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Dynamic stability in cutting
Citation for published version (APA):Kals, H. J. J. (1972). Dynamic stability in cutting. Technische Hogeschool Eindhoven.https://doi.org/10.6100/IR38767
DOI:10.6100/IR38767
Document status and date:Published: 01/01/1972
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DYNAMIC STABILITY IN CUTTING
H.l. 1. Kals
DYNAMIC STABILITY IN CUTTING
DYNAMIC STABILITY IN CUTTING
PROEFSCHRIFT
ter verkrijging van de graad van doctor in de technische wetenschappen
aan de Technische Hogeschool te Eindhoven, op gezag van de rector
magnificus, prof. dr. ir. G. Vossers, voor een commissie aangewezen door
het college van dekanen in het openbaar te verdedigen op vrijdag 19 mei
1972 te 16.00 uur
door
Hubert Jan Jozef Kals
geboren te Heerlen
Dit proefschrift is goedgekeurd door de promotoren
PROF. DR. IR. A.C.H. VAN DER WOLF
PROF. DR. P.C. VEENSTRA
aan mijn ouders
aan Leus
I
CONTENTS
INTRODUCTION
t.t. Background of the problem
1.2. Ristorical review
1.3. Situation of the problem
II THE MACHINE TOOL STRUCTURE AND ITS INTERACTION WITH
THE CUTTING PROCESS
2.1. The dynamic response of the machine tool
2.2. Simplification to a one-degree-of-freedom system
2.3. Structural damping in machine tools
2.4. Special toolholders used for experiments on dynemie
cutting
2.4.1. The Vanherck-Peters cutting stand
2.4.2. The influence of the moving bedslide on the
dynamic compliance of the taalholder
2.5. A model of the dynamic cutting process
III ON THE CALCULATION OF STABILITY CHARTS ON THE BASIS OF
THE DAMPING AND THE STIFFNESS OF THE CUTTING PROCESS
C.I.R.P. Ann. 19 (1971) 297.
3.1. Introduetion
3.2. The incremental cutting stiffness
3.3. Dynemie approach of the cutting process
3.3.1. General
3.3.2. Process damping and specific cutting stiffness
as basic quantities for stability charts
3.3.3.
3.3.4.
3.3.5.
Experimental approach of the problem
The calculation of ki
The influence of the wear of the tool on both
the process damping and the cutting stiffness
3.4. Conclusions
9
21
43
IV PROCESS DAMPING IN METAL CUTTING
Fertigung 5 (1971) 165.
4.1. Introduetion
4.2. Determination of the transfer function of the cutting
process
4.3. Experiments
4.4. Results
4.5. Process damping and its influence on the threshold
of stability
4.6. Conclusions
V THE CALCULATION OF STABLE CUTTING CONDITIONS WHEN TURNING
COMPLIANT SHAFTS
To be published.
5.1. An analysis of the vibration
5.2. The stability criterion
5.2.1. Orthogonal cutting
5.2.2. Introduetion of a cutting edge angle K + 90°
5.3. Experimental verification
5.4. Conclusions
VI DISCUSSION OF RECENT RESULTS FROM LITERATURE
Report WT 0287, Eindhoven Univarsity of Technology,
presented to the C.I.R.P. Teehuical Committee Ma, Paris
61
89
(1972). 102
6.1. Introduetion
6.2. Discussion of the results
APPENDIX I 112
APPENDIX II 113
APPENDIX III 114
SAMENVATTING 118
CURRICULUM VITAE 119
manuscript closed september 15th, 1971
I INTRODUCfiON
Nomenclature
b
b g
Width of aut
Limit value of width of aut
cc Coeffiaient for the damping of the autting
pPoaesa
ChatteP fPequenay
Relevant dynamia aamponent of the autting
foPae
6F Dynamia aamponent of the main autting forae V
6Ff Dynamia aomponent of the feed fopae
óh Chip thiakness modulation
h Nominal undefomed ahip thiaknesa 0
k1
Chip thiaknesa aoeffiaient
k2 Penetration aoeffiaient
k3
Cutting speed aoeffiaient
kd Dynamia autting aoeffiaient
K Penetration aoeffiaient
t Time
v Cutting speed
óvf Variation of the feed speed
y Defleation of the tool with respect to the
woPkpieae
a Angle between the p!'inaipal diPeetion of
motion and the diPeetion of ahip thiakness
modulation y
Q
60
Rake ang le 1)
Angula:r> tpequenay of workpieae or tool
Va:r>iation of angular frequenay
m
m
Ns/m
Hz
N
N
N
m
m
N/m
Ns/m
Ns/rad
N/m
Nrad/m
s
m/s
m/s
m
0
0
rad/s
rad/s
tool geometry is defined according to the recommendations
vf ~he C.I.R.P. Technical Committee c.
9
l.I. Background of the problem
During machining of metals, different kinds of vibrations
will occur in the machine tool structure. These vibrations
will lead to more or less periodical deviations in the cutting
geometry. Among other , such as the noise and the
increasing tool wear, the vibrations result in a wavy surface
of the workpiece and in this way the quality of the product
is impaired.
From their nature we can distinguish two major kinds of
vibrations, viz. forced vibrations and vibrations induced by
the cutting process itself.
In the case of forced vibrations a dynamic force is acting
which is independent of the vibration itself. These vibrations
can be caused by the machine tool, or come from outside being
transferred by the foundation. This type of vibrations may arise
from
irregularities in driving elements such as gear-wheels,
hearings, guide ways
- mechanica! and electrical unbalance
hydraulic devices such as gear pumps
- mechanical impulses.
Vibrations caused by the cutting process can be distinguished
into free vibrations and those which are self-induced.
"Free-type" vibrations generally are of minor importance,
because they will be damped in a very short period of time.
Hence the deflections of the tool with respect to the workpiece
are small. This type of vibrations can be caused by the shearing
process, the instability of the built-up edge and the inhamo
geneaus nature of the workpiece material. In this case the dynamic
cutting force is not well defined.
As distinct from the vibrations mentioned before, self
-induced vibrations are caused by a dynamic force, generated
by the vibration itself, which becomes extremely violent.
Characteristic for this type of vibrations is that the
frequencies are always approximately equal to the natural
frequencies of the machine tool structure. Commonly known as
10
"self-excited chatter", particularly this type of vibrations,
which is of a very complex nature, should be avoided.
1.2. Ristorical review
The first important investigations in the field of chatter
vibrations have been performed by Doi in 1937. He aseribes the
vibration to resonance due to fluctuations in the cutting
action synchronizing with a natural frequency of the lathe.
After World War II the first fundamental werk was done by
Arnold (1). He carried out his experiments under extreme
conditions, applying high Erequencies for a flexible tool. He
found chatter to be the result of the cutting force as a
function of the cutting speed showing a falling characteristic,
in cooperation with small variations of the cutting speed.
Chisholm (2) investigated the same type of chatter and re
ported also the self-excited vibration to be caused by the
deseending characteristic of the cutting force depending upon
the cutting speed.
In 1953 however, Hahn (3) showed that dynamic instabili
ty is also to be observed in materials which have no falling
cutting force characteristic and concluded the negative force
-speed relationship not to be the centrolling factor. The
latter was affirmed by Doi and Kato (4), (5). They consi
dered that chatter due to a flexible tooi and chatter due to
the deflection of the workpiece or the main spindie have the
same origin. They found that chip thickness variatien is
important as to the chatter phenomenon. They regarded the time
delay between the fluctuations in the feed force and the vi
bration with respect to the same direction as the feed force
being a fundamental effect, since the available energy for
chatter and the estimated energy for dissipation have approxi
mately the same value.
At the same time other investigators were working in this
field and developed separately two important theories, enabling
11
to establish the threshold of stability during cutting.
In the United Kingdom, Tobias and Fishwick (6) assumed
the dynamic force to be a function of three indepen-
dent factors according to
( 1.1)
Apart from the variables already mentioned, moreover they
considered the feed rate vf as a major quantity with respect
to the cutting force variation. The values of k 1, k2 and k3 can only be determined by dynamic experiments. Tobias and
Fishwick used eq. (1.1) in order to solve the differential
equation of an elementary vibratory system which stands for
the tool. In this way they found a stability chart showing
stable and unstable regions.
The Czech investigators Tlusty, Polacek, et al. (7)
distinguished two causes for the chatter phenomenon. On the
one hand they define the "mode-coupling" effect which arises
from two different directions of vibration resulting in an
elliptic path of the tooi point. On the ether hand they
consider the "regenerative" effect, which is caused by a
chip thickness variation. This variatien results from both
the instantaneous deviation of the tool with respect to the
workpiece and the surface of the workpiece which has already
been cut during the preceding machining operation. Starting
from a simple force relation
(I .2)
where kd was supposed to be a real number and consiclering a
vibratory system of two degrees of freedom, they solved the
equations of motion for both cases to find a polar stability
chart showing the minimum value of kd at which instability
may occur for different directions.
The last named investigators studied more in detail the
transfer function of the machine tool, whereas Tobias and
Fishwick studied this function of the cutting process.
12
Peters and Vanherck (8) succeeded in combining both
theories about chatter and presented a comple.tely graphical
salution of the problem which is of surprising simplicity.
It is obvious that in the theories mentioned before the
transfer function of the machine tool has to be known. In the
case of existing machine tool structures it is necessary to
use experimental methods, to determine this transfer function
(9), (10), (11). For constructions in the design stage
numerical methods are required since model techniques for
machine tool structures are of limited power. However,
although the development of the analytica! approach bas
accelerated considerably and the determination of the fre
quency response of low damped structures is quite possible
(12), up to now there is a lack of data on structural
damping. Moreover, the present knowledge about the dynamic
behaviour of guide ways and hearings must be improved in
order to be able to simulate in a dynamica! way machine
tools in werking conditions.
As far as the transfer function of the cutting process
is concerned, several investigators carried out dynamic
experiments trying to measure directly dynamic cutting forces
as aresult of periodic chip thickness variations. Although
their contributions to the solution of cutting dynamica may
be considered to be very important, in general the results
lead to two different conclusions. So, the u se
of chatter theories is limited by a detective knowledge of
the transfer function of the cutting process.
It is the aim of the present work to obtain a better in
sight in the mechanism of cutting dynamics.
1.3. Situation of the problem
It has been observed by Doi (4) that in the case of an
- with respect to the tool - oscillating workpiece, the force
variatien lags the vibration of the tooi. Experiments carried
13
out by HÖlken (13) showed also the dynamic cutting force
lagging the vibration of the tool. However, from the
measuring methods used by the investigator.s mentioned, it
can be concluded that in general the results were highly
affected by an outer modulation of the chip. This modula
tion caused by the previous cut shows a phase shift with
respect to the inner modulation. So, from the results of
Doi and HÖlken no important conclusions can be drawn.
Experiments carried out by among others Smith, Tobias
(14), and Kegg (15) revealed results which are contra
dictory to the conclusions of Doi and HÖlken, At low
frequencies the results of Smith and Tobias confirm those
of Doi, which have been obtained from experiments applying
an extremely low frequency of 1.5 Hz. For increasing
frequencies however, the lag of the cutting force with
respect to the vibration of the tool decreasas to zero,
changes sign and finally bacomes a phase lead (Fig. 1.1.).
Fig. 1.1.
14
8 ph os ongle
(o) 6
e
0
4
2 0
~ /0
mot , mild steel ho =0,127 mm v = 2.2 m/s
~ ~
~~ /
/0
LoF~· ----(/'0 ---::--u
OI;,/
0 -2 0 100 200 300 400 f (Hz)
Experimental results of Smith and Tobias (14)
concerning the phase relation between the dy
namic components of the cutting force - 6Fv and
6Ff- and the vibration of the tool, as a
function of the frequency of oscillation.
It bas also been observed that the phase lead of the cutting
force decreases with increasing cutting speed, This is in
agreement with assumptions made earlier by Tobias and
Fishwick (6), which lead to the explanation of an increasing
process stability for lower machining speeds according to
the equation
~F = kl ~h-! ~V + ,,, n f (I • 3)
In this equation, n stands for the rotational speed of either
the workpiece ar the cutting taal.
Kegg reported also a leading cutting force as a result
of a detailed study made on the same subj eet. Same of his
results are shown in Fig. 1.2. As can beseen from the figure,
Fig. 1.2.
0 phase i
0 angle(0 )'
g
5
3 0 4 t 0 i"
0
2) e-
o. 5 {;.
I
0.
0 . 2f----' .
100
a
4
/;
4 4
v=1.37m/s _ ho"'0.191 mm -
Y .e· mat, 81 B 45 45. 101im<Ah< BO.!dim a .o•
4 ·····-
/;
4
.
I • 200 300 400 500 500
f(Hz)
Experimental results showing the dynamia autting
aoeffiaient kd (modulus and argument) as a
funation of the frequenay of osaillation as
found by Kegg.
Kegg is able to measure the dynamic cutting coefficient. For
that very reason it is a pity that he only carried out measure
ments for one cutting speed.
IS
Starting.from results concerning the influence of the
clearance angle upon the dynamic cutting coefficient, Kegg (16)
also developed a theory for low speed stability which is
partly analogous with Tobias' theory. Different from the assumptions made by the latter, Kegg supposed the cutting
speed itself to he the basic quantity and assumed the out
-of-phase component of the dynamic cutting force being inver
sely proportional to the cutting speed. The reason for this
is that in Tobias' way of thinking, for certain chatter
situations referred to as "digging-in", the orientation of
the depth-proportional and rate-proportional farces are forced
to act in the same direction. This should result in a decreasing
stability at lower speeds which is never reported for practical
machining operations. Thus, Kegg applies
liF = k t.h- ~ I V dt (1.4)
Concerning the cutting process damping value c , the next two c
requirements are left. Firstly, it must depend upon material
strength, but for a glven material strength it must vary neither
with the cutting speed nor with the feed. Secondly, it must
strongly depend on the cutting tool clearance angle. As for a
number of quantities mentioned in this chapter, there are no
reliable data available for c • c In this stage of progress about the salution of dynamic
stability, some experimentally orientated cooperative work has
been started within C.I.R.P. (17). The aim of this work was to
measure the critical depth of cut under well defined working
conditions in order to be able to compare materials on chatter
susceptibility. Using a special toolholder, most investigators
found the limit width of cut as a function of cutting speed
passing through a minimum as can beseen in Figs. 1.3., 1.4.
At about the same time two new theories reported a
relation between the static and dynamic parameters in the
cutting of metals. Starting from a shear plane model, Das,
Tobias (18), (1,9), and Knight (20) used the so called
Universa! Machinability Index for predicting cutting stability.
16
8 bg
(mm)
-----Eindhoven Unlver1lty of Technology
==n~:!: ....
Fig. 1.3.
bg (mm)
Fig. 1.4.
($
4
2
~\ I I
~I ,' \ " ... \
\, I I I' I
.......... \ ~-...... _
I I !' \ ·"' .I J ", ''f.< ' ~ ' '·' '., ",
I ~ 1.0 ~ 20
v(mls)
The limit width of cut b versus cutting speed g
v as found by several investigators applying the
C.I.R.P. cutting stand and using the same work
materiaL
2
The Eindhoven results concerning the limit width
of cut versus cutting speed v for severaZ
kinds material applying the C.I.R.P. cutting
stand.
2.5
40 v (ml•l
17
Peters and Vanherck (21) found the incremental cutting stiffness,
which can be obtained from static cutting tests, suitable
for computing stability charts. Where both, Knight and Peters,
are able to explain only a small increas.e of the limit value
at higher cutting speeds, the results of the cooperative work
cannot be explained either by them or by any other of the
investigators mentioned.
Using the special toolholder mentioned, some investigators
(22) found the chatter frequency varying more ore less with
the limit width of cut. (See Fig. l.S.) On the other hand,
30 2 f
(Hz 2
)
-~·
21
"''
1
17
16
I 0
Fig. 1.5.
•-• limit value bg •-• frequency (b•llg> B-El fre uency (b•Clmm)
~
\· C5
\ El .. .}_, ~
' V !: "-i ' ---~ -"· ''I
~ -~ 1-1 _ _,
4
3
2
I
0 1.0 15 2D 2.5
v (m/s)
The chatter frequency f on the threshold of
stability, and for a constant width of cut b,
versus cutting speed v.
Van Brussel and Vanherck (23) concluded from their results
to a constant chatter frequency. However, platting the
results and relating them to zero, they easily can ignore
an important change in frequency and consequently in process
damping. Moreover, revealing the dynamic cutting stiffness
being caused by the direct chip thickness modulation, they
find this quantity to be out of phase with tool deflection.
