facilitated with the rolling
bearing when its rotating speed is higher, and thus a facing problem is towhat extent we have the engineering data for the rigidity and damping of
the rolling bearing, which are available for the higher rotational speed.
In this context, we must be aware that the available data are for the speed
less than 2,000 rev/min, and that even the rolling bearing manufacturer has
not arranged and systematized yet any engineering data with respect to therigidity and damping capacity while rotating the main spindle more than
2,000 rev/min, and up to 40,000 rev/min. This is because of too much222

difficulty in measuring the corresponding data as will be discussed later.
Of course, we need to have such data to advance the chatter theory andcontrive the innovative remedy for chatter suppression hereafter, and thus
some quick notes will be stated in the following to give a clue for
conducting a forerunning research.

7.2 Fundamentals of Main Spindle Supported by Rolling
Bearing
In discussing the main spindle of rolling bearing type, we must first be awareof the three leading design attributes, i.e. “Higher-accuracy Machining”,“Higher-speed Machining”, and “Heavy Machining Capability”, whichshould be actualized in the main spindle. In general, in accordance with thedesign specifications, we design the main spindle in consideration of (1) only
one attribute, (2) combination of two of three attributes, or (3) all the three
attributes simultaneously. Obviously, we must understand that the chatter
behavior depends considerably upon the characteristic features given by thedesign specifications. Figs 7.1 (a), (b) and (c) show thus the typical structuralconfigurations with respect to each leading attribute.

(a) High rigidity main spindle - For heavy machining223

(b) For higher-speed machining(c) For higher-accuracy machining
Fig. 7.1 Different features in structural configurations of main spindle
depending upon weighing design attribute
As can be seen from these, there are considerable differences in the kindsof bearing, arrangement of bearings, mechanism of pre-load, and so on,
depending upon the design requirements. In this context, we must

224

furthermore be aware of the following.
(1) As a rule of the machine tool joint, the rigidity and damping in the main
spindle are in reciprocal relation.
(2) The leading three design attributes are in reciprocal relation one
another; however, to respond properly the machining requirement of theuser, we often consider simultaneously three or two of these as
mentioned beforehand. Obviously, it is very difficult to optimize thestructural configuration of the main spindle in such a case, and thus we
can realize the preferable one, even which results in very complicatedspindle configuration.
(3) Although the main spindle of two-point supporting type has beenprevailed, in certain cases, we used to design the main spindle ofthree-point supporting type to reinforce furthermore the rigidity.
(4) We can estimate the design qualification of each machine tool
manufacturer by the integrated rate of the three leading attributes, andas reported elsewhere, German and Japanese manufacturers can manage
the design with utmost integrated rate, i.e. structural configuration
simultaneously integrated three attributes.
Importantly, both the main spindles for heavy machining and higher-speed
machining are at burning issue from the viewpoint of the chatter. Apartfrom the main spindle for heavy machining, the main spindle forhigher-speed machining appears not to be related to the chatter; however,such a main spindle can facilitate the large chip removable rate per unit
time, i.e. another type of heavy machining, which is applicable tomachining of the aircraft component made of Al alloy and concerns.
In discussing the discrepancy between the theoretical stability charts andexperimental one while machining either with lower-speed or withhigher-speed, it is thus reasonable that such discrepancies are derived fromthe dynamics of the main spindle; however, it is too much difficult to detailwhat is the determinant. Paraphrasing, we must exploit the correlation
between the structural configuration of the main spindle and its influencing
magnitude to the chatter, and a recommendable research is to estimate both225

the rigidity and damping capacity distribution diagrams within a main
spindle system (see Fig. 3.9).
Of course, such a research is very valuable; however, in this context, we
may assert first the urgent necessity of the measurement of the rigidity anddamping of the main spindle while rotating the higher-speed more than
2,000 rev/min in consideration of the details of the design attributes shownin Table 7.1. Table 7.1 summarizes in detail the design attributes of themain spindle, and as can be seen, we can obtain the suitable structuralconfiguration as per the machining requirement by the preferable
combination of these attributes.

Table 7.1 Leading attributes to be considered in design ofmain spindle system
On the basis of the basic knowledge mentioned above, we must discuss
another fatal issues in the main spindle as follows.