18
From a physical point of view this can only be described by
process damping.
Some recent work on the same subject carried out by the
present author shows the change in frequency to be essential.
From this a new method was developed to measure the process
stiffness and the process damping. Chapter III will deal with
this method (24).
References
(I) Arnold, R.N., Proc. Inst. Mech. Engrs. 154 (1946) 261.
(2) Chisholm, A.J., Machinery 75 (1949) 51.
(3) Hahn, R.S., Trans. A.S.M.E. 75 (1953) 1073.
(4) Doi, S., Memoirs of the Faculty of Engineering.
Nagoya University. 5, No. 2 (1953) 179.
(5) Doi, S. and Kato, S., Trans. A.S.H.E. 78 (1956) 1127.
(6) Tobias, S.A., Fishwick, W., Der Maschinenmarkt No. 17
(1956) l.'i.
(7) Tlusty, J., Polacek, M., Beispieleder Behandlung der
Selbsterregten Schwingungen der Werkzeugmaschinen.
3. FoKoMa, Vogel Verlag, Coburg (1957).
(8) Peters, J., Vanherck, P., lndustrie-Anzeiger No. I I
(1963) 1~8 and No. 19 (1963) 342.
(9) Rehling, E.R., Entwicklung und Anwendung elektrohydrau
lischer Wechselkrafterreger zur Untersuchung von Werk
zeugmaschinen. Doctor's thesis, T.H. Aachen (1965).
(10) Van der Wolf, A.C.H., The development of a hydraulic
exciter for the investigation of machine tools.
Doctor's thesis, Eindhoven University of Technology
(1968).
( 11) Knight, W.A., Sadek, M.M., Tobias, S.A(, 11 th Int.
M.T.D.R. Conference, Manchester (1970).
(12) KÓenigsberger, F., Tlusty, J., Machine Tool Structures,
Vol. I., Pergamon Press, Oxford (1970).
(13) Opitz, H., HÖlken, W., Untersuchungen von Ratter
schwingungen an Drehbanken, Forsch. Ber. Laudes NRhein
Westf. (1958).
19
(14) Smith, J.D., Tobias, S.A., Int. J. Mach. Tool Des. Res.
I (1961) 283.
(IS) Kegg, R,L., A.S.H.E. paper No. 64 WA/Prod.-11 (1965) 283.
(16) Kegg, R.L., C.LR.P. Ann. 17 (1969) 97.
(17) Peters, J., Vanherck, P., Report on a new test-rig to
carry out comparative tests of "susceptibility to chatter"
of materials. C.I.R.P. Report presented to Group Ma,
University of Louvain, 19 sept. (1967),
(18) Das, M.K., Tobias, S.A., Proc. Sth Int. M.T.D.R. Conference
(1965) 183.
(19) Das, M.K., Tobias, S.A., Int. J. Mach. Tool Des. Res.
7 (1967) 63.
(20) Knight, lv.A., Int. J. Mach. Tool Des. Res. 8 (1968) I.
(21) Peters, J., Vanherck, P., C.I.R.P. Ann. 17 (1969) 225.
(22) Tlusty, J., Koenigsberger, F., Specificatien and tests
of metal cutting machine tools. Proc. of the Conference,
19th and 20th Febr., The University of Manchester
Institute of Science and Technology (1970).
(23) Van Brussel, H., Vanherck, P., IJth Int. M.T.D.R.
Conference, Manchester (1970).
(24) Kals, H.J.J., C.I.R.P. Ann. 19 (1971) 297.
20
II THE MACHINE TOOL STRUCfURE AND lTS INTERACTION WITH THE CUTIING PROCESS
Nomenclature
a.. Real part of dynamic complianee. The suffix i lJ denotes the direetion in which the defleation is
measured, j denotes the direetion of the excita-
tion force m/N
b.. Quadrature component of dynamie eomplianae lJ
(For the suffix, see a .. ) m/N lJ c Structural damping coefficient Ns/m
Coefficient of the damping in the autting proaess Ns/m
c.o.m. Coeffieient of merit N/m
E Young's modulus of elasticity N/m2
m
Friction force
Coulomb friction force
Dynamic component of the cutting force
Amplitude of the excitation force
Structural stiffnees F
Ratio of motion defined as c ~ thicknees eoefficient 0
Maas
q Amplification factor
R n
Maximum negative in-phase component
Maximum in-phase component of the transfer
function of a single-degree-of-freedom
system
R_ Maximum negative in-phase component of the
transfer function of a single-degree-of-
-{reedom system
Time of revoZution of the workpiece
Directional
Direction of exaitation
N
N
N
N
N/m
N/m
kg
m/N
m/N
m/N
s
21
v NominaZ carriage speed 0
w Direction of exeitation
x8
t Static defleetion
x Amplitude of displacement
x Amplitude of displacement at natural frequeney 0
x1
Half peak-to-peak value of displacement
X Amplitude of velocity at naturaZ frequency 0
y Deflection of the tooZ perpendieular to the
cut surface y Peak value of y
Delayed chip thickneas modulation
a Angle between the direction v, respectively v,w w, and the direction of the chip thiakness
modulation
6 Angle between the direction v, respeatively v,w w, and the direction of the dynamic eutting
forae
Fraation of aritiaal damping at natural
frequenay ( = 2r;)
Damping ratio F
Coulomb damping ratio, defined as ;(f 2V • mk}
Cutting edge angle o
A Logarithmie decrement
v Dimensionless frequency
~ Phase angle between force and tool
displacement
w Angular frequenay
w0
Undamped naturaZ frequenay
wn Angular chatter frequenay on the threshoZd
of stability
wR_ frequenay aorresponding with R
22
m/s
m
m
m
m
m/s
m
m
m
0
0
0
rad
rad/s
rad/s
rad/s
rad/s
2.1. The dynamic response of the machine tool
In order to predict the machining conditions giving rise
to chatter, first the dynamic behaviour of the machine tool
structure has to be investigated. The structural characte
ristics are required in terms of the relative displacement
between the cutting tool and the workpiece as a response to
a force acting between them. Although the knowledge of the
influence of the various structural elements on the machine
tool receptance is required for the impravement of a
machine structure with respect to chatter, when computing
the threshold of stability'of an existing tool, only the
overall receptance has to be known.
For cutting conditions, where the dynamic cutting force
may be considered to act always in the same direction, the
method of measuring the relevant receptance is rather straight
forward. In this case, it is sufficient to measure the rele
vant displacement between tooltip and workpiece in the
direction perpendicular to the cut surface, whilst the
machine tool is excited sinusoidally by a force acting between
the same elements in the direction of the dynamic cutting
force. The locus of the harmonie response is obtained by
recording the in-phase and quadrature components of the dis
placement for each frequency on an XY-plotter (1).
A difficulty arises due to the fact that the direction of
the dynamic cutting force, and in some cases also the normal
to the cut surface, varies between certain limits. As will
be shown later, both the cutting speed and the feed influence
the direction of the dynamic cutting force in a physical way.
Moreover, when horizontal milling for instance, both
directions mentioned can vary depending on the depth of cut,
tool geometry, and whether the process is up or down milling.
But it is quite adequate to measure the direct and cross
receptance loci of the system for any two directions v and w
enclosing an angle which is preferably chosen to be 90°. From
these receptances the cross receptance between any two
arbitrary directions ~F and y can be found. If the direct
23
and cross receptances in the directions v and w are
represented by (avv + i bvv), (aww +i bww)' (avw +i bvw)
and (awv + i bwv), the real part of the opeFative recep
tance (ayF + i byF) is given by
+ avw cos ~v cos Sw (2. I)
See also Fig. 2.1.
V
Fig. 2.1. Geometry of the operative receptance.
Applying Maxwell's theorem and introducing the directional
factors
u cos ~ cos B (2.2) WW w W'
u cos ~ cos sv (2.3) wv w
uvv cos a cos (2 .4) V
u == cos a cos Sw (2.5) vw V
it follows
(ayF + i b F) = a u + a (u + u ) + a u + y ww ~vw vw vw wv vv vv
+ i [ b u + b (u + u ) + b u ] WW WW VW VW ·wv VV VV
(2. 6)
24
In Chapter IV it will be shown that, from the point of
view of chatter, only the particular part of the receptance
locus which corresponds to phase angles larger than 90° is
of interest. Hence, only the negative values of ayF are of
importance. Experience shows that the conditions are at its
worst when the system happens to operate in point C of the
curve of Fig. 2.2.
Fig. 2. 2.
1 cm • 4,6 11m/kN
39
i f 1 ~cm ___ JRnJ __ -1
out·of·phaH axl•
F
The oper>ative 1~eoeptanae of a ver>tioaZ di>iUing
machine make llettner, type Il.R.ll. 50/li:JOO.
The so called coefficient of merit is defined according to
c.o.m. 2
where R is the real part of the locus in point C. n
(2. 7)
25
The greater the value of the c.o.m., the better the
resistance against chatter of the machine under the
conditions investigated.
2.2. Simplification to a one-degree-of-freedom system
For experiments aften special tools are used which
approximate a single-dl[!gree-of-freedom system. Bearing
in mind that the harmonie response locus of a system of
one degree of freedom with hysteretic damping is a circle,
its centre lying on the out-of-phase axis (2), it is easy
to understand that in the case of a low-viscously damped
system with ene degree of freedom the response locus
approximates an are of a circle in the region of resonance.
In this way it is aften possible to simplify the description
of the operative response of a machine tool by replacing it
by one or more equivalent systems which are characterized
by a circular are (3).
With m the mass, k the static stiffness and c the
coefficient of viseaus darnping, the harmonie response of a
single-degree-of-freedom system can be written as
x 1 1
~ k 1 m 2 i c w - k U) + k (2.8)
We define
w =~ 0 (2.9)
c w 0 ~ 2 ç __ o_ =
k q (2. 1 O)
w v~-
w (2. 1 1.) 0
w'hen substituting eqs. (2.9), (2.10) and (2.11) respectively
in eq. (2.8) we find for the real part of the dynamic compliance
26
(2. 12)
The extreme values can be determined when putting
[
aRe {; } ]
av I dk = do
0
0
(2. 13)
This leads to
(2. 14)
and subsequently for the roots
± 6 (2. IS)
Thus, it results for the extreme values
[Re {; }] I I k ö (2 - o)
I (})2
(2. 16)
[Re{;}] I I =k ö (2 + ó)
I (v2) I
(2.17)
From the ratio
(2. 18)
and from eq. (2.10) it fellows for the damping ratio
(2. 19)
From eq. (2.15) a different relation, also giving the damping
ratio, can be obtained. Therefore, we define
27
2 (2.20)
In this way eq. (2.15) yields
(2.21)
Using the eqs-. (2.19) and (2.21) it is possible to
check whether a structure can or cannot be considered as
a single-degrèe-of-freedom system.
Eq. (2.21) can be written as
(2.22)
Actually, this equation implicates the limit conditions on
the threshold of stability during machining as will be shown
later.
2.3. Structural damping in machine tools
Three different causes for damping forces acting in
machine tooi structures can be distinguished:
damping inherent to the material
- viscous damping and damping due to frictional forces
caused by guide ways and hearings
- friction forces acting in joints.
Material damping is caused by hysteresis. Of this damping it
is known that it depends upon the E-value and decreases with
increasing value of E. However, this type of damping accounts
for only 10- 20% of the total structural damping (4). A more
substantial contribution to structural damping is made by guide
ways and hearings. These parts being lubricated, the damping
introduced by them should be preponderant viscous. As a result
of the low speeds between the mating parts of a guide way, as
normally is the case for machine tools in cutting conditions,
28
however, the viscous type damping becomes merged with Coulomb
friction. The Coulomb friction becomes of great importance
when the velocity amplitude of the vibration exceeds the nomi
na! sliding speed.
kX 5 --• F1 lOkW lathe
41--------l.mak e LAI\IGE -1----1----#-+ type L 8
F1: SIG N
Vo (mmls): • : 0.36 "' 0.72 0: 1.44 x 2.88 c: 4.32
3~--~---1----l----l--~1--~1~---l----4 v· 7.20
Fig. 2. 3. Resonanoe euYves of a bedstide for severat
vatues of V0
.
m"" 140kg lil 'l:<580rad/s.
Fig. 2.3. shows some experimental results on the dynamic
behaviour of the moving bedslide of a medium size lathe. The
computed results of an analogue model (5), where Coulomb
friction as well as viscous damping is taken into account,
are given in Fig. 2.4 •. Fig. 2.5. shows the analogue model.
From Fig. 2.4. it fellows that the influence of Coulomb
friction increases with increasing values of the quantity
kw= F1/(cV0)r'The quantity V
0 represents the nomina! sliding
speed, F1 is the excitation force and c stands for the vis
eaus damping. Actually, the dynamic behaviour of the system
will not be influenced by friction if kw< I. In this case
29
5 t = 0.1 k x,
tw= 0.4 F, 4
F1 kw•-eV0
3 tw·aim
0
t. 2rmt
Q2 Q4 0.6 1.4 I 1.6
Fig. 2.4. Resonance curves of an analogue model for
various values of the ratio of motion kw.
I
·~=96N - 11 =48~ "..,..-/. v• v· / 4
I /. - ~~· !/ V tran
headstoek ~R 3 I re ,- eod·se -
2 I V i! -
{I 1 ,--.. • kw• 1 kw• 1
~48N) ~ISN)
0 '( 1 2 3 4 5 6 7
Fig. 2.6. The half peak-to-peak-value to force ratio
kX 1/F 1 at natural frequency (v = l) versus
carriage speed V0
for two values of the
e:xaitation force.
8 9 1 0 _V0 (mm/S)
30
it holds for the amplitude of velocity at natural frequency
X < V • This is confirmed by the experimental results shown 0 0
in Fig. 2.6.
Fig. 2.ó.
oo. ~ 13. ~ kkw tv 10tw
02· 0.2" 14· ~ 100tw
Q3, 2tv 15· 1
2kkwtw 06· 19 23} as required for
27 31 ompl. limits 10. 0.2"
_h_ 21· 0.002 12·
10tw
Bleekdiagram and analog"tte 1110de1. of the moving
~e inctuding Coulomb f:r.'i.cti.on.
31
The proof that for k = I the relation.X /V = I is valid w 0 0
can be seen from the next derivation. X0
being the amplitude
of displacement at natural frequency and q being the ampli
fication factor at resonance, it fellows for the static
deflection xst (see eq. (2.10.))
x -lx st q o
(2. 23)
Hence the amplitude of the harmonie force F1 is covered by
Fl k xst 2 l; k xo (2.24)
or
c{fx c x m o 0 (2.25)
Finally it yields
Fl x k 0
w c V v (2.26) 0 0
However, since eq. (2.10) is only valid in the case of pure
viseaus damping, eq. (2.26) will hold only for kw (I.
Although the influence of Coulomb friction on the amplitude
of vibration is considerable, one has to be carefull when
estimating its influence on chatter, since the vibration is
non-linear.
Generally, the relative high damping introduced by
guide ways makes the dynamic compliance helenging to machine
parts supported by means of guide ways of minor importance.
However, the dynamic properties of slides can lead to
stick-slip. For the description of this phenomenon it is
not allowed to isolate its dynamic mechanism from the other
parts of the machine tooi structure. Because all parts of
machine tools show a certain compliance, during machining
they will vibrate and introduce small inertia farces acting
on the sliding system. These dynamic farces will excite the
guide-way systems since the driving spindies or other
driving systems will take care of the nominal driving farces.
32
In some cases this behaviour may introduce instability of
the cutting process. A similar dynamic behaviour may be ex
pected for hydraulic drives.
The damping introduced by joints will rather be dry
friction than be of viscous nature. This can also be observed
for a workpiece clamped in a chuck. When the compliance of
the workpiece exceeds that of the machine tool structure, the
dynamic stability will strongly be influenced by the contact
damping between the shaft and the clamping devices. This
damping, which can even exceed the usual values known for
machine tools, will depend on the micro-slip in the clampings
(10). It is obvious that the damping depends on the compliance
of the workpiece and it can be understood that the less stiff
a shaft is, the more independent the stability conditions are
of both the machine tool and the clamping devices.
Generally the dynamic behaviour of joints will approximate
that of guide ways. Th~ main difference is that the damping
introduced by them will be smaller. An experimental study in
the field of joints has been made by Thornley and Koenigsberger
(6). It was concluded that the damping decreases with increasing
prelead and increases by the presence of oil or greases in the
joint interface. Next it was established that rougher surfaces
introduce a slightly higher damping than smooth surfaces.
Loewenfeld (4) carried out experiments measuring the
logarithmic decrement after excitation of machine tools
as a whole, as well as that of the single elements. Some
of his results are given in the Figs. 2.7. and 2.8.
A conclusion, which has also been drawn by
Peters (7), is that ~-values for machine tools rarely
exceed 0.03. So, as a rule, damping in machine tool
structures is quite low. In this context, ~ represents
the damping ratio of an equivalent system of one degree
of freedom, which is supposed to substitute the relevant
part of the receptance locus of the machine tool.
33
Fig. 2. ?.
~.~-------r-------r-------r-------r------~
A
Bed Bed• bedslide
Bed • Becl+bedslide+, Complete heodstoek headstock+ machine
foot
The contribution of the single elemente.to
machine tool damping.
Q3r-----r-----~~~----~----~----~
A
Fig. 2. 8. The damping of maahine toole aompared to their
aasted elemente.
2.4. Special toolholders used for experiments on dynamic
cutting
34
2.4.1. The Vanherck-Peters cutting stand
The dynamic experiments, being dealt with in the
Chapters III and.IV, have been carried out with special
toolholders. The design of these toolholders originates
from Peters and Vanherck (8), who proposed a special test
for comparative tests within C.I.R.P. on susceptibility
to chatter of materials. (See Fig. 2.9.) Approximating
Fig. 2.9. The C.I.R.P. cuttir~ stand.
a system with one degree of freedom, the tooiholder consists
of a mass which is linked to a base plate by two leaf
springs. An adjustable damper is inserted between the mass
and the base plate.
In the beginning of the experiments the direction of
the principal degree of freedom, i.e. the direction perpen
dicular to the leaf springs, has been chosen at 32°30' with
the horizontal. In this direction the dynamic cutting force
was supposed to have its optimum action.
In order to reduce the influence of its vibration a tool
shank with 25 x 25 mm2 cross sectien is used. The natural
35
frequency of the toolholder, being about 150Hz, is lying
within the range which covers small and medium size machine
tools (100 + 200Hz). To diminish the influence of the dynamic
properties of the lathe on which the test rig is mounted, the dynamic stiffness of the tooiholder is chosen to be low.
Moreover a tool with K = 90° is used, which makes the
radial component of the cutting force negligible. This
reduces the influence of the bending flexibility of the
head spindie of the 1athe to a minimum.
One of the toolholders used in this work is similar
to the rig above. The secoud one differs only by the
direction of the principal mode which coincides with the
horizontal direction.
2.4.2. The influence of the moving bedslide on the dynamic
compliance of the tooiholder
In order to carry out the experiments on the dynamic
cutting data, the toolholders mentloned have been mounted
on a 10 kW-lathe, make Lange. Although the dynamic stiff
ness of this lathe is high with respect to the toolholder,
under werking conditions the compliance of the latter is
affected by the moving carriage (9).
Fig. 2.10. shows the receptance loci of a tooiholder
for several values of the carriage speed and for two
different values of the excitation force. From the figure
it eau be seen that the damping increases with increasing
carriage speed. For the same value of F1/V0
the curves show a
very similar course. This is entirely in agreement with the
influence of the frictional coefficient kw' which bas been
described in Sectien 2.3.
The explanation for the changing compliance of the tooi
holder is that the moving bedslide acts as an auxiliary mass
damper. Thus it will also be clear that the influence of the
carriage speed on tooiholder receptance decreases with
increasing value of the damping of the toolholder. With respect
36
to this Fig. 2.11. represents some experimental results.
Summarizing one can conclude that for V0
= 0 the harmonie
forces between carriage and frame are not sufficient to
exceed the Coulomb friction forces in order to cause a
relative displacement. The coupling between carriage
and frame is rigid. So, iwith respect to the toolholder,
no damping action will be added.
o--o:v0 .o mm/s. Cl=144.6Hz. ed58Hz. t.-LI:V0•0.43 mm/s . .t.= 145.1 Hz. ~>•170Hz. c-o:v0 .Q.66 mm/s. 11:146.7 Hz. • .170Hz.
v-v: V0 =1.15 mm/s. v ·148.6 Hz. •· 17 1 Hz.
F,. 25N.
o-o: Vo•O mm/s. t>• 1446Hz. •· 158Hz. t.-t.: Vo•0.14 mm/s. •· 146A Hz. •· 17-::l Hz. D-C:V.,.OA3 mm/s. 11• 148.4Hz. •= 170Hz.
M.P., 100 120 130 140 150 160 180 200 HZ. Scale:0035 !Jm/N/dtv. Bedslide driven by leodscr-.
Fig. 2.10. The inf~uenae of the aarriage speed V0
on the
transfer funation of Vanherok's too~holder for
~o vaZues of the exoitation forae F 1•
37
Q4
~ ~~ r---- x~ -- Vo Fi
--_:::_;:.:::._77?7»'7777~ fr
-Q2 ~14
Ql Z::_t•Q35
CS4 128 192 255 320 384
Fig. 2.11. The amplitude to force ratio of the reZative
displacement between tooU1oZder and carriage
l<x 1- x2)/F 11f versus carriage speed for
VaPious values0 of the damping of the toclholder.
If, however, the carriage is moving by the action of
the lead-screw or the screw-spindle and even though the
amplitude of an excitation force is small, this dynamic
force will cause vibration of the carriage.
If the speed is high, the coupling between
carriage and frame is viscous and almost independent of
the carriage speed. In this case the damping between
carriage and frame shows a minimum and thus the absorbing
action on the vibration of the tooiholder reaches its
maximum.
V0 (l'm/s)
In the case of the velocity amplitude of the carriage
being equal or larger than the nominal speed V0
, the
coupling between carriage and frame is periodically rigid.
Because V0
is small, the damping between frame and bedslide
will be rather Coulomb friction than viscous. This results
in a decreasing vibration amplitude of the carriage with
decreasing carriage speed.
38
In common practice we can divide the velocity range in
:wo parts.
For carriage speeds exceeding the velocity amplitude of
the vibration, the compliance of the tooiholder will not
depend upon the carriage speed.
For lower values of the feed rate, however, we have to
take into account the change in dynamic response of the tool
holder.
2.5. A model of the dynamic cutting process
Starting from a single-degree-of-freedom system, we
assume that the cutting process adds damping and stiffness
to the equivalent quantities of the structure. As a first
approach the orientation of the principal direction of
motion of the structure is chosen in the direction of chip
thickness modulation, i.e. perpendicular to the cut
surface.
In the case of self-excited vibrations caused by
undulations on the workpiece, the undeformed chip thick
ness is a result of the instantaneous deflection of
the tool and the ordinate of the workpiece surface, which
has been generated the previous cut. Thus, in the
dynamic cutting model a time de1ay has to be introduced.
In the case of turning, the time delay is identical to
the time for one revolution of the workpiece Tr.
When we suppose the real component of the dynamic
cutting force to be proportional to chip thickness roodu
lation and the process damping to be strictly proportional
to the vibrational speed, the model can be represented
diagrammatically according to Fig. 2.12.
k1
the process stiffness, i.e. the chip thickness
coefficient, cc the process damping coefficient, y the direct
chip thickness modulation and y* the delayed chip thickness
modulation, the differential equation of the motion becomes
39
cuttin machine tooi
Fig. 2.12. A simplifi~d model on autting dynamias.
where
y'" Re{Y exp(i w t)}
k y* I (2.27)
(2.28)
Bearing in mind that the surface undulation y* shifts the
deflection y by an angle ~. it holds on the threshold of
stability
y* '" Re { Y exp[i (w t - <!~)]} (2.29)
Consequently it follows for the equation of motion
- m wn2 + i (c + cc) w
0 + (k + k 1) '"k1 exp(- i <f!)
(2.30)
or
w .. n
In this equation. oon stands for the angular frequency
during cutting on the threshold of stability.
40
(2.31)
From eq. (2.31) it fellows
k + k 1 (I - cos ~)
and
m w n
2 0 (2.32)
0 (2.33)
The latter equation contains the limit value of k1
,
i.e. the lewest positive k 1-value which may cause chatter.
This value occurs when ~ = Î rr + 2 rr p (p =I, 2, 3, •••• ).
It yields for the limit value
(2.34)
The chatter frequency at the limit conditions can be
derived from equation (2.32) according to
lll = ~ k + kl n m
Raferences
(1) Van der Wolf, A.C.H., The development of a hydraulic
exciter for the investigation of machine tools.
Doctor's thesis, Eindhoven Univarsity of Technology
(1968).
(2) Bishop, R..E.D., J. of the Royal Aeronaut. Soc. 59
(1955) 738.
(3) Tobias, S.A., Machine tool vibration. Blackie & Son,
Glasgow (1965).
(4) Loewenfeld, K., Der Maschinenmarkt, Nr. 10 (1957) 11.
(5) Hoogenboom, A.J., Some dynamicaspects of the Cou
lomb friction combined with relative velocity.
Report WT. 0248, Eindhoven Univarsity of Technology
(1970).
(6) Thornley, R.H., Koenigsberger, F., C.I.R.P. Ann. 19
(1971) 459.
(2.3"5)
41
(7) Peters, J., Proc. of the 6th Int. M.T.D.R.·Conference,
Manchester (1965) 23.
(8) Peters, J., Vanherck, P., Report on a new test rig to
carry out comparative tests of "Susceptibility to
chatter" of materials. C.I. R.P •. Report presented to
Group Ma, University of Louvain, 19 sept. (1967).
(9) Kals, H.J.J. and Hoogenboom, A.J., The influence of the
carriage speed on the compliance of the toolholder.
Report WT 0227, Eindhoven University of Technology.
Note presented to the C.I.R.P. Technica! Committee M~
(1970).
(10) Lindström, B., C.I.R.P. Ann. 20 (1971) 5.
42
lil ON THE CALCULATION OF STABILITY CHARTS ON THE BASIS OF THE DAMPING AND THE STIFFNESS OF THE CUTTING PROCESS
Abstract
This chapter d.eals with a new method for calculating
stability charts. 1) Simple experiments, based on frequency
measurements only, yield the data of the workmaterial neces
sary to establish the threshold of stability. From this the
dynamic cutting coefficient can be determined. A close
agreement between the calculated values and the experimental
results is shown for cutting speeds exceeding I m/s.
Nomenclature
A Constant
A /A Amptitude ratio n o b Width of eut
b g
c
Limit value ofwidth of eut
Struetural damping eoeffieient
cc Damping eoeffiaient of the eutting proaess~
m
m
Ns/m
related to the main direetion of motion Ns/m
cmt Equivalent struçtural damping aoeffiaient
defined as c t = 2 1;, t/(m k ) Ns/m m m m Overall damping eoeffiaient of the maehining
system Ns/m
Frequenay of the system pulse response duFing
eutting Hz
1) A more detailed description can be found in "Stabiliteit van de
verspanende bewerking". Dictaat nr. 4.024, Eindhoven University
of Technology.
43
fmt Frequency of the system pulse response without
cutting, but with moving earriage Hz
F Cutting force N
Ff Feed force N
Fv Main cutting force N
bF Dynamic component of the eutting force N
1:1F f Dynamie component of the feed force N
&F Dynamie component of the main cutting force N V
h Nominal undeformed chip thickness m 0
bh k
kd
k. l.
k m
Chip thickness variation StrueturaZ stiffness
Dynamic eutting aoefficient
Speaifie process stiffness
Equivalent stiffness of the machine tool
strueture in working aonditions
kst Ratio between the increments of cutting force
and thickness per unit of width of cut
k1
Chip thiekness coeffiaient
k2 Penetratien coefficient
k3
Cutting coeffiaient
1:1k Inerement of stiffness
m Maas
n Number of periode
R In-phase component of the reaeptance locus of
the structure
R n
T c
T m
V
VB
y
'l'oo l tip radius
Maximum negative in-phase component
function of the cutting process
Transfer function of the machine tool
Cutting
Width of the flank wear land of the tool
Instantaneous deflection of the tool
a Angle between the principal direction of motion
and the direetion of the chip thickness modula
tion; clearance angle of the cutting tool
Angle between the dynamia autting force
and the principal direction of motion
44
N/m
N/m 2
N/m
Ns/m
Ns/rad
N/m
kg
m/N
mm
m/N
N/m
m/N
m/s
mm
m
0
0
y Rake angZe
ç Damping ratio of the strueture
çmt Damping ratio of the strueture in working
eonditions (b • O)
À
w c
Damping ratio of the syatem during eutting
Cutting edge angZe
Minor edge angle
Cutting edge inelination AnguZar frequeney of system puZse response
du.ring eutting
Damped natural angular frequeney
wmt AnguZar frequeney of system pulse response
without eutting, but with moving
wn NaturaZ angular frequeney of the whole
maehining system
w0
Undamped naturaZ angular frequenay
w Undamped natural angular frequeney of the om tooZ in working eonditions (b 0)
6n Variation of angular frequeney workpieee
or tooZ
3.1. Introduetion
0
0
0
0
rad/s
rad/s
rad/s
rad/s
rad/s
rad/s
rad/s
There are two current ideas in the field of performing
dynamic stability tests of machine tools (l).
The first method is characterized by measuring the
transfer function of the machine tool. The critical depth
of cut is obtained by using Tlusty's equation (5) 2 )
b B __ ;__
g 2 (3. I)
The quantity kd is called the dynamic cutting coefficient
which depends upon the cutting conditions. Rn is the maximum
2 ) A derivation of this relation is given in Chapter IV.
45
real part of the polar curve, showing the dynamic
compliance of the machine tool as a function of frequency.
The secend methad simply consists of carrying out
experiments in order to establish the critical depth of cut
for standardized conditions.
The progress in the investigations concerning cutting
stability is mainly hampered by an insufficient knowledge of
the kd-value. This value depends on many quantities such as
feed, cutting speed, tool wear, geometry of the tool and
workpiece material. The influence of the various parameters
on cutting stability makes it difficult to campare results
obtained from either of the methods.
3.2. The incremental cutting stiffness
Peters and Vanherck (2) assume that it is allowed to take
the incremental cutting stiffness ki for the kd-value already
mentioned. Thus, they calculate the critica! depth of cut
applying the relation
(3.2)
,The numerical values of ki are obtainable from static cutting
Fig. 3.1.
46
Determination of the irwrementat outting stiffness
ki according to the methad of Peters and Vanherck.
tests. Fig. 3.1. shows, in the case of orthogonal cutting,
a change 8F of the resultant cutting force due to an increase
8h of the chip thickness. The incremental cutting stiffness
is defined as
(3. 3)
where
(3.4)
and B represents the angle between the vector 8F and the
direction of motion of the tool. It is clear that in this way
the dynamic cutting coefficient introduces no phase shift.
Peters and Vanherck compared the calculated b -values with g
experimental data obtained by using a special tool holder (see
Chapter II). A fairly good agreement was found. However, experi
ments carried out in the Labaratory of Production Engineering
of the Eindhoven University of Technology, applying the same
tool bolder, did not confirm the reliabil of the methad to
the same extent (3) .. Our results are shown in Fig. 3.2. In
general, the calculated b -values are considerably smaller g
than the experimental data. Appendix I deals with the experi-
mental set-up for measuring the limit width of cut.
Among other things, to be explained later, Fig. 3.3. shows
the curves for ki according to the metbod of Peters and Van
herck. The cutting forces have been measured with a three
-component dynamometer having its first natural frequency at
approximately 1.5 kHz. The experiments have been carried out
applying the following conditions:
orthogonal cutting
- workpiece material C45N
- tool: standard carbide insert P30
geometry: a= 6°, y
K1 = 30°,
- nominal feed: 0.072 mm/rev.
, K 90°,
= 0.4 mm, À
47
Fig. 3.2.
F'ig. 3. 3.
48
bg (mm)
mot:C45N w h0 • 0.072 mm CmtA~~0.15 t.:experimentol volues i
' a I -\
\ 4
À \
\~\. / 2 ,,
~? ... o ......
1.0
'-A \
c: 0.075 < h ( 0.100 } eok:uloted o:0.050<h <0.075 volues
_/_
V" \r./
-o~ -
I I
I
__.o- -20
v (m/&)
The experimenta~ and the caleulated stability
chart of the special taalholder (Peters' methadJ.