226Thrust supporting at front main bearing and chuck mounting at spindle
nose
In the main spindle, a crucial issue in the design is thrust (axial load)
supporting including the pre-loading mechanism, and in general, the front
main bearing can facilitate such a function together with maintaining theallowable rigidity and damping capacity. Paraphrasing, the machineattachment-tool-work system can be characterized by the front main
bearing, which plays the role of quasi-fixation point, to a large extent.

Fig. 7.2 Further crucial issue in main spindle from viewpoint of chatter
For the sake of further convenience, Fig. 7.2 shows another structuralconfiguration of the front main bearing, and as will be clear from it, wemust be aware of the considerable influence of the front bearing to thechatter. Importantly, the chatter behavior depends upon the reference length
shown also in Fig. 7.2 to certain extent together with mounting method of
the chuck at the spindle nose. In this context, we must remind again that thejaw traveling mechanism in the jaw chuck affects considerably the227magnitude of mingling parametric vibration within the regenerative chatteras already suggested by M. Doi (see Figs. 1.8 and 1.9). In addition, Feng
(2003) suggested the better applicability of the jaw traveling mechanism ofwedge bar type to the higher-speed main spindle, although not verifying its
validity to satisfactory level.
Relieving mechanism of thermal elongation of main spindle at rear bearing
In general, the rear main bearing cannot support any thrust, so that thethermal elongation of the main spindle can be in free. Otherwise, asreported elsewhere and will be shown later, the temperature rise of themain spindle varies the pre-load, fitting tolerance between the bearing andthe quill, and so on to a large extent, resulting in the considerable
influences on the chatter. We must thus pay the special attention to thestructural configuration of the rear main bearing.
Of special note, a noteworthy contrivance is also shown in the left of Fig.
7.2, which is for the tapered roller bearing and called “High-speed
Adapter”. The tapered roller bearing is often employed in the main spindle,
because of its simplicity in load supporting, although the temperature rise islarger than those of other kinds of the rolling bearing, and thus Timken
contrived, as well known, the high-speed adapter. More specifically, thethermal deformation of the outer ring is quicker in radial direction than thatin axial direction, and thus small clearance at the land B can be facilitatedthe release of the dynamic constrain of the outer ring, resulting in the
reduction of heat generation.
To this end, it emphasizes that a suitable setting of mathematical model is
at issue in the chatter analysis. Paraphrasing, to what extent we must
replace the structural configuration of the main spindle with a model. Fig.
7.3 shows a grinding wheel spindle, which is similar configuration to thatfor higher-accuracy machining because the grinding machine is, in general,
228

for finish machining. In fact, we cannot see a large differing feature in the
bearing arrangement in Fig. 7.3 as compared with that shown in Fig. 7.1 (c).
More specifically, both the front and rear bearings are with “Back-to-Back”
arrangement for Fig. 7.1(c) and “Back-to-Face” arrangement for thegrinding wheel spindle. Thus, we need not to consider such an arrangement
difference when producing the mathematical model; however, in theanalysis of thermal deformation, we must consider its meaningful
influences.

Fig. 7.3 Grinding wheel spindle of built-in motor type
We learn also the same story in the case of the angular contact thrust ballbearing shown in Fig. 7.2. With the advance of the due technology, we canuse such a new bearing without the distance collar (spacer) together with
the improved raceway finish and concerns to reduce the friction as shownin Fig. 7.4. Obviously, we can expect the large effect on the reduction of
the heat generation, whereas we may not expect the sufficient increases inboth the rigidity and the damping capacity to suppress the chatter.
As will be clear from the above, the new angular contact thrust ball bearing
is compact, because of having no distance collar, and thus capable of229

replacing the application sphere of the cross-ball bearing. For example, thecross-roller bearing has prevailed in table supporting of the quinaxial-controlled MC of trunnion type.

Fig. 7.4 Improvement in angular contact thrust ball bearing – SKF-brandOf special note, we must take into consideration of the influence of thebuilt-in-motor, which is shown in Fig. 7.3, in the analyses of the chatter andthermal deformation, and as can be readily seen, the spindle configuration
should be modeled by weighing the essential features of the chatter.