The quantity represents the average value of
the damping ratio when the carriage is moving.
!).---.----.--~--~~--.---,---~--~--------~ mot: C 45 N <1•32°30'
1,0 2.0 v(m/s)
The ineremental outting stiffness ki vs. eutting
speed v aoeording to the methad of Peters and
Vanherok (I) and aeoording to the new methdd (II).
I
•
3.3. Dynamic approach of the cutting process
3.3.1. General
Many investigators in the field of dynamics of the
cutting process.have already observed the existence of
damping in the cutting process. In this context the best
known relatión
I'.F (3.5)
is given by Tobias (4). However, performing experiments in
order to obtain numerical values for the damping phenomenon is
found to be difficult. Therefore it is not amazing that reliable
values for the process damping caused by the workpiece material
are not available at the present time.
The test rig which is used in cooperative work in the
C.I.R.P. Ma-Technica! Committee for investigations into suscep
tibility to chatter of work materials, allows the carrying out of
experiments to obtain data of the damping ratio during turning
operations.
3.3.2. Process.damping and specific cutting stiffness as basic
quantities for stability charts
Tlusty et al. (5) derived
T c I
2 (-R) (3.6)
where Tc represents the transfer function of the cutting process
and R is the real part of the transfer function Tm of the machine
tool. When ki is supposed to be independent of the depth of cut b
the dynamic cutting force can be written as
T Llh "' b k. Llh c ~ (3. 7)
49
Hence, it follows on the threshold of stability (see also Chapter
IV)
b k. g l.
(3.8)
For a single-degree-of-freedom system it can be derived
R n
I 1 k '4__,..1;--;(..,.1_+_1; ... ) "' (3. 9)
where k is the stiffness, c the damping constant, 1; the
damping ratio and w0
the angular velocity at natural fre
quency. Now, it can be written
c w 0
(3. JO)
If, during turning operations on the threshold of stability,
a process damping cc is added to the system it will be necessary
to increase b in order to achieve instability. In the case of
the principal direction of motion being the same as the direction
of the maximum chip thickness modulation (see Fig. 2.12.),
the limit condition can be expressed as
b k. g l. (3. 11)
Exciting the tool by a pulse during cutting, it is
possible to measure the displacement response before regenera
tien occurs. From this we can calculate the damping ratio of
the system with
A n
[
1f n c J exp- ---5
wd m (3. 12)
where n is the number of periods, and m is the mass. The value
wd is characterized by
w = w j (I - 1,; 2 )
d 0 1 s (3.13)
Thus, the amplitude ratio becomes
50
or
A n
A"' 0 [
11 n c exp - _w_o_m_s_ -~-(-1-----:- (3. 14)
1l n (3 .15)
It should be noticed that the overall damping ratio of the
system çs can be written as
(3. 16)
and consequently
(3. 17)
Then on the threshold of stability the following relations
will be valid
b k. c w = 2 çs ~k (k + b ki) g ~ s 0 g (3 .18)
b 2 'k[,·Rz] g (3. 19)
b = 2 k (ç5
+ 2
+ ••. ) g çs (3.20)
, with
(3.21)
it is found
(3.22)
If the stability chart under certain conditions is known,
it is possible to calculate the values of ki with the aid of the
ç5-values obtained from the logarithmic decrement. The ki-value
will not be influenced by the dynamic behaviour of the tool.
51
Fig. 3.4.
Fig. 3. 5.
52
16 5
16 0
15 5
15 c)
> 0.2 0
~ s
0.1 5
0.1 0
<!> 0
0
/ /
--0.5 1.0
/ /'
-- ;..----; --1.5 2.0 2.5
-3.0
b(mm)
3.5
The overall damping ratio ç5
and the frequency
fc of the pulse response during cutting vs. depth
of cut b.
0 17 fc (Hz
16 )
16 0
15 5
150
5 0.1
~s 0.1 0
ao
B
V
2 fl " é " " " "
'V
~ g 'V ~ 'V
0.5 1.0 1.5
mot,c 45 N h0 :0.072 mm b: 1.5mm -
~mf>:0.080
a ll " " " 2 " "
'i 'V
'V 'i n 'V
" 'IJ
2JJ 2.5 v (m/s)
The damping ratio ç5
and the frequency fc of the
machining system vs. cutting speed v.
The k.- and ~ -values obtained can be used for predicting sta-l s
bility charts for machine tools of which the transfer functions
are known. It should be noticed, however, that when the direction
of ~F is unknown, it is impossible to extrapolate ki to any other
direction than the principal one of the rig.
The assumption that the cutting process adds damping to the
vibratory system is confirmed by the results as shown in
Fig. 3.4. and Fig. 3.5. From Fig. 3.4. it can be derived
that both the process damping and the process stiffness
are almost proportional to b. The results are obtained from
experiments measuring the pulse response of the test rig
during cutting. Fig. 3.6. shows a typicai_ example of such a
response. The measuring set-up is describ~--1n Appendix_II.
Fig. 3.6.
reference: 1000 Hz IIIIIIIIIIIH!IIII!III 1111111 IIUIIIIIIUII
V• 1.5 m/a b• 1.5 mm
tmt=o.oeo
ho•0.072mm
Exampte of a putse response. The mass of the taal
holder a 16.6 kg.
3.3.3. Experimental approach of the problem
Consiclering the system on che threshold of stability, it
53
fellows that
b k. g 1
In general, the following relation exists
Ak = m (w 2 - w 2) n o
(3.23)
(3.24)
where Ak is the amount of stiffness to be added to the system
in order to shift the natural frequency to wn.
After pulse-excitation during cutting, the angular frequency of
the motion of the tool will be
w c (3.25)
In Chapter II it has been discussed that the transfer function
of the Coolholder depends upon the velocity of the carriage.
The magnitude of the change in compliance will be influenced
by ç and tosome extent also by the carriage speed (6).
If b = 0 and the carriage of the lathe moves, the next relation
is assumed to hold
w =w -'1- 2 mt om 1< çmt) (3.26)
where w0m represents the natural angular frequency of the
machine tool whilst the carriage moves at a given speed. The
corresponding damping ratio is smt" Next it is defined
k = m w m om 2
(3.27}
A • 2 ssum1ng l;;s << 2 and Çmt <<I, it is possible to
approximate the process stiffness by the equation
(3.28)
Thus, on the threshold of stability the next relation must
be valid
54
(3.29)
where
= c + c mt c (3. 30)
The feedis considered to be a parameter in eq. (3.29). Analysis
yields
m (w 2
c
2 2 lll - (;,)
c mt
w _/ m (k + b k.) om1 m g1
2 A w w c mt
This gives
2 As A<< I, it fellows
w mt
w _c_" I +A w mt
1;$ I + ------'-------
/(I - I;; 2) /(I r 2) s - "mt
and for practical applications
w r; =~-
s w mt
. (3. 31)
(3.32)
(3.33)
(3.34)
(3.35)
(3. 36)
If the experimental results satisfy this equation at the
threshold of stability, the validity of the theory in the
preeedins pages is proved.
55
E'ig. 3.7.
0.2
~s bmt
0.2
5
0
0.15
0.1 0
0.0
E'ig. ;), 8.
mat: C 45 N tmt<~~O.OSO
--t---t---+--1 h0 • 0.072mm a= 32°30'
1.5 2.0
v (m/s)
2.5
The experimentaî atability ahart of the toolholder
for = 0.080.
mat C 45 N \ h0 =0.072 mm :----
1\ a.. 32° 30'
\ \~ A/
~\ A ~ A
g -}-. '\
/A "
A ·~ '\. A
'? .... ~L -
* \lm.t A
I t.ovalues from eq.3.15 o, va lues from eq. 3.36 -v, values for b • 0 (eq. 3 15)
1.0 1.5 2.0
v (mis)
2.5
The overall damping ratio at the threshold of
stability ç8
and the damping ratio ~rot va. autting
speed v.
Concerning the results, Fig. 3.7. shows the stability
chart for 'mt = 0.080. The values of ~s' computed with the
aid of the logarithmic decrement for the corresponding cutting
56
data of the stability chart of Fig. 3.7., are shown in Fig.
3.8. A curve in the same diagram shows the values calculated
with wc and wmt' For reasous of experimental practice, the values
of the parameter b have been chosen a shade smaller than the
critical ones. The secoud curve in Fig. 3.8. shows that çmt
slightly depends upon the cutting speed. A very good agreement
between experiment and theory can be observed for cutting speeds
exceeding I m/s. Fig. 3.9. gives one curve showing the dependenee
of the frequency fc on the cutting speed at the threshold of
stability.
185 f mt,fc
(Hz. )
17 5
16 5
15
•
\ -·-·
\ ~ :\ A
'A ~~~
I 1
mo''"'" b h 0 • 0.072mm , a. 32°30' 'LJ.:fc {b•bg)
' V : fmt (b• 0) '
I A _A..
~Jt/ "' ~ A
""' t.,..l I I u
'-~/
!
i-I 14 0 Q5 1.0 2.0 1.5 2.5
Fig. 3.9.
v (m/s)
The frequency at the threshold of stablZity fc and
the frequency fmt vs. cutting speed v.
From the results it becomes clear that the right-hand slope
of the stability chart (Fig. 3.7.) is not only defined by both
the compliance of the tool and a specific cutting stiffness ki'
but also by process damping. Although at low cutting speeds the
calculated values of ç6
do not agree well with the experimental
results, it is to be expected that process damping will also
have considerable influence in this range of cutting speeds.
A quantitative analysis on this subject will be given in
Chapter IV.
57
3.3.4. The calculation of k. ~
If the direction of motion does not coincide with the
direction perpendicular to the cut surface, Rn is reduced by a
factor cos a. The angle a represents the angle between both
directions. Then, relating the process damping to the
direction of motion, eq. (3.23) has to be replaced by
w 0
cos (l. (3.37)
Now, it fellows for the damping ratio of the machining system
Applying the preceding
c w b k. -~ g ~ cos Cl
b k. - 2 g ~
theory it
c s l(m k ) m
m w w c om
cos Cl
yields
2 m w om cos (l.
Thus, introducing the approximation wmt
will be valid
2
m w c w om
(3.38)
(3.39)
(3.40)
(3.41)
wom' the next relation
(3.42)
Fig. 3.3. shows the ki-data calculated from the results
presented in the Figs. 3.7. and 3.9. A considerable difference
is to be seen between the data obtained in this way and those
calculated according to the Peters method.
58
3.3.5. The influence of the wear of the tool on both the process
damping and the cutting stiffness
Fig. 3.10. shows the values of the overall damping ratio
and of the frequency, as obtained from the pulse response, versus
tool wear when the cutting speed v = 1.25 m/s and b 1.25 nnn.
1~0 fc (Hz)
17 0
150
0.3
ts
0.2
0. 1
0 0
V
/:. /:.
0.2
V V
mat: C 45 N ho= 0.072 mm
tmt~0.060
iv 1.25 m/s ! b = 1.25 mm
I
I Ä t:.
Ä! /:.
"'' /:.
0.4 O.G o.a VB (mm)
Fig. 3.10. The damping ratio ç8
and the frequency fc of the
machining system vs. tool wear VB.
It appears from the results that the damping does not increase
up to 0.2 mm flank wear (VB) of the tool. However, for values
exceeding 0.2 mm, the damping will rapidly increase up to several
times its initial value. This influence is taken into account
during all experiments mentioned by restricting the wear of the
tool to the range 0.1 + .0.2 mm.
According to Fig. 3.10., the change frequency corresponds
with an increase of the cutting stiffness of about 125 %,
while the damping increases until five times its initial value.
59
3.4. Conclusions
For cutting speeds exceeding I m/s, the methad proposed
seems to offer a reliable tool in the ánalysis of cutting
dynamics. Only measurement of frequency bas to be performed
in order to get all the information necessary for the
calculàtion of both the specific cutting stiffness and the
process damping. The values obtained in this way will not
be influenced by the dynamic data of the tool. The dynamic
cutting coefficient can be derived as well materials can
be compared on susceptibility to chatter. It is possible to
use the results for predicting the dynamic behaviour of
machine tools during cutting, if either the direction of
motion of the tool and of the test rig coincide or the
direction of the dynamic force is known.
In practice the ki-values will not be strongly influenced
by the wear of the tool, while the damping may increase to
high values.
References
(I) Vanherck, P., C.I.R.P. Ann. 17 (1969) 499.
(2) Peters, J., Vanherck, P., C.I.R.P. Ann. 17 (1969) 225 •
. (3) Hijink, J.A.W., Bepaling van de incrementele stijfheid in
het gebied van lage snijsnelheden en kleine aanzetten door
middel van statische beitelkrachten, gemeten met een stijve
beitelkrachtmeter met een eigenfrequentie van 1.5 kHz.
Report WT 0243, Eindhoven University of Technology (1970).
(4) Tobias, S.A., Machine tool vibration. Blackie & Son (1965).
(S) Tlusty, J., Polacek, M., Danek, 0. and Spacek, L.,
Selbsterregte Schwingungen an Werkzeugmaschinen. V.E.B.
Verlag Technik (1962).
(6) Kals, H.J.J. and Hoogenboom, A.J., The influence of the
carriage speed on the compliance of the toolholder. Report
WT 0227, Eindhoven University of Technology. Note presented
to the C.I.R.P. Technical Committee Ma (1970).
60
IV PROCESS DAMPING IN MET AL CUTTING
Abstract
In the previous chapter, a new method was presented to
determine the relevant components of the transfer function of
the cutting process. Using this method, the specific cutting
data obtained make it possible to calculate a series of in
fluences on the dynamic stability of machine tools in cutting
conditions. Up to now, these influences could not be explained.
Moreover, the method is also shown to be valid for lower cutting
speeds.
Nomenclature
a In-phase component
a . M=imum negative in-phase component m1n b Width of cut; quadrature component m, m/N
bcr Critical width of cut taking into account the phase
equation
b Limit value of width of cut g
c,cmt Structural damping coefficient
c Coefficient of the damping in the cutting process c
ei Specific process damping coefficient
c~ ResuZtant specific pPooess damping l.
f Frequency of the pulse response during cutting c
f0
Natural frequency
Ff Feed force
Fv Main cutting force
óFa Projection of the dynamic component of the resultant
cutting force on the principal direction of motion
h Undeformed chip thickness
h0
Nominal undeformed chip thickness
óh Chip thickness modulation
m
m
Ns/m
Ns/m
Ns/m2
Ns/m2
Hz
Hz
N
N
N
m
m
m
61
m
n
Stiffneaa of the machine too~ atruature
Dynamia autting aoeffiaient
Speaifia proaeaa stiffneas
Resu~tant speaifia proaeas atiffnesa Equivalent stiffness of the machine tooZ stPUa
ture with moving aarriage (b = 0)
Chip thiakness aoeffiaient
Damping aoeffiaient
Maas
Rotationa~ speed ofworkpieae
p In-phase oomponent
q Quadrature oomponent
R
R n
t
T c
In-phase oomponent
Maximum negative in-phase oomponent of Tm
Time
N/m
N/m
Ns/m
kg
rev/s
m/N
m/N
s
~~~ ~m a T Limit va~ue of Tc on the thresho~d of stabi~ity N/m cg Th Transfer funation of the regenerative effect
T Transfer tunetion of the machine too~ stPUam
V
VB
y
* y
Y'
y*
ture, t:.y/t:.F
•-u.""'-rw speed Width of the fiank wear Zand of the too~
Instantaneous defleetion of the too~
Delayed chip thiakness modutation
Peak value of y
Peak value of y*
a Angle between the prinaipal direetion of motion
and the direetion of the chip thiakness modu
~ation; alearanee angle of the autting toot + Angte between ki and the prinaipal direction
of motion
13 for a = 0
Angle between c: and the principal direction of l.
motion; rake angZe of the autting tool
y for a = 0
Damping ratio of the stPUcture
62
m/N
m/s
mm
m
m
m
m
0
0
0
0
0
çmt Damping ratio of the machine tooZ struature
with moving carriage (b G 0)
çs Damping ratio of the syetem during autting
K
À
Cutting edge angle
Cutting edge inclination
~ Phase shift between the two chip thickness
moduZations
AnguZar frequency
Angular frequenay of pulse response during cutting
Angular chatter frequenay
wmt AnguZar frequency of pulse response without
autting
Angular frequency during autting
Angular naturaZ frequency
Angular velocity of workpieae or tooZ
4.1. Introduetion
0
0
rad
rad/s
rad/s
rad/s
rad/s
rad/s
rad/s
rad/s
The dynamic behaviour of a cutting process can be described
by a closecl-loop model as shown in Fig. 4.1. The machine tool
is characterized by its transfer function Tm' being in fact the
y•-y Tc Tm --
~
y-y* Th
Fig. 4. 1, Fig. 4.2.
y
F
Btoak-diagram representing the
dy~~ia autting process.
Diagramrnatic representation of
the chip thiai<ness modu Zation.
63
dynamic compliance of the structure, whilst Tc de~cribes the
transfer function of the cutting process. The loop is closed by
the function Th' which introduces the wave-on-wave chip thickness
modulation (Fig. 4.2.). The quantity y represents the relative
motion between tool and workpiece i.e. bhe direct. chip thickness
modulation. In the case of turning the delayed chip thickness
modulation y~ caused by the feedbackpathof the previous.cut,
can be written as
y*(t) = y(t - ~) (4. 1)
where t stands for time, and n is the rotational speed of the
workpiece. Assuming a harmonie motion between tool and workpiece,
it follows
y y cos w t (4. 2)
and
y* = y* cos (!) t (4. 3)
where a phase shift exists between Y and Y~. Thus we find for the
resulting chip thickness modulation
llh = y* - y (4 .4)
When introducing llF~ for the relevant part of the dynamic
cutting force, we can write
and
T c
The function Th is defined by
* T =I- L h y
(4 .5)
(4 .6)
(4.7)
Next we define the threshold of stability with the aid of the
equation
(4 .8)
64
h out·?f-phase --!>'j OXIS
-F Tc
h•ho• y'-y b "'!l Ci
q in-phase oxis
400Hz
Fig. 4. 3. Fig. 4. 4.
Polar plot of the transfer function C:ompoeition of the function
Th on the threehold of etability.
Fig. 4~3. shows the function Th on the threshold of stability.
Just like Th, the transfer functions Tc and Tm are complex.
The functions can be represented by a polar curve, as Fig. 4.4.
shows. Where Tm aften can be described with the aid of a compound
of some single-degree-of-freedom systems, the complex nature of
Tc is not well known. In general, Tc is supposed to cause a
phase shift. The phase shift can be introduced by the assumption
of an interaction between T and T due to the relative motion c m betwe'en tool and workpiece. · According to Tobias' theory (I) we
de fine
; kl 8.h - (4. 9)
The polar curve of (Tc Tm Th) is represented in Fig. 4.5.
The curve consists of an infinite set of circles, all passing
through zero and rotating clockwise around zero with increasing
value of the angular velocity w (Fig. 4.5.). The diameter of every
circle is only determined by the value (Tc Tn,). The Nyquist
curve intersects the negative part of the in-phase axis in a great
number of "-1" points Pj, where on the threshold of stability the
next relation is a condition to fulfil the Nyquist criterion
65
We define
and
TT =a+ib c m
According to Fig. 4.3. the next relation will be valid
2 q = p (2 - p)
(4. 10)
(4. 11)
(4.12)
(4. 13)
Under limit conditions, the quadrature component of the function
(Tc Tm Th) equals zero. It yields for the in-phase component
2 a (4. 14)
In the case that the force is in phase with the chip thick
ness modulation we can derive
a . R = _m_1.n_
n Tc (4 .15)
The quantity Rn stands for the minimum part of the in-phase
component of the transfer function Tm. The equations (4.10),
(4.14) and (4.15) lead toTlusty's (2) relation for the limit
value
(4. 16)
However, for complex numbers of Tc' the limit value will not
occur for R = R . n
For the determination of Tm reliable methods are available
(3), (4). The measurements with respect to the function Tc' however,
have been presented but have not been proven to be satisfactory. As
both quantities are essential for an accurate prediction of machine
tool chatter at the design stage and for predicting the stable
working conditions for numerically controlled machine tools, the next
aim must be to obtain a fair knowledge of Tc' Extensive work in
this field has been done by a.o. Smith and Tobias (5), Kegg (6) and
Albrecht (7).
4.2. Determination of the transfer function of the cutting process
Among the methods applied for the determination of the
process transfer function one can distinguish:
I. methods where the regenerative effect is taken into
account (8)
2. methods excluding the regenerative effect (6), (9).
In the previous chapter it was shown, that, if a machine tool is
represented by a single-degree-of-freedom system showing an angular
natural frequency w0
, during cutting at speeds exceeding I m/s,
the next relation will held
w n
(4.17)
In this equation, wn stands for chatter frequency and çs repre
seuts the damping ratio of the complete system on the threshold
of stability.
Equation (4.17) allows to obtain values for the specific process
atiffness ki as wellas for the specific process damping ei'
Actually, the word "specific" refers to a width of chip of I mm.
Equation (4.17) leads to the next relation, which for ç2
<< I s
will be valid on the threshold of stability. (See Chapter III)
ç = [~ - IJ cos a s wmt
(4.18)
The angular frequency wc goes with a pulse response of the
overall system during cutting under limit conditions whilst
avoiding the regenerative effect, whereas wmt represents the angu
lar frequency of the pulse response for a width of cut b = 0 and a
moving carriage. The angle a gives the orientation of the princi
pal direction of the vibratory system with respect to the
direction of the chip thickness modulation.
67
From Chapter III the next relation for the specific process
stiffness results
(w - w ) c mt
where b represents the limit width of cut. g
(4.19)
It has to be noticed that, with c as the damping constant,
km as the equivalent spring constant in werking conditions and
m as the mass of the structure, the overall damping ratio of
the system can be written as
whilst the damping ratio for b
c mt
0 equals
(4.20)
(4.21)
From the equations (4.19), (4.20) and (4.21) it follows for the
specific process damping
c. l ~ [[~ (w - w )] cos a. b wt c mt g m
(4.22)
Actually, the values of ki and obtained in this way are +
projectionsof the original vectorial quantities and ei.
Both the process stiffness and the process damping are coeffi-
cients. Thus we have to define the vectorial quantity as the
variatien of the resultant dynamic cutting force per unit of
displacement of the tool normal to the cut surface, related to
a width of cut of I mm. The quantity c~ is defined as the incre-l
ment of the resultant dynamic cutting force per unit of velocity-
-increase in the direction perpendicular to the cut surface and is
also related to a width of cut of 1 mm. In the case of orthogonal
cutting
68
and c~ will act in a plane through the direction of the J.
cutting force F and the feed force Ff' Tc determine both k; and c: V 1 1
we need two different toolholders with a different principal
direction. Fig. 4.6. shows diagrammatically the composition of k: 1
and c:. These quantities are supposed to.be independent of the 1
vibrational direction of the structure. An experimental evidence
for this will be shown in Chapter V.
2" principal direction of motion
Fig. 4.5. Fig. 4.6.
Nyquist plot of (Tc Tm Th). The aomposition of k; and 1
..,. c .•
1
We assume on the threshold of stability the delayed chip
thickness modulation to have no influence on the process damping.
As a matter of fact it is assumed that this modulation has no
further influence on the transfer function than its geometrical
contribution to the chip thickness modulation (IJ). The back
ground for this assumption will be discussed in detail in
Chapter VI. The relation between the relevant component of the
dynamic cutting force and the quantities derived previously in
this chapter can be expressed now with the aid of equation (4.9)
as
6F Cl.
(4.23)
In this relation, both the process stiffness and the process
damping are proportional to b. This assumption is justified by the
69
17 5
) fc
(Hz
170
15 5
150
15
150
0.2 0
s 0.1 5
0.1 0
>
~
mot:C45N h 0 :0.072 mm
v: 233 m/sec
I V I /4
:/
/ /
/' /
/
--- ,_....... -- :...-----o.5 1.0 1.5 20 25 3.0 35
b (mm)
Fig. 4. 7. Cheak on linearity of the proaess damping and the
proaess·stiffne~s with respeat to the width of aut.
experimenta1 results shown in Fig. 4.7. (Compare eq. (4.28))
For an ang1e S, respective1y y, between the principa1 direction
of a given structure and k~, · + · f 11 ~ respect~ve1y ei' ~t o ows
/1F
b [1<1 /1y] ct cos s - i (I c~l tJl = wk ~
cos y) /1h (4.24)
Bearing in mind that
/1y - - I /1h - (4.25)
it finally yie1ds
T b [1<1 cos s + i wk I c~l cos r] c ~ (4.26)
70
Among ether things, the composition of this function for a given
situation is shown in Fig. 4.4.
4.3. Experiments
For the measurements special toolholders are used, originating
from a model similar to the C.I.R.P, cutting stand as accepted
for cooperative work in the Teehoical Committee Ma (JO).
In order to obtain all the information necessary to compose the + +
resultants ki and ei' measurements are carried out regarding the
two principal directions of the vibrating system, viz. a = 0° and
a= 32°30'. (See Fig. 4.8.)
Fig. 4. 8. The two different designs of the toothoZder.
In accordance with the conditions for which equation (4.18)
is valid, the measurements are carried out using cutting data on
the threshold of stability. So, befere starting measurements, the
experimental stability charts are required. In order to avoid the
regenerative effect, data are only taken within the time for one
revolution of the workpiece, whilst during the time preceding the
pulse the toolholder is locked in order to exclude any contribu
tion from the delayed chip thickness modulation. In spite of the
precautions mentioned, at low cutting speeds this method is not
quite satisfactory, due to an increasing noise level in the dyna
mic cutting force. When slightly decreasing the width of cut we
can evade this problem. The values of w1 measured in this way
have to be extrapolated to wc. From the relation
71
(4.27)
which can be replaced by
+ ••• J (4.28)
it fellows that within a small range the angular frequency is
approximately a linear function of b.
To avoid lobes in the stability charts the experiments have
been carried out for Wmt > 30. This implicates that b = b . n cr g
The pulse response of the teelholder is measured by strain
gauges mounted on the leaf springs. The signals are fed into a
high speed 8 bit x 1024 word core memory to store the digital
equivalent of thé analogue signal as a function of time. The
values of 4 and w are computed with regressive techniques. It
turns out that the measured data of the frequencies show consi
derable less scatter than these of ç. The measurements during
cutting were carried out .three times. The results have been
corrected with respect to the change in the damping of the
teelholder due to the static deflections changing the oil slit.
In order to eliminace the influence of the nose radius of
the cutting tool, all experiments have been carried out on
tube-shaped workpieces. Standard carbide inserts (P 30) have
been used. The following tool geometry has been applied
SKF 1550 steel as the werk material, the experiments have
been carried out for two different values of the feed viz.
0.072 mm/rev and 0.208 mm/rev, within the range of 0.5 -
- 2.3 m/s for the cutting speed, Tool wear was restricted to
0.15 mm VB.
72
4.4. Results
For a = 32°30' the results of the measurements concerning
wc' wmt and Çmt as well as the computed values of and wc
have been plotted in Fig. 4.9 .. Fig. 4.10. shows the computed
values of the specific process stiffness ki' according to eq.
(4.19), for the two different feeds of 0.072 mm/rev and 0.208
1260~--~--._--~--~~=r~~---4--~~~~~~a
liJmt
(rod/s)
ts ~-4--~r-~--~~~~+-~~_,---+--~
tmt
900o~--~--~~5----L---1~~--~----t~5---L---2~~--~---zs~
v (m/s)
Fig. 4.9. The plots of the experimental results of wc, çmt
and wmt, and the aomputed va lues of 1; s and wc.
The aomputed values of wc are obtained with the aid
of the graphiaat salution method Appendix III deals
Uïith.
73
3 kj
(N/ni
x10~ 2
-· mat , ~KF 1550 ,. .goo a.• 60 'I· 5 À• cf
h0 • 0.070 {' • • • >2'~ mm o,a..r:f'
ho=0.208 {"'' a.·3t3d mm <>'a.= 0 ,
/ '"\:.
1.
c \ ~~ I
/ \ /[ ,c ....
\ \Vt.~
z \ lil to, 'L , ., \ I
c ... a...., 1\ ~·- -·-b ~) if
U<
CS \J I
.-<>'\ \ 11 -o.
) o...._
'-·- .o/
Q5 \0 2.0 v (m/s)
Fig. 4.10. J."he "!Ja"/.ues of the specific pr>ocess atiffnesa ki
ver>sus cutting speed v .
. 5 Cj
) .o
(N s/rrf x106
1
5 0.
0
-o ~
' I
\à . I \.
-,~.
\'
\j
mot, SKF 1550 ~~<>"0,072 {O:a=32 30 ~t•90: mm o:a:d' Q.: 5 0 I
5o :ho·0.208 {"'' a:32 30 1: Oo : mm <>:a.= (f
l i /l' "' · ....
fÎ'-.. I ···a. ,/
I l'.t.-~ •O -·-.. v·-/ ,
~iJ 0
o' I
r·'r ~ 7 1/ 0'-.,1\ i j
~-~ o"'
I
•
I
-1 0 Q5 \0 2.0 2.5 v(m/s)
Fig. 4.11. The values of the specific process damping ei ver>sus
cutting speed v.
74
mm/rev. In this figure the curves concerning SKF 1550 steel show a
course which is equal to that of the curve concerning C45N
steel presented previously in Fig. 3.3.
As for 4.11. which represents the computed values (eq.
(4.22)) of the specific process damping, the results show the
damping varying within a wide range as a function of the cutting
speed. A minimum is shown for about the same cutting speed where
the b -value has its minimum. (See Fig. 4.14.) The c.-values g l
near the minimum are negative, whilst for increasing cutting speeds
the process damping cc may rise to high positive values even
exceeding the structural damping of the Vanherck teelholder
for ~mt a 0.10. This may be the reasen why some investigators
found the dynamic cutting force leading the direct chip thickness
modulation, while ethers observed a lagging dynamic force, since
cutting conditions, especially the cutting speed, were very
different (11). As it is already shown by the eqs. (4.19) and (4.22),
when comparing the results of • 4.10. and 4.11. it can
be seen that the graphs concerning the process damping and the
process stiffness show a very similar course. This is confirmed
by Kegg's results (6). (See Fig. 4.18.) Apparently, both quanti
ties are controlled by a same physical mechanism.
In Fig. 4.12., the quantity ci/ki is shown diagrammatically
as a function of the cutting speed, The quantity is basic for
the vector reprasenting the damping in a graphical solution
me~hod for the limit value b , which originates from Gurney's g
metbod (1). (See Appendix III) Using the data of Fig. 4.12.,
the theoretica! stability charts, shown in Fig. 4.14., have been
computed. A very good agreement with the experimental results is
to be seen in this figure.
Experiments carried out previously (see Chapter III), using
measurements based on the consecutive amplitude ratio of the pulse
response as a reference, did show the method not to be valid for
cutting speeds less than I m/s. However, the results of Fig. 4.14.,
based on the maasurement of frequencies, prove the metbod to hold
within the whole range of cutting speeds applied in this investiga
tion.
75
10~--~--,---~--~--~--~----.---.----r
c;Jk;
(sxlo-4) 5~--~~~~~~~---+-~---4--~~~~
--+-~f---1-----m--lot~,-=sK~ 1550! :: gg: y " 50 À::: 0°
0 ' ho• 0.072 { c, a •32 30
-30f----+--·········+·····+----'-~~cc-!-+ mm o dl .<:f' h0 ·0.208 {L>' a •
mm ():a"'
-J!i•+---~--o-.Ls--~----w----~--1L5--~---20~--_L--~25
v(m/sl
Fig. 4.12. The aourse of the quantity ci/ki versus autting speed v.
Fig. 4.13.
76
mot, SKF 1550 ho•O.g08mm ". 00 Cl • 5° y • 50 A • 0°
0.57 mis ~00
m/s -V
o.83m/s
1.00 m/$
Diagramrnatic representation of k~ and c~ jor l. l
different m<tting speeds;
+ In Fig. 4.13., the vectorial quantities k. and . are shown for
-~ l
different cutting speeds and a feed of 0.208 mm/rev. As can be seen
from the figure, the directions of both quantities depend strongly
upon the cutting speed. With respect to the process damping, this
behaviour is contradictory to assumptions made by Andrew (12) and
by Kegg (I 3) .
The final results are shown in Table 4.1., containing the nume
rical values bath for the length of the veetors mentioned and for
the augles of the veetors with respect to the direction perpendi
cular to the cut surface.
A remark has to be made as to the reproducibility of the
method. For a feed of 0.072 mm/rev for instance, and for all
the cutting speeds applied, the scatter in remains within a range
of 1.9 %. So the metbod proves to be very reliable.