7.3 Difficulties in Measuring Rigidity and Damping
Capacity of Rolling Bearing While Rotating Main Spindle
Reportedly, damping in the rolling bearing may be caused by the following
factors.
(1) Deformations of rolling elements and rings.
(2) Deformation in ring fits.
(3) Friction of rolling elements against rings especially arising the change
of inclination angle of the shaft.
(4) Friction of rolling elements against the cage.
(5) In the tapered roller bearing, the friction between the roller ends and theribs of inner ring, resulting in the appearance of peculiar behavior.
Obviously, these factors are also related closely to the rigidity, it is thus
recommendable to produce both the “Rigidity Distribution Diagram” andthe “Damping Capacity Distribution Diagram” within the230machine-attachment-tool-work system as already mentioned above.
Admitting the importance for producing such distribution diagrams, thefacing problem is however the too much difficulty in the measurement ofboth the rigidity and damping capacity in the main spindle system whilerotating higher-speed as exemplified by the test rig shown in Fig. 2.17 (Ito,2008).

Fig. 7.5 Variations of interface pressure with running time
A root cause of further difficulty lies in the changes of the dynamic
characteristics of the main spindle system with the laps of time after
starting the rotation. In fact, a crux is the variation of both the fitting
tolerances between the outer race and the housing, and between the shaft
and the inner race. More specifically, Inaba et al (1995, 2000) measured thechanges of the interface pressure between the quill and the outer ring of the
bearing by using the test rig shown in Fig. 7.5, where the ultrasonictransducer of focus type (Oscillating frequency: 5 MHz, incident area at

231

joint: 5 mm) detects the interface pressure.
It is very interesting that the interface pressure varies obviously with therunning time, showing a peak value immediately after starting the run andgradually decreasing value with the running time as also shown in Fig. 7.5.
This behavior can be interpreted as an obvious evidence to demonstrate aconsiderable difference in fitting tolerance between the still and running
conditions. In short, the main spindle system varies its dynamic
characteristics by running, and thus we can suggest the considerable
change in the stability charts between in still stand and in cutting.

7.4 Concluding Remarks
As will be clear from the above, people in chatter concerns are not familiar
to the fundamental knowledge about the machine tool joint so far
accumulated; however, they must recognize that the main
spindle-attachment-tool-work system is one of the machine tool joints, andthat to what extent the knowledge so far obtained are available. In fact, atburning issue is to measure the rigidity and damping capacity of the main
spindle system while rotating more than 2,000 rev/min, and in thepreferable case, we must estimate the “Rigidity Distribution Diagram” and“Damping Capacity Distribution Diagram” within the main spindle system.
More specifically, we must also measure both the dynamic joint stiffness,for example, between the chuck and the spindle nose of main spindle, andbetween the tool shank and the tapered hole in the main spindle.
To this end, it is worth suggesting that we must identify roughly the
structural configuration of the main spindle, which is beneficial to interpretthe characteristic features in the stability chart for higher-speed cutting. In
short, the more clarification the structural configuration of the main spindle,
the more accurate is the chatter analysis; however, from the practicalapplication point of view, at facing issue is how far and to what extent wemust replace the actual structural configuration with the simplified model.

232

In this context, people in chatter concern might assert that the structuralexpression can represent satisfactorily the characteristic feature of thespindle configuration even when rotating with higher-speed, provided that
we can measure the equivalent mass, damping coefficient and spring
constant while spindle rotating. In contrast, we must eye why we
differentiate the structural configuration of the main spindle in accordancewith its design specifications as shown in Fig. 7.1. We have so far paid any
attention to such differences; however, we could observe the differing
chatter pattern when machining the work by MC with spindle forheavy-cutting from that machining by MC with spindle for higher-speed(see Figs. 7.1 a and b).

Fig. 7.6 Generalized warm-up characteristics of main spindle andestimation factors for structural configuration in main spindle
To enhance the chatter theory, intuitively, we need to deploy such a new
vision, and it is thus recommendable that we must eye the effective use of
the “Thermal Stability Chart” shown in Fig. 7.6 to estimate roughly the
structural configuration of the main spindle. More specifically, the thermal
stability chart reveals indirectly the characteristic feature in the structuralconfiguration of main spindle, and thus we must measure the temperature233

rise and its rising rate every certain time interval as schematically shown in
Fig. 7.4 while maintaining the rotational speed in constant. Of course, it isbetter to change the rotational speed in various steps.