10 I mat: SKF 1550 I
g \ tmt~O.l ll rrl_
\ / 8
\~ • .. .J/ i ···r---\ \ I r 7~--.
I 5
t\ tN_:/ ! / i V Cl ...
\I\g\ I l/ .
4 I i"-~~~ __..c-' lir ....
.... : . i j'./ "' 0 • i -· L~L-
I 11 •32o30{c: ho=0.072mm
A: ho=0.208 mm
1--•.•. e,+: eomputed volues 11
_ 0
o \ o: h0 =0.072 mm
I I · . <>: ho•0.208 mm I
0.5 10 15 2.0 2.5 v (mlsl
Fig. 4.14. The aomputed bg-values in aompaPison with the expePi
mental stability oharts.
77
v (m/s)
o.so
h • 0.012 mm 0
h0
• 0.208 mm
(10_""•'>1 (0
_)_j__lk_7_1 -(-I0-9
N_/_,i_).j..e_0_(
0
_).J-I_C:_·I_(_I_o6_N_•'_"'_
2l+-Y0
-(-
0
)--i!
1.56 73 1.03 112 I 0.67 !.08 41 0.45 334 1.80 74 !.43 118
0.83 1.59 81 1.77 100 1.41 26 0.19 274
f.-:.:I.:::_OO:ê_+--:_:1._:::45::__~8:;2-J.-_;:0,:.:•6:;:6 __ ~1:::38::_1-.:2:..:.2:.:.6 __ ···_···· 66 1.35 91
~~·~1~7~~-----~---+-------~---~~1:..:.9~8-----~72~~--~:..:·:..:26 ___ ~_92 __ ~ t.2s 1.o1 79 o.69 211 - I .. -· 1.33 2.36 85 1.19 121
1.67 2.16 69 0.75 78
2.00 2.57 61 1.25 71
2.33 1.69 59 0.67 50 - ..
Table 4.1. The numePiaal representation of and
material: SKF J 550
frequency of
excitation:- l75 Hz
-+ c .• ~
4.5. Process damping and its influence on the threshold of stability
Ignoring the process damping in a single-degree-of-freedom
system, eq. (4.16) results in
T = 2 I; k cg (4.29)
If the process damping is taken into account, we can write
c + c k = ___ c k • (c + cc) w
1 l(m k) 0 (4. 30)
where cc represents the projection of the process damping vector
in the principal direction of the vibratory system.
Writing
and
c c b c. 1
it fellows from eq. (4.30)
78
b k. = 2 [l; g 1
(4.31)
(4.32)
(4.33)
or
b g
2 1; k
k. ~
c. (jJ J ~ 0
~
(4.34)
Hence, b can be considered as a function of the dimensionless g product (c.w )/k. and the quantity 2~;k/k .• Fig. 4.15. shows b
~ 0 l. l g versus 2~k/ki for several values of (ciw
0)/ki. From this graph
it becomes clear that for (ciw0)/ki + I the influence of the pro
cess damping on the threshold of stability increases strongly.
For (ciw0)/ki fo I there will be unconditional stability for any
b-value as far as regeneratien is the major centrolling factor.
From Fig. 4.15. it appears that the influence of a variatien of
(ciw0)/ki on bg increases with increasing 2~;k/ki-value. (See
also Fig. 4.16.) Analytically, this result is obtained by
differentiating eq. (4.34) with respect to (ciw0)/ki.
2 <;; k
const.
c. w ]2 l 0
ki (4.35)
Fig. 4.15. The injïuenee of ~k/ki and (ciw0)/ki on the Zimit
value bg.
79
With respect to the shape of stability charts of machine
tools some important conclusions can be drawn:
1. when (c.w )/k. « I, there will be no significant :L 0 1
influence of the process damping on the threshold of
stability
2. when (ciw0)/ki ~I, unconditional stability is to be
expected
3. when (c.w )/k. +I, we have to make a distinction 1 0 1.
between
machine tools with a high çk-value:
process damping will have major influence
-machine tools with a low ~k-value:
process damping will have minor influence.
For structures with different dynamic properties, some
remarkable differences in the course of the limit width of cut
as a function of cutting speed can now be explained. For
instance, camparing stability charts of machine tools and of
the Vanherck-Peters toolholder, when adjusted to high ~-values,
the stability borderline of the tooiholder shows a remarkable
increase of the limit width of cut with increasing cutting speed
(Fig. 4.20.). This cannot easily be made in agreement with Knight's
theory of a small increase of the level of stability with
increasing cutting speed, based on the behaviour of the mean shear
angle (14).
Peters (15) found that ~-values for conventional machine tools
seldom exceed 0. 03, whilst in the case of single elements, value.s
of 0.002 are quite normal. (See Chapter II) In chatter research
also special toolholders with low ç-values are often used. Two
more examples will be mentioned here: a boring bar used by Peters
(16) with a ç-value of 0.008 and a tooiholder used by VUOSO (17)
with ~ " 0.01.
Not much information is available concerning values of the
stiffness of medium size machine tools. In the case of a less
stiff machine tool like a radial drilling machine, Landberg (18)
found spring constauts averaging I .5 x 107 N/m at a distance of
I m from the column center line. The natural frequencies of
medium size machine tools are commonly in the range of 100
80
+ 200Hz (17). The mass of the system will seldom exceed 100 kg.
Thus a spring constant of 50 x (2TI x 150) 2 ~ 5 x 107 N/m will
represent a reasonable average value for this type of machine
tools.
As to the Vanherck-Peters toolholder, at standard conditions
as proposed for cooperative work in the Technica! Committee Ma
of the C.I.R.P. (19) and in werking conditions (20), it fellows 7 k 1.5 x 10 N/m and ç ~ 0.15. Thus it yields çk 0.15 x 1.5 x
x 107 = 2.25 x 106 N/m. Gomparing data, for the boring bar we
find çk ~ 0.008 x 1.6 x 107 = 1.3 x 105 N/m and, on the
average, for a medium size machine tool çk ~ 0.02 x 5 x 107 = 106 N/m.
Consequently, applying SKF 1550 steel, the limit values of
the boring bar (f0
= 168 Hz) will hardly be influenced by the
process damping, as is shown in Fig. 4.16. This has been
confirmed by experimental results (16). In the case of the
Vanherck teelholder (f0
~ 160Hz), even with respecttoa medium
size machine tool, a considerable influence on the threshold of
bg14
(mrn)
12
mat, SKF 1550 h 0 : 0.07!'!mm
0 a • 32" 30 ; >d}O meen values for kj: 1.5•109 NJm2(f0 .,150 Hz)
10r-__ ,Q~o~g~,,To_9N~/~m~2?~~=~52~0r7~+---~~~--7T+-+-~~-4
Fig. 4.16. The influence of the procese damping on the limit
values for some experimental toolhoZdere mentioned in
li terature.
.. "' c f
'
81
stability is to be expected. (See 4.16.) This has been con-
firmed by many investigators using the same teelholder and
applying several workmaterials (19). Thus process damping will
cause an increase of stability with increasing cutting speed (Figs.
4.14. and 4.20.).
A comparison as has been made will only be valid for about the
same value of the quantity (ciw0)/ki. When orthogonal cutting SKF
1550 steel, supposing a natural frequency of the tool of about
150 Hz, the value of the dimensionless quantity will increase up
to 0.5. Taking into account the change of ki according to Fig. 4.10.
then, with respect to the Vanherck toolholder and for the conditions
as mentioned, a working area under limit conditions can be given as
indicated in Fig. 4.17.
According to Kegg's results (6), for frequencies beyend 150Hz,
Tc decreases slightly with increasing frequency whilst the out-of-
bg (mm)
8
Vonherck-Peters' tooiholde mot: SKF 1550 ho·0..072mm 4 •32* 30'; ..... go• w0 .,10' rad I see tJ< • 2.25x 10• N/m
61------t
2
if2tk), ~kï mln
8 10 2tk kj'"" (mm)
Fig. 4.17. The working area under limit conditions for the Vanherok
Peters toolholder when maohining SKF 1550 steel.
82
g()
6
3
phase
0 ongle (o)
0 ~ I--
0
I
)
' 8
.6
" t !
0
0
!
......._
-~
~ "·
-- 1-
~
v= 1.37m/s ho=tO.Iglmm V :8o mat: 81845 45.'$m< .. h< 80.1ö6m a•
'"' "' f~
out-of-phase oxis
.2
~ 0
0 0.2 OA 0.6 08 x 109 N/m2
0 0 100 200 300 400 50C 6 00
150 f (Hz) 520
Fig. 4.18. The influence of frequency on the cutting stiffness
and the process damping according to Kegg's results.
-phase angle increases up to 75° for f =450Hz. (See Fig. 4.18.)
This can only be due to both a considerable decrement of ki and
an, at the same time, practically unchanging out-of-phase compo
nent. It should be·noticed that the increase of the quadrature
component with increasing frequency is considerable less than
proportional.
With respect to the VUOSO toolholder, a next conclusion
can be drawn. Although the çk-value of the VUOSO toolholder 5 is low (f
0 ~520Hz, çk ~ 5 x 10 N/m), the 2çk/ki-value is
high whilst the (ciw0)/ki-value will still easily attain to I.
So, starting from a certain value of ei' a minor increase will
cause quite high values for the limit width of cut, finally
resulting in unconditional stab.ility of the cutting process.
Fig. 4.16. shows the influence of the process damping on the
limit value as predicted for SKF 1550 steel, taking into
account a change of ki according to Fig. 4.18. Since the
results for several kinds of mild steel are comparable, we
assume this extrapolation to be allowed (19). It can be ex
plained now why experiments,. carried out under standard condi
tions in different laboratories with the aid of the VUOSO tool
holder, did not produce comparable results with respect to the
83
limit values (17). An important conclusion has to be .drawn: both
toolholders mentioned are not suited to carry out comparative
tests on susceptibility to chatter of materials by comp.aring
stability charts.
Next it is also possible to explain the· considerable diffe
rences between the influence of tool wear on the threshold of
stability when using the VUOSO tooiholder or when eperating normal
machine tools, as is shown in Fig. 4.19. (17). Where for the tool
holder, contrary to the machine tool (f0
= 130Hz), the values
of (c.w )/k. are rather high, an increase of process damping 1 0 l
caused by developing tool wear (9) easily results in considerably
different values of the limit width of cut. Experiments performed
with the Vanherck tooiholder did also show a substantial increase
of the level of stability with increasing tool wear (21). In this
case, however, the high ~k-value of the tooiholder will be res
ponsible for the influence of process damping on the threshold of
stability.
With the foregoing, the presence of damping in the cutting
process enables us to predict a flat course of the limit value
versus cutting speed of the Vanherck-Peters tooiholder as the
structural damping decreases to small values. (See Fig. 4.20.) In
this figure it has to be noticed that the differences at the left
hand side of the borderline are not as great as these at the right
hand flank. In of a less increasing process damping at lower
cutting speeds, the influence of the damping proves to be
more important here, due to low values of ki. This is also
the reason why, even in the case of machine tool structures
with low ~k-values, always a substantial increase of the limit
value of width of cut at lower cutting speeds is present. It is
evident now that the specific process stiffness ki is primarily
responsible for "low speed stability".
The curves of Fig. 4.20. represent the experimental stabili
ty charts where the black marks indicate the predicted limit
values. For çmt 0.055 and ~ 0.17 the limit values are deter
mined with the aid of eq. (4.34), taking into account the eerree
tions of çmt induced by both the varying oil slit of the damper
and the varying carriage speed. The derivative of the limit value
84
4.---.---.---.---,---,---.---.---,---,---, bg
(mm) ..
240 1200 cutting tim.. (s)
Fig. 4.19.' The influenae of toot wear on the threshold of sta
biZity for different machine toot structures.
mat' SKF 1550 11 h0 =0,072 mm
) ct= 32°3o' A:Cmt::><0.17 1/ v:Cmt""O.lO
c:~mt"" 0.055
I • \ ·-········
8
\ \\. I \ \\ i'... / / ,,..... ... V
6
\ \ '\; ( ...
_e/ ., 1\
\ ~~, ~\~ :-/~ ;:' i'"
1\
4
--
'\. ~ l I • ........ . ................. L--I
~'-..__ ~~Y~ --, ·- D-
2
•,•:computed values (eq. 4 .34)
"': H " (Gurney's method)
•
0.5 1.5 2 2.5 v (m/sl
Fig. 4.20. The influence of proaess damping on the threshold
of stability for different values of the structuraZ
damping.
85
b has a high value in the region of the high cutting speeds as g
well as in the region of the low cutting $peeds. For this reason,
the process damping and the process stiffness being not quite pro
portional to b may cause great deviations between bath results,
particularly for high values of bg. Moreover one has to pay
attention to the fact that for high values of ~mt it is nat per
mitted to ignore deviations due to neglecting ~ 2, In this way mt a fair agreement between the predicted results and the experimental
ones is attained.
4.6. Conclusions
Process damping in metal cutting is proved to be a basic
quantity in order to describe cutting dynamics. The influence of
process damping increases strongly when (c.w )/k. +I. In the case l 0 l
of a highly damped stiff machine tool, eperating on the .threshold
of stability, the adding of a small process damping will result in
"' "' 1~·~--~--~~~~~---+---+---+---4--~~~~~ "'
0.15f---+---+--l--+--+--+--l,_-+--+-__.; ç. o.10f--+.r-4-zrf--tr-.!r--:---l---.lr~IH~-t--+----i
2.0 2.5
v (m/s)
Fig. 4.21. The overall damping and the frequenay of the puZse
response fo:r• a aonstant width of out as a funation
of autting speed.
86
a substantial increase of the limit value. lt should be noticed
that the influence of negative damping is considerably smaller than
the influence of positive damping.
The results presented in this chapter are supported by the
fact that some important influences on the limit width of cut can
be explained now.
The results of Fig. 4.21. concerning the damping during cutting
of C45N steel do not show an increasing damping for very low
cutting speeds. This proves again, that the specific process stiff
ness is primarily responsible for "low speed stability".
In this view we have to reconsider Tobias' theory (I) con-
cerning the increase of stability at lower cutting speeds
as wellas Kegg's "low speed stability" solution (13).
References
(I) Tobias, S.A., Machine tool vibration. Blackie & Son,
Glasgow ( 1965).
(2) Tlusty, J., Polacek, M., Danek, 0., Spacek, L.,
Selbsterregte Schwingungen an Werkzeugmaschinen. V.E.B.
Verlag Technik, Berlin (1962).
(3) Koenigsberger, F., Peters, J. and Opitz, H., C.I. R.P.
Ann. 14 (1966) 96.
(4) Van der Wolf, A.C.H., The development of a hydraulic exci
ter for the investigation of machine tools. Doctor's
thesis, Eindhoven University of Technology (1968).
(5) Smith, J.D. and Tobias, S.A., Int. J. Mach. Tool Des. Res.
I (1961) 283.
(6) Kegg, R.L., A.S.M.E. paper No. 64 - WA/Prod.-1 I (1965) 464.
(7) Albrecht, P., A.S.M.E. paper No. 64 - WA/Prod.-11 (1965) 429.
(8) Van Brussel, H. and Vanherck, P., A new methad for the
deterrnination of the dynamic cutting coefficient. llth
Int. M.T.D.R. Conference, Manchester (1970).
(9) Kals, H.J.J., C.I. R.P. Ann .. 19 (1971) 297.
(10) Peters, J., Vanherck, P., Report on a new test rig to carry
out comparative tests of "susceptibility to chatter" of
materials. C.I.R.P. Report presented to Group Ma, Univer
sity of Louvain, 19 sept. (1967).
87
(11) Vanherck, P., C.I.R.P. Ann. 17 (1969) 499.
(12) Andrew, C., Proc. Inst. Mech. Engrs. 179 (1965) 877.
(13) Kegg, R.L., C.I.R.P. Ann. 17 (1969), 97.
(14) Knight, W.A., Int. J. Mach. Tool Des. Res. 8 (1968) l.
(15) Peters, J., Proc. of the 6th Int. M.T.D.R. Conference,
Manchester (1965) 23.
(16) Peters, J., Vanherck, P., Industrie-Anzeiger No. IJ
(1963) 168 and No. 19 (1963).
(17) Tlusty, J.,-Peters, J., Matthias, E., Report on cutting
tests of stability against chatter. C.I.R.P. Group Ma,
Doe. No. 3/67, University of Louvain, 20 april (1967).
(18) Landberg, P., Metaalbewerking No. 9 (1959) 173.
(19) Peters, J., Vanherck, P., Program of chatter-susceptibi
lity tests. C.I.R.P. Group Ma, Doe. No. 69 Ps, Universi
ty of Louvain, 15 aug. (1969).
(20) Kals, H.J.J., Hoogenboom, S.A., The influence of the
carriage speed on the compliance of the toolholder. Re
port WT 0227, Eindhoven University of Technology. Note
presented to the C.I.R.P. Technica! Committee Ma (1970).
(21) Tlusty, J., Koenigsberger, F., Specificatiens and tests of
metal cutting machine tools. Proc. of the Conference, 19th
and 20th Febr., The University of Manchester Institute of
Science and Technology (1970).
88
V THE CALCULATION OF STABLE CUTTING CONDITIONS WHEN TURNING COMPLIANT SHAFTS
Abstract
It is shown that when turning shafts, the stable cutting
conditions are controlled by a simple relation. In the case
of compliant shafts, elementary experiments supply the value
of the dynamic stiffness. It is demonstrated that the dynamic
quan,tities of the work material borrowed from the previous
chapter are reliable values with respect to different appli
cations.
Nomenclature
b
b g
Width of cut
Limit width of cut
Equivalent damping coefficient of the ctamped
shaft
cil'ci2 Specific p~ocess damping coefficient (K 90°)
ci 1 Specifia p~ocess damping aoefficient (K ~ 90°) _,.
c. ~
F
Resuttant specific process damping
ResuZtant cutting force
Amplitude of the excitation force
Dynamia component of the resuZtant cutting force
öF0
Component of the dynamic autting force coinci
ding with the direction perpendicuZar to the aut
surface
h 0
NominaZ undeformed chip thickness
llh CMp thiakness moduZation
ke Equivalent stiffness of the cZamped shaft
kil'ki2 Specific proaess stiffness (K 90°)
kil Speaific process stiffness (K ~ 90°)
m
m
Ns/m
Ns/m2
Ns/m2
Ns/m2
N
N
N
N
m
m
89
s
t
V
xi ,x2
* XI
xo XI ,X2
y
Resultant specific prooess stifjness
Equivalent maas of the olamped shaft
Nose radius of the tool
Feed
Time
Cutting speed Displacement
Delayed ohip thiokness modulation
Amplitude of displacement at natural frequency
Amplitude of displacement
Deflection of the tool perpendicular to the
mm
mm/rev
s
m/s
m
m
m
m
cut surface m
K
Angle between the prinoipal direetion of motion
and the direction of the chip thiokness modula
tion; clearance angle of the outting tool
Angle between k~ and the direction perpendioular l.
to the out surface
Rake angle of the outting tooZ
Angle between c7 and the direction perpendicuZar
to the out surface
Damping ratio of the olamped shaft
Phase shift
Cutting edge angZe
Phase shift
Natural angular frequenoy
Chatter frequenoy
5.1. An analysis of the vibration
0
0
0
0
rad 0
rad
rad/s
rad/s
It can often be observed that when turning shafts, the
workpiece itself causes chatter. In that case the dynamic stiff
ness of the machine tool exceeds the stiffness of the shaft. The
vibration of a shaft clamped between live center and chuck gene
rally can be described by the motions in any two different
directions. If the angle between both directions mentioned is
90
I
:1-Fig. 5.1. Fig. 5. 2.
An analysis of the vibration of the shaft. The "mode-aoupling" effeat.
equal to 90°, there is no significant cross compliance between
them. The two principal directions are preferably chosen aceer
ding to the situation given in Fig. 5.1.
It is assumed that the vertical component of the vibration
does not affect the dynamic cutting force. Consequently, the
influence of a torsio~al vibration can also be excluded. Thus,
the dynamic cutting force will only be generated by the compo
nent of the vibration in the horizontal direction.
An introduetion of the equivalent dynamic quantities of the
clamped shaft me•· ke and ce enables a simple description of the
vibration on the threshold of stability according to
m e xl + c e xl + k e xl = bg kil (x~ - x 1) - b ei! XI g (5. I)
m xz + c *z + k xz b k.2 (x; - XI) - b ci2 x! e e e g -1 g (5.2)
where
x*-I XI llh (5.3)
XI = XI exp(i ûlk t) (5.4)
x*= I XI exp[i (wk t - <!>)] (5.5)
91
(5.6)
The angle e represents the phase shift between the vertical and
the horizontal vibration.
From the eqs. (5.1) and (5.2) it follows
(5. 7)
b J.:.!l [k [exp(- i <j>) - 1] - i wk c~ 2] exp(i e) g 1x21 i2 L
(5. 8)
Consiclering separately the vertical motion (eq. (5.8)),
this system cannot become unstable by itself since it is assumed
that the dynamic cutting force is not influenced by that motion.
Contrary to this, the horizontal vibration is directly re
lated to the chip thickness modulation and the resulting dynamic
cutting force is only controlled by the displacement x 1. Thus
the stability of the system is defined by eq. (5.7). This
equation yields the chatter frequency when equaling the real
part zero.
0
From this it follows
It appears that the chatter frequency always exceeds the
natural frequency of the vertical vibration which can be
92
(5.9)
(5. I O)
written as
(5. 11)
It becomes clear now that generally the values of the phase
shift between force and displacement are different for both
directions of vibration. From this and from the phase shift
between the excitation farces represented by the right hand
termsof the eqs. (5.7) and (5.8) the phase shift e results.
For given conditions the value of e results from eq. (5.8).
If e ~ p TI (p = 0, I, .•. ), the pathof the taaltip with
respect to the workpiece is elliptical. (See Fig. 5.2.) The
direction in which the ellipse is described depends on the sign
of e. Apart from the energy which is generated by the wave-on
-wave chip thickness modulation, the vibrational energy is
controlled by the nature of the elliptical motion. If the
displacement of the shaft in the direction of the cutting force
goes simultaneously with the maximum penetratien depth and at
the same time with the maximum cutting force, extra energy is
supplied to the system for chatter. The area of the ellipse is
a measure for that quantity of energy. Whether the energy re
sulting from the elliptical motion is positive or negative de
pends on the sign of e. Tlusty (I) reported this "mode-coupling" effect for the
first time. He showed that in situations where the wave-on-wave
chip thickness modulation is excluded, the cutting process can
yet become unstable as aresult of "mode-coupling". Generally
"mode-coupling" occurs simultaneously with the regenerative
effect. However, it has also been observed by Tlusty that when
turning shafts, the threshold of stability is mainly controlled
by the regenerative mechanism. This has been confirmed by
experiments carried out by the present author. Hence, the
"mode-coupling" effect has been neglected when
the threshold of stability in the next sections.
93
5.2. The stability criterion
5.2.1. Orthogonal cutting
Regarding the assumption made in Sectien S.J. and neglect
ing the influence of "mode-coupling", it has been shown that
only the horizontal vibration of the shaft is of importance
with respect to the stability of the cutting process. As a
matter of fact, the shaft can be considered as a one-degree
-of-freedom system. Thus, the situation is quite analogous
to the one that occurs when applying the special toolholder
for ~ = 0° Chapter IV dealt with (2). Fora radial feed the
relevant component of the dynamic cutting force can be
written as (see eq. (4.23))
or
Next the stability limit can be calculated with the aid of
eq. (4.34) according to
(5. 12)
(S. 13)
(5. 14)
The dynamic behaviour of the lathe being of minor importance,
the dynamic quantities of the clamped shaft can be obtained by
performing relatively simple experiments directed towards the
measuring of the dynamic compliance at natural frequency X0
/F 1•
Then, it follows for a one-degree-of-freedom system
An approximation for both the 2~k-value and the
natural frequency can also be obtained by measuring the
94
(5. 15)
8 7 6
5
4
3
2
5
~
1.
0
70
10 0
15 0
20 0
damping (%) ;,.,."_
~V. r---- ...........x.:.._ r-._~
'Á
600 700
L---P-~ -notural frequency fo
Hz
I I = ...._ ........_
.......... r ............. r--J -.....
x r----r--... ........_ r.......... ,_ r--xR
800 900 1ooo 1
-1--,- - I--1-- I-X ~ 1---1---'
,_ L---
Fig. 5. 3. ExperimentaZ diagram aft er Lil".dstr>öm (3),
shoûling the dynornie pr>operties of a aZamped
Fig. 5.4.
shaft a function of its îength.
Cutting geometry when turning with a cutting
edge angle K 45°,
x..., x
x -.... 1100
)\-1
static stiffness of the clamped shaft. From this, the
damping ratio can be obtained with the aid of experimental
diagrams as applied by Lindström (3). (See Fig. 5.3.)
shaft I ength (mm)
95
5.2.2. Introduetion of a cutting edge angle K ~ 90°
When a cutting edge angle K ~ 90° is applied, we have to
take into account the changed cutting geometry. The undeformed
chip thickness h0
is considered to be a basic quantity.
Accordingly, equivalent cutting conditions are achieved for
S COS K (5.16)
Fig. 5.4. shows the situation for K = 45°. The relevant compo
nent of the vibration is represented by y.
It is assumed that the component of the displacement coin
ciding with the direction of the cutting edge does not affect
the dynamic cutting force. Therefore, the variation of the
width of cut caused by this vibration is neglected. When
also neglecting the non-linear influence of the nose radius of
the tool, it follows that the direction of the dynamic cutting
force 6F is perpendicular to the cutting edge. So, the relevant
component of the dynamic cutting force is given by
6F b [kil 8h ' dy J 0 - cil dt
where ki_l kÏ] COS K
ci_l cil cos K
The effective dynamic stiffness of the shaft is now
or 8F 0
T.Y
COS K
COS K
Thus we can derive the stability equation
96
(5. I 7)
(5. 18)
(5. 19)
(5.20)
(5 .21)
2 ç k b e (5.22) g
kil COS K ci1 I - w
0
b 2 ç ke
(5.23) 2 g kil cos K cil
w 0
5.3. Experimental verification
Experiments to measure the limit width of cut b have g
been carried out for bath situations mentioned in Sectien 5.2.
The experiments have been performed while using the work mate
rial SKF 1550. A feed was applied of 0.07 and 0.22 mm/rev in the
case of orthogonal cutting and 0.10 and 0.30 mm/rev respective
ly in the case of K = 45°.
The corresponding calculated stability charts are obtained
with the aid of the eqs. (5.14) and (5.23). The specific
cutting data of the workmaterial are taken from Chapter IV (2).
The results shown in the Figs. 5.5. and 5.6. are obtained with
orthogonal cutting, excluding the influence of the nose radius
of the tool. The dynamic stiffness of the shaft at natural
frequency (ISO Hz) was approximated to be 0.6 x 106 N/m.
The data of the Figs. 5.7. and 5.8. result from cutting
experiments with a cutting edge angle K 45° and a longitudinal
feed. In order to reduce the influence of the nose radius of the
tool, the dynamic stiffness of the clamped shaft was increased
up to about three times its initia! value by decreasing the
length of the shaft. In doing so, the minimum limit width of cut
increases approximately proportional.
Measurements showed that the natural frequency of the
clamped shaft increases to about 190 Hz. to results
on the dynamic cutting force as a function of frequency as
obtained by Van Brussel and Vanherck (4), where it appears
that within the range of 130 T 180 Hz the overall dynamic
97
.,. ...
2.5
g b
fm ml
.,,
t5
1.0
~
0
Fig. 5.5.
75 1. bg lm
1. mi ~~
1. 25
.00
0 .75
0. 5
0 .25
0 0
Fig. 5.6.
98
mat:SKF1550 s:O.O!.mr:'~~ev_ 0 I .'lb 90 ,o:-6 •6Y- 5 i2tk.:::0.6x10 N/m e:vibr. ampl. > 1011 0: • " <1011 x:stable conditions c:compute.d values
.. \ • .. ~uTr~ \ J t1' - - -0
\o [J
• 0 • • ;/ ;., -''-.. / )( ~o-e/ x "
i
0.5 1.0 1.5 2.0 2.5 v I mis}
The limit û!idth of aut b g as a funat~:on of autting
speed v appZying a radial feed of 0.07 mm/rev.
mat:SKF 1550 s= 0.22 mm/rev 0
'M.=90°; a: 6°: v= 5
I' I 2tkt0.6x106 N/m
e: vibr. ampl.> 1011
\ o: .. • ( 1011 x: stabie conditions c: computed values
\ .-
\ /~/ V
~-[I
_/ ...
~ D.25 0.50 0.75 1.00 125 1.50 1.75
v( m/s)
The limit width of aut b as a funation of autting g
speed v applying a radial feed of' 0.22 mm/rev.
10 ~------------m-a_t_:_S_K_F_1_5_S_o __ s---~~~~O~m-m~/~r-ev----~
bg rE= 0.4 mm x" 4So; a=6o; 'V" so (mml 2tk.,.1.46x106 N/m
F'ig. 5. 7.
9~----l-----r------t e: vibr. om pl.> 1011 ., • • <1011 •: stabie condit i ons o: computed val u es
e~----_.~~--~------4
~ ~ 71---------+-11\----+----~---1
_\ l 6~x ~x~ ~---1--/-1--------1• 5~---.--~~~~~.----+~~~~~~~~
x x\~ • /-/x 4~----~----~~,~.0.-.~ ~~~*-~----~
b ...... li! .... ~--sc-1--.. ~--
31------..,~
x x
x ~ x x x
~·~----~----~------~----~----~
1~------~----~------~-------+------~
0 0.5 1.0 15 2..0 2.5 v ( m/sl
The limit width of out bg as a functicn of autting
speed v a longitudinal feedof 0.10 mm/rev.
cutting force remains unaffected by the modulating frequency,
it is assumed that for both shafts mentioned the vah.1es of k.
and of ciwo are identical. This implicates that the damping
coefficient ei is a function of frequency (see also Section
4. 5).
~
In general a good agreement is shown between the calculated
and the experimental results.
When cutting in unstable conditions it has been observed
that the phase of the vertical displacement of the shaft is
that of the horizontal one. This results in a nega-
ï .:e of the phase shift 8, indicating au additional ener-
99
Fig. 5.8.
10
bg (mml
9
8
7
6
5
4
3
2
0 0
x
0.25
mat: SKF 1550 s = 0.30 mm/rev rE= 0.4mm 'M.= 45o: <1=60: v= so
2tk.% 2.00 x10 6 N/m e: vibr.ampl.> 1011 Cl: " • < 10 1L x: stabie conditions c: computed values
~ I
\ /_ ~ I ~ • • x
\ ., ,:/ x
x x x
050 0.75 1.00 1.25 1.50 1.75 v ( m/s)
The limit width of cut b as a function of cutting g
speed v applying a longitudinal feed of 0.30 mm/rev.
gy supply for chatter. However, the results of the Figs. 5.5.,
5.6., 5.7. and 5.8. show the influence of "mode-coupling"
on the threshold of stability being of minor importance.
5.4. Conclusions
The fair agreement between the theoretical and the exp.eri
mental results, as shown in the previous section, confirms the
usefulness of the method for measuring the dynamic quantities
of the cutting process, which initially has been described in
100
Chapter III (5). There are no doubts about the relevancy of the
influence of process damping.
In order to calculate the stable conditions when turning com
pliant shafts, the system can be reduced to a one-degree-of-freedom
system, its direction of vibration coinciding with the direc-
tion of chip thickness modulation.
From the results it also appears that the data obtained from
orthogonal cutting can be applied in the cases where K I 90°
if the undeformed chip thickness is considered to be a basic
quantity.
Finally it has been proved that, according to the assumption
madeinSection 5.1, the vibrational component in the direction of the cutting speed does not affect the dynamic cutting
force. With respect to this it should be mentioned that
experiments show that during cutting under unstable condi-
tions the vibrational component in the direction of the
cutting speed generally exceeds the component in the direc-
tion of chip thickness modulation. Moreover the assumption
is confirmed that both quantities k; and c: are independent l. l.
of the direction of vibration. This assumption has been
introduced in Chapter IV (2).
It shÓuld be remarked that the economie cutting speeds,
being usually applied'in turning, easily exceed those
used here. However, within the scope of studying the chatter
phenomenon the cutting speeds applied are of special interest.
References
(I) Tlusty, J., Polacek, M., Danek, 0. and Spacek, L.,
Selbsterregte Schwingungen an Werkzeugmaschinen. V.E B.
Verlag Teehuik (1962).
(2) Kals, H.J,J., Fertigung 5 (1971) 165.