References
Feng P F. (2003) Berechnungsmodell zur Ermittlung von Spannkräften bei
Backenfuttern. Dr. Dissertation of Technische Universität Berlin.
Inaba C et al. (1995) In-Process Measurement of Contact Pattern
Variations in Rotating Main Bearing of Machine Tools Using UltrasonicWaves Method. Trans. of JSME (C); 61-588: 3375-3381.
Inaba C et al. (2000) Estimation of Contact Pressure between Bearing andBearing Housing by Means of Ultrasonic Waves. Trans. of JSME (C);66-645:1674-1680.
Ito Y. (2008) Modular Design for Machine Tools. McGraw-Hill, New York.

234

Chapter 8 Thought-provoking Quick Notes for
Characteristic Factors in Metal Grinding
In the block diagram shown in Fig.1.19, the regenerative effect (1/(1-e-TwS)),(1/(1-e-TsS)) and the ratio of the vibration amplitudes to the amplitudes ofgenerated waviness in the workpiece surface g mean pure geometrical
relationships. The other transfer functions have deep relationships withgrinding wheels, which are affected by truing and dressing operations, andare varying with the progress of grinding operations.
Grinding wheels consist of abrasive grains, binders and pores, and these
constructions are determined in industrial standards. In these standards,
however, the grains geometries and the distributions of binders frommicro-viewpoint are not prescribed and these distributions are standardized
statistically. And then even if the specification is same, two grinding wheels
with same characteristics do not exist. From such a viewpoint, the transfer
functions consisting of Fig. 1.19 have to be evaluated as functions of thesurface conditions of grinding wheels. In this chapter, considering therelationships between these transfer functions and grinding wheels, thecharacteristics of grinding wheel surfaces affecting the chatter vibrations
are described.

8.1 Contact Compliance of Grinding Wheel to WorkpieceContact compliance of grinding wheel 1/Kcon shown in Fig.1.19 is aparameter meaning the elasticity of grinding wheel at contact area to theworkpiece. Since metal workpieces are harder than grinding wheels, theirelastic deformations can be negligible, and then Kcon means thedeformations of only grinding wheels at contact area to the workpiece.
Kcon have been discussed experimentally so far (Snoys,1968) (Hoshi,1983).
A measured example of Kcon is shown in Fig. 8.1 (Fukuda,1974).
According to this example, the approaching amount between the grinding
wheel and the workpiece increases with the pushing force, i.e. normal
force, and tends to hard spring characteristics. Therefore, this character235

Fig. 8.1 A measured example of contact stiffness ofgrinding wheel (by Fukuda)

Fig. 8.2 Measured cutting edge distributions in
grinding wheel surface (by Izumi)
can be explained by Hertz contact theory depending on the assumption thatboth the grinding wheel and the workpiece are uniform bodies from236

macro-viewpoint.
Grinding wheel consists of abrasive grain, binder and pore. In the surfaceof grinding wheel, since cutting edges distribute as shown in Fig. 8.2(Izumi, 2002), their contact model shown in Fig. 8.3 was assumed (Brown,1971). In this model, it is considered that the deformation in contact area
consists of two deformations between the grains and the workpiece andbetween the grinding wheel and the workpiece. In this case, Hertz contact
theory is also applied and the experimental results of the deformation incontact area are explained in macro-viewpoint.

Fig. 8.3 Grinding wheel contact model to workpiece (by Brown)
Taking into account discontinuous grains as shown in Fig. 8.2, someapproaches to explain the dynamic phenomena in contact area have been
proposed. Yamada et al (2003) tried to explain the static phenomena by
assuming that rigid grains are suspended by the binder with linear springcharacteristics (Yamada,2003). By this assumption, the static stiffness ofbinders connecting grains can be obtained as shown in Fig. 8.4. From thistrial, it is clarified that the static stiffness k connecting grains is
independent from the grinding wheel geometry and determined only by thegrain size. Furthermore, it is shown that the amount k is inverse proportion
to grain size as;
K = 470 / #------(8.1)

where, # is grain size.
This tendency can be understood under the assumption as follows.
Supposing that the binder is a simple beam, its stiffness increases with theincrease of grain size. However, since the grain distribution is assumed to237

be body centered cubic in this model, the constant shown in Eq. (8.1) varies
by the model arrangement 1. And then, this shows only the qualitative
tendency.

Fig. 8.4 Static stiffness of a binder suspending grains (by Yamada)
According to this model, one grain suspended binders with stiffness kcontacts to the workpiece at first contact as shown in Fig. 8.5. With
increasing the normal force, the number of grains contacting to theworkpiece increases. Since these springs have parallel distribution, thecontact stiffness Kcon increases with the increase of the number of contact
grains. By this discussion, hard spring characteristics frommacro-viewpoint shown in Fig. 8.1 experimentally obtained can be
explained.
In the same experiment, it is confirmed that the experimental relationshipbetween the normal force and the approaching deformation varies with thecontact direction. Hence, it is experimentally clear that grinding wheels
have some anisotropic characteristics. However, no theoretical models of
Yamada et al calculated the static stiffness of binder connecting abrasive grans under an assumption thatthe grains are connected in body centered cubic mode. Another type of grain model is also proposed in thefollowing paper;
Salem S H, Russell J K. (August 1980) Applying finite element method in the solution of the elastic
problems in grinding wheels. International Conference on Manufacturing Engineering, Melbourne. p.150.

1238

the grinding wheel, which can explain such phenomena, has been proposed
yet.

Fig. 8.5 Grain contact to workpiece (by Yamada)

8.2 Grinding Stiffness
Grinding forces in metal grinding can be divided into the normal force andthe tangential force as shown in Fig. 8.6. In these forces, since the normal
force has deeply related with static and dynamic phenomena, experimental
discussions on Fn are carried out. The relationship between the normal
grinding force and the actual depth of cut u is shown as;
Kg = Fn / u（N / µm） -----(8.2)

Since this ratio has the same unit with stiffness, it is called as “Grinding
Stiffness”. It is known that it can be a function of grinding condition as
follows;
K g = Fn / u= k b v / (V+v)----- （8.3）

where, k is specific grinding force, b is grinding width, v is workpiece
speed and V is grinding wheel speed.

239

In short, this Kg means the “Grindability”.

Fig. 8.6 Normal and tangential components of grinding force

Fig. 8.7 Relationship between grinding stiffness andratio of flank wear (by Inasaki)
In grinding the metal with a grinding wheel just after dressing, Kg is almost
proportion to the actual depth of cut. This amount has a tendency toincrease with the increase of the grade of grinding wheel hardness. On theother hand, it is known that the ratio of flank wear η becomes a certain
amount, Kg increases rapidly as shown in Fig. 8.7 independent from thegrade of the grinding wheel hardness experimentally (Yoshikawa, 1962 and240

Inasaki, 1970). Furthermore, the non-linear soft-spring characteristics also
appears. It is known that with increasing the ratio of flank wear, grazing
takes place in the surface of grinding wheel and then normal grinding
cannot be continued. This is the reason that rapid increase of Kg shown in
Fig. 8.7. However, the physical effect of flank wear on Kg and thequantitative discussion on the amount of η and its effect have not been
conducted yet.

8.3 Unbalance of Grain Distributions in Grinding Wheels
After combining abrasive grains and binder, grinding wheels are fabricatedby pressing and sintering. Therefore, the unbalance of the abrasive grains
distribution in grinding wheel cannot be avoided. Since the grinding wheelwith unbalanced grain distribution has not even elastic modulus, theunbalance of grain distribution and cutting edges can be evaluated bymeasuring the elastic modulus 2. Fig. 8.8 shows a measured example of theelastic modulus E of a vitrified-bond grinding wheel WA46 (Umino, 1983).
This figure shows that E is small at the point 1 and is large at the point 5.
This means the unbalance of grade of hardness in this grinding wheel, andconsequently the dynamic characteristics are not even in this grinding
wheel.
After balancing this grinding wheel carefully, carrying out dressing, large
acceleration synchronizing with grinding wheel rotation is monitored. This
acceleration is large at the position where the elastic modulus is large. This
experimental results show that the place where elastic modulus is large, thegrade of hardness is large and the density of abrasive grain distribution ishigh. The density of cutting edge distribution after dressing is shown in Fig.
8.9. This figure shows that the cutting edge density is high at the position 5
where elastic modulus is high. And at the position 1, the cutting edge
density is low. This result means that at the position 1, the cutting edge
density is small and the grade of hardness is also small. Then the wheel
It is reported that both the grain distribution and the grade of hardness of the grinding wheel can beevaluated by measuring the lastic modulus of grinding wheel as exemplified by the following papers. In
these papers, the elastic modulus is evaluated by the use of ultrasonic pulse.
Umino K, Shinzaki N. (1979) Study on Grading of Grinding Wheels by Ultrasonic Pulse Method (Part 1
and Part 2). Precision Engineering; 36-8: 538, and 36-9: 608.
2

241

wear tends to proceed fast at the point 1, and it causes uneven grinding
wheel wear. Consequently, the variation of elastic modulus in the grinding
wheel is the cause of uneven grinding forces synchronizing with grinding
wheel rotation, and the forced chatter vibration synchronizing withgrinding wheel rotation takes place.

Fig. 8.8 Distribution of wheel grade of hardness (by Umino)

Fig. 8.9 Distribution of wheel cutting edges density (by Umino)242

Fig. 8.10 Maximum waviness amplitude in grinding wheel (by Inasaki)
Inasaki et al investigated on the waviness generated in the grinding wheel
surface and its effect on the chatter vibration in spark-out grinding processexperimentally (Inasaki, 1968). In this study, it is shown that the amplitudeof waviness rapidly increases at the cumulative metal removal becomes acertain amount as shown in Fig. 8.10. The number of generated wavinessis;n = f ( N / 60 )----- (8.4)

Since this number depends on the grinding wheel rotating cycle N, thisvibration is due to the grinding wheel run-out and/or uneven grinding
wheel wear depending on the uneven grade of hardness or distribution ofabrasive grains.
These phenomena described above means that the wear compliance Ka
shown in Fig.1.19 varies by synchronizing with the grinding wheel rotating
cycle.
Some trials to fabricate even and balanced grinding wheels started. At
present, however, any production methods of the balanced grinding wheel
have not been proposed.
243

8.4 Grinding Damping
Grinding damping is a parameter induced from a geometrical analysis ofregenerative effect as shown in Fig.1.18 (Ohno, 1969 and Lee,1982). This
analysis is carried out under the condition that the grinding force is
proportional to the removed volume of ground workpiece per unit time. As
the analyzed result, it is clarified that the radial component of grinding
force Pn(s) has a term proportional to the relative vibrating speed as shownin Eq. (1.18).
Grinding damping Cg shown in Fig. 1.19 is described in Eq. (1.20), it is
proportional to the length of arc contact Lc ;%&'𝐿𝐿" = $&(' 𝑢𝑢* (𝑡𝑡)----- (8.5)

Therefore, with increasing the depth of cut u, the effect of Cg is increasing,and consequently the chatter does not occur. On the other hand, the chattertends to take place when u is small, for example, in spark-out process. Thistendency cannot be observed in metal cutting, and importantly is a
representative characteristic in grinding operation. This effect has been
understood that the grinding wheel contacts to the workpiece at a surface,which consists of the length of arc contact and grinding width, and thissurface suppresses the vibration in its normal direction. However, the actual
contact surface is not flat as shown in Fig. 8.2, but some plural points arecontacting to the workpiece. From such a viewpoint, grinding damping Cg
may be induced from the damping effect of binders. However, any
discussions on the damping effect of binder have not been carried out sofar.

8.5 Effects of Parameters Being Current Issues on ChatterIn order to describe the effects of previously shown parameters on chatter,both the stability limits for types I and II shown in Fig.1.19 are reproduced
herein. Characteristic equations for types I and II are described respectivelyas;2441 + {Gm(s) / km + 1 / kcon}{Cgs + kg (a-e-Tws)} = 0
Gm(s) / km + 1/Kg = 1 / Ka (1-e-Tss)
From these equations, asymptotic stability limits for both the type can beobtained as follows;
Re{1 / (Km(s) / Gm + Cgs) } = -1/(2Kg)
Re{Gm(s) / Km + 1/Kg} = -1 / (2Ka)----- (8.6)----- (8.7)

Fig. 8.11 Stability limits in Type I and II chatter vibrations
Figures 8.11 (a) and (b) illustrate the meanings of these expressions. Morespecifically, in Fig. 8.11(a), when the left hand side of Eq. (8.6) coincides
to right hand side of it, chatters take place under frequency ω1 or ω2. In caseof type II shown in Fig. 8.11(b), as well.
Since these characteristic equations and stability limits consist of theparameters Kcon, Kg, Ka and Cg mentioned above, the stability limits areaffected by these parameters. Supposing that the machining speed V is 10
times faster than previous amounts, the effects of these parameters on
chatter vibration will be discussed in the following.
On the effect of contact stiffness Kcon, it is experimentally shown that its
magnitude varies depending on the contact direction to the workpiece as

245described in Section 8.1. For example, in case of that grinding wheel
diameters are 100 to 300 mm, the grinding spindle have to be rotated in N
= 28,000 to 84,000 rev/min, i.e. 480 to 1,440 rev/sec, i.e. Hz, and these arewithin similar area of natural frequencies of grinding machine
constructions. Considering that Kcon varies with grinding wheel direction, it
induces the forced vibration synchronizing with spindle rotation. Fromsuch a viewpoint, the effect of Kcon is very important to discuss the forced
vibrations.
The grinding stiffness Kg shown in Eq. (8.3) is a function of specificgrinding force k, and so far investigated experimentally, and thus
qualitative amounts are obtained. It is known that this amount varies withgrinding wheel speed V, workpiece speed v, depth of cut u, wheel diameter
D, workpiece diameter d, successive cutting-point spacing a, and cuttingedge density n, and duly some experimental expressions were proposed. In
these parameters, a and n vary with the surface condition of the grinding
wheel. As shown in Fig. 8.2, the cutting edge distribution in grinding
surface varies randomly in grinding process. Furthermore, since the cuttingedge distribution is not uniform, Kg is also the vibration source as similar
as Kcon.
The wear stiffness of the grinding wheel Ka is a parameter concerning type
II chatter as shown in Eq. (8.7) and Fig. 8.11(b). In case of that it isconstant, the stability limit based on the chatter due to the regenerative
effect of the waviness on the grinding wheel surfaces can be induced.
However, considering the unevenness in the cutting edge distribution asshown in Fig. 8.9, and the sudden increase of the amplitude in grinding
wheel waviness as shown in Fig. 8.10, the stability limit for type II cannot
be explained only by Fig. 8.11(b).
In order to explain the stability for type II chatter developing relatively
long time as shown in Fig. 1.17, the wear process of grinding wheel and its
effect on the chatter of regenerative type have to be discussed.

References
Snoys R, Wang I C. (1968) Analysis of the static and dynamic stiffness ofthe grinding wheel surface. Preprint of the 9th MTDR Conference, p.1133.
246

Hoshi T, Koumoto Y. (1983) Mechanism of vibration in plunge-cut
cylindrical grinding. JSPE; 49-12: 1680.
Fukuda R, Tokiwa T. (1974) A study on the contact stiffness between
grinding wheel and workpiece. JSPE; 40-10: 809.
Izumi M, Lee H S, Inoue S. (2002) Development of an instrument for grain
cutting edges in a grinding wheel profile. JSGE; 46-2: 189.
Brown R H, Saito K, Shaw M C. (1971) Local Elastic Deflections in
Grinding. Annals of the CIRP; XVIV: 105.
Yamada T, et al. (2003) Study on elasticity of grinding wheels (Modelingand calcutating the stiffness of grinding wheels). JSME; 69-683: 1933.
Yoshikawa H. (1962) Criterion of between–dressing tool life of grinding
wheel. JSPE; 28-5: 286.
Inasaki I, Yonetsu S. (1970) Experiments on the grinding stiffness. JSPE;
36-3: 207.
Umino K, Tooe S, Shinozaki N. (1983) Study on irregularity of wheel
grade (1st report) Irregularity of wheel grade and grinding phenomenon.
JSPE; 49-6: 741.
Inasaki I, Yonetsu S. (1968) Surface waves generated on the grinding
wheel. Bulletin of JSME; 11-47: 922.
Ohno S. (1969) Study on self-excite vibration in cylindrical grinding.
JSME; 35-276: 1806.
Lee H S, Furukawa Y. (1982) On the dynamic grinding process undervibrating condition. JSPE; 48-5: 603.

247