(3) Lindström, B., C.I.R.P. Ann. 20 (1971) 5.
(4) Van Brussel, H., Vanherek, P., Maasurement of the dynamic
cutting coefficient and predietien of stability. Report
70e16, University of Louvain, presented to the C.I.R.P.
Teehuical Committee Ma, Tirrenia (1970).
(5) Kals, H.J.J., C.I.R.P. Ann. 19 (1971) 297.
I OI
VI A DISCUSSION OF RECENT RESULTS FROM LITERATURE
Nomenclature
Width of eut
Limit value of width of eut
Amplitude of the foree component eaueed by
the inner modulation of the chip
F10
Amplitude of the foree component eaueed by
the outer modulation of the ehip
öF. Component of the reeulting d.ynamie eutting J
m
m
N
N
foree aarreeponding with the direetion j N
h 0
öh 0
Dynamie oomponent of the feed
Dynamie component of the main eutting force
Nominal undeformed ahip thiakness
of chip thiekness modulation
Ii,I0
Quadrature component of the dynamia stiffneee
per unit of width of eut eaueed by the inner
chip modulation and the outer chip modulation
k
k. ~
of the machine tool struature
;;n,"m,.T7 . .r• proeess stiffness
klv Chip thiekness coeffieient of the main
autting
Chip thickness coeffieient of the feed force
N2j Inner modutation coeffieient of the component
of the speeifie dynamia eutting in the
direation j
Ri,R0
In-phase aamponent of the d.ynamie
R .. ~J
102
per unit ofwidth of eut eaueed by the inner
chip moduZation and the outer ahip modula
tion respeetively.
Coeffieient the oomponent of the s"'"m,.n.r•
N
N
m
m
N/m
N/m
s Feed
V Cutting speed
Y Amplitude of the innep ohip thiokness
modulation y* Amplitude of the outer ohip thiokness
modulation
Phase shift (See eq. (6.1))
Phaee shift (See eq. (6.2))
~ Phaee shift between the inneP and outer
ohip thicknese modulatione
w
Mean shear angle
Damping ratio of the machine tool stPUcture
Angular frequeney
6. 1. Introduetion
mm/rev
m/s
m
m
rad
rad
rad 0
rad/s
Over the last number of years there are investigators
who advocate a more detailed approach of the dynamic cutting
process. The introduetion to this is found in the work of
Das and Tobias (1). Starting from a shear plane model of
the cutting process, these investigators present a pure
geometrical analysis of the wave-on-wave cutting process
that occ1.1rs when the tool vibrates. They consider separately
the influence of the inner and the outer modulation of
the undeformed chip and derive the relations on the dynamic
cutting forces on the basis of static parameters only. From
this it follows that a phase shift is introduced by both
the dynamic force component of the inner modulation and the
force component of the outer modulation.
In this way of thinking, Polacek (2) developed a method
to measure the various dynamic components of the cutting
force applying a dynamometer.
Van Brussel and Vanherck (3), (4) carried out experiments
on the same subject. They propose a metbod yielding the
dynamic stiffness of the cutting process. As a matter of fact
103
this method is basically analogous to the one described in
Chapter III (5). The observations of Van Brussel and Vanherck
confirm the theoretica! results of Das·and Tobias concerning
the inner and the outer modulation forces behaving independent
of each other.
6.2. Discussion of the results
With respect to the different edges of the modulated chip,
Van Brussel applies the following force equations
F1i exp[i (w t + a)] ~ b Y (Ri + i \) exp(i w t) (6. I)
(6.2)
where
index i refers to the direct chip modulation
- index o refers to the delayed chip modulation
R is the in-phase component of the dynamic cutting stiffness
- I is the quadrature component of the dynamic cutting stiffness
- ~ is the phase shift between the two chip thickness modula-
tions
- 8 and E are the phase shifts of the dynamic cutting force
components with respect to the direct and the delayed chip
thickness modulation respectively.
The results of the various parameters obtained by the author
mentioned are shown in Table 6.!. The data show different values
of Ri and R0
• However, from a point of view it may be
expected that for any cutting condition the values of R~ and R0
are equal. The outer modulation will only change the resulting
depth of cut. This has been proved by Van Brussel and Vanherck to
have no influence on the overall dynamic cutting force (4). Thus,
104
one may conclude that the· values of ki obtained by the present
author can stand for Ri as well as for R0
•
In this way of thinking, differences between the inner modula
tion stiffness and the outer modulation stiffness can exist only on
account of the quadrature components Ii and I0
•
According to the theory of Das, Van Brussel explains the
existence of the leading quadrature component I0
with the aid of
a shear plane model (3}. Das derived the next equation of the
phase shift g between the outer modulation and the outer modulation
force component
e: "' ~ h cot i!> v o m
where il>m represents the average value of the shear angle.
Eq. (6.3) shows that for small values of the chip thickness
(6.3)
and for values of the cutting speed the influence of e: can
be ignored.
The stability criterion as applied by Van Brussel and
Vanherck yields the.limit value b when the dynamic stiffness g of the machine tool equals the negative value of the resulting
stiffness of the cutting process. Fig. 6.1. shows the graphical
salution method based on the stability criterion mentioned. A
straight line approximates the dynamic stiffness of a low damped
machine tool. The intersectien of this line with the out-of-phase
axis of the polar diagram represents the dynamic stiffness 2sk
s -0.22 mm/rev frequency of excitation 150 Hz
v (m/s) R. ( 109 N/m2) R (109 N/m2) I. (l o9 N /m2) I (JOg N/m2)
1 0 1 0
0.37 1.83 0.48 1.58 0.73
0.47 1.33 0.73 1.05 o. 70
0.58 1.08 0.78 0.65 0.35
0.75 1.28 0.93 o.ss 0.45
0.93 1. 15 0.95 o.ss 0.35
1.5_0 1.35 I. 15 0.60 Ó.40
Table 6.1. Results afterVan Brussel and ïanherok.
105
dynamic stiffness of t he cutt i ng proc ess
out-of-phase axis
dynamic stiffness of the machine tool
Pig. 6.1. The graphioal so~ution methad for the limit width
of out by Van Brussel and Vanherok.
b'g aoo.---~----.----.--~r----.---,,----.---,
(%)
700
600
5001-----.-----,
4001-----+----1
I
0~--~--~----~--~----L---~--~--~ 0 0.25 0.50 0.75 Ii _!g_ 1.00
Rî • Ro
Pig. 6.2. The influenae of the ratios I0
/R0
and Ii/Ri on
the limit value b for a generalized situation. g
106
at natural frequency. The inclination at the point of inter
sectien approximates 2ç rad. The components Ri and Ii are
plotted with reversed sign, whilst a circle, having 0' as centre
and the radius /(R 2 +I 2), gives all the loci of the stiffness 0 0
of the cutting process per unit of width of cut. In order to
obtain the limit value bg' straight lines are drawn passing
through the origin 0, intersecting the circle and the machine
tool stiffness locus. The minimum of the ratios AO/BO, A'O/B'O,
etc. brings in b • The different lines correspond to different g relative phase shifts between the inner and the outer modulation
as indicated by the angle ~.
With the aid of this method, the influence of the quadrature
components I0
and on the threshold of stability has been
calculated. Fig. 6.2. shows the influence of the ratios I0
/R0
and I./R. on the limit value for the generalized situation that 1 1
R = R'., R /r;,k being constant, and;; = 0.025. From the figure 0 1 0
it can be concluded that the influence of I. on b is considerably 1 g
stronger than the influence of I on b . It should be noticed 0 g
that in general Van Brussels' results leadtoabout the same
values of I0
/R0
and Ii/Ri.
With respect to the parameters of the outer modulation an
other important remark can b~ made. Contrary to what may be
expected from the foregoing, the results of Table 6.1. show
values of R0
which are considerable smaller than those of Ri. However, it is remarkable that for all the different cutting
speeds applied, the values of R. and /(R 2 + I 2) are approxi-1 . 0 0
mately the same. This is diagrammatically shown in Fig. 6.3.
With the aid of the graphical salution method of Fig. 6.1.
a s~bstitution of R and I by the real it can be seen that
component /(R 2 0
2 0 0 + 1
0 ) does not affect the limit value. The
introduetion of I0
will only increase the phase shift between
R0
and Ri.
Van Brussel and Vanherck (4) computed stability charts for
their special toolholder. Gomparing the theoretica! and the
experimental values of $, it follows that the discrepancy
between the values of both series of results generally is
of the samemagnitude as the influence of 10 on~ (5%).
107
Fig. 6.3.
2
~ x109
N/m2
/~c / /
/.~/ /c
1 ,çc c
# /
v~ /
V x109N/m2
2
The agreement between the resulting Çynamia
stiffness of the outer modulation I(R2 + r 2
) and 0 0
Ri' afterVan Brussel's results.
The facts mentioned do not support the relevancy of 10
,
the more so as the present author's results, which have been
obtained excluding the influence of a quadrature outer roodu
lation component, show a very good agreement between various
series of practical and theoretica! results. With respect to
this it is mentioned that in some cases the E-values can
increase up to 0.7 rad.
Table 6.2. shows the results obtained by Polacek. The equa
tion of the dynamic cutting force derived by the latter can
be written as
(6.4)
Polacek shows this equation to be similar to Van Brussel's
equation with the only difference that the parameters in eq.
1_08
s (mm/rev) 0.05. 0.1 0.2
v (m/s) 0.47 0.83 1.67 0.47 0.83 I. 67 0.47 0.82 1.47
freq. of f1Ff l1F l1Ff l1F l1Ff l1F t,Ff l1F t,Ff t,F t,Ff t,F t,Ff t,F t,Ff t,F t,Ff t,F exc1tat.
50:200 Hz V V V V V V V V V
RI (I 09 N /m2) 0.09 0.67 0.27 1.05 0.59 1.51 0.29 0.85 0.36 1.45 0.54 1.34 0.08 0.88 0.32 1.24 0.15 1.09
R2 (I 09 N /m2) o. 17 0.23 0.05 -0.11 0.06 0.08 0.19 0.26 -0.04 -0.24 -0.03 -0.3 0.11 0.24 -0.08 -0.08 0.05 -0.49
N2(105 Ns/m2) 1.5 0.54 0.84 0.74 1.17 1.07 4.7 1.07 1.93 2.35 6.62 6.62 4.9 1.07 2.78 1.49 I. 71 5.87
R3 (I 09 N /m2) o. 12 0.5 o. 17 0.88 0.4 1.36 0.09 0.57 0.27 1.41 0.21 1.08 0. 13 0.67 o. 19 I. 21 0.17 I. 14
R4 (109 N/m2) 0 0.08 0.07 -0.11 0.13 -0.15 -0.05 -0.18 -0.11 -0.17 -0.09 -0.24 0.02 0.04 -0.11 -0.05 -0.08 -0.04
Table 6.2. Results after Polacek.
(6.4) relate toa particular direction j, whilst the parameters
of the eqs. (6.1) and (6.2) stand for the resulting dynamic
force. The analogy is
A direct comparison of all the results mentioned with
the author's findings is not possible since Polacek used a
different work material and Van Brussel did not mention
any materials specification.
With respect to Polacek's results it draws attention that
positive as wellas negative values·of R4 are obtained. It is
obvious that the negative results do not fit the theory of
Das. Moreover, according to Das' theory, the values of R4
/R3
should show an increase with respect to an increasing feed.
This, however, is also not confirmed by the results pf Table
6.2.
Resuming one can conclude that, at least for feed values
up to 0.2 mm/rev, the physical meaning of !0
in relation to the
shear plane theory is very doubtfull. At this stage one can
make objections agains~ the assumption made by Das that the
orientation of the shear plane will remain unaffected by the
vibration. Physical considerations suggest that the direction
in which the shearing zone propagates will be controlled by
the stress conditions close to the tip of the tool. Thus,
the variatien of the cutting force will be strongly affected
by a dynamically changing shearing process.
In reference to the inner modulation damping, Polacek's
results as well as the results mentioned in Chapter IV (6)
show that in the directions of both the feed and the cutting
speed, the damping can be positive and negative as well.
Das' results only permit a negative damping with respect to
the dynamic component of the force in the direction of the
main cutting force and a positive damping related to the
component in the direction of the feed force, according to
110
where
l::.F V
8Ff • kif 8h0
sin w t + k h ~ 8h cos w t lv o v. o
klv is the chip thickness coefficient of the main
cutting force
- kif the chip thickness coefficient of the feed force
- 8h0
the amplitude of chip thickness modulation.
(6.5)
(6.6)
Finally, it should pe mentioned that the assumption that the
component of a vibration in the direction of the main cutting
force has no influence on the dynamic cutting force (see Chapters
IV and V) is confirmed up to a great extent by experiments carried
out by Polacek (2).
References
(I} Das, M.K., Tobias, S.A., Int. J. Mach. Tool Des. Res.
7 (1967) 63.
(2) Polacek, ~!., Slavicek, J., Messen des Dynamischen
Schnittkraftkoeffizienten und Berechnung der Stahilitäts
grenze. Bericht des Forschungsinstitutes fÜr Werkzeug
maschinen und Zerspanungslehre, VUOSO, Prag (1971).
(3} Van Brussel, H., Vanherck, P., !lth Int. M.T.D.R.
Conference, Manchester (1970).
(4) Van Brussel, H., Vanherck, P., Measurement of the dynamic
cutting coefficient and prediction of stability. Report
70cl6, University of Louvain, presented to the C.I.R.P.
Teehuical Committee Ma, Tirrenia (1970).
(5) Kals, H.J.J., C.l.R.P. Ann. 19 (1971} 297.
(6) Kals, H.J.J., Fertigung 5 (1971) 165.
lil
APPENDIX I
The experimental set up for measuring the limit width of cut
When during cutting the critical width of cut is reached,
the amplitude of vibration increases suddenly to a high value.
(See Fig. I. a.) Therefore the cri ti cal width of cut is def.ined
at the bending point of the curve showing the amplitude of
vibration versus width of cut. As for very low cutting speeds
this point cannot be clearly distinguished, in this case the
critical value is supposed to be achieved when the amplitude
of vibration exceeds JO ~m.
0 2 v, (~m
1
)
6
2
0
8
6
4
2
~~
I '
u !
I I I i -- EN 8 stnl
-- CZE CH. 12050.1 stool
--- c 45 N ctnl
---- UHB 11 st .. !
' v ;; Q.9 m/s ho" 0.072 mm I Y1= half p.p~volue
'I I
1!!.
V" /'
<~ .. : ~
I . ·"' 9'
l/~ lv- i- . ........... / ~ /
. ........-::: ~ ,-' -~ -
~~~-- -~ --p ·-
2 3
Fig. I.a. The amplitude of vibration Y1
as a_funetion of
u;idth of eut b.
112
I
i I
' f I I
' ; I
; /
I
I
4 b (mm)
The displacement of the tool is measured by strain gauges
mounted on the leaf springs of the special toolholder. (See
Fig·. 4.8.) The strain gauges are connect:ed to a strain-gauge
amplifier (Hottinger). After being led through a wave ana
lyzer, the output signal of the amplifier is conducted to a
AC/DC converter and next the mean value of the vibration
amplitude is mechanically recorded. The wave analyzer is mainly
used to eliminate the influence of low-frequency oscillations
of the tool due to other mechanisms than regenerative chatter.
The electronic equipment is shown in Fig. I.b.
Fig. I.b.
I eaUbration unit
I L_ __ _j
a.c.-d.c.
The electronic equipment for measuring the limit
width of cut b • g
APPENDIXII
The experimental set-up for measuring the pulse response
The experimental set-up consists of the special teelholder
which has been mounted on a 10 kW-lathe, make Lange. The natural
frequency of the toolholder was chosen to be about 160 Hz, while
the damping ratio was adjusted to approximately 0.08.
113
stroingouges
amplifier
<
sine generotor
Fig. II.a. The eleetronie equipment for measuring the pulse
response.
The displacement signal from the strain gauges is led through
a strain-gauge amplifier and a galvanometer amplifier, and next
fed into a U.V. recording-oscillograph (C.E.C. type 5-124). The
value of the frequency of the pulse response is obtained by cam
paring the response signal with a reference signal of 1000 Hz
being recorded simultaneously on the U.V. oscillograph-paper.
The electronic equipment is shown diagrammatically in Fig.
II.a. The pulse is simply generaeed by a stroke of a rubber
hanuner.
To eliminace the influence of the delayed chip thickness
modulation during cutting, data are only taken within the
time of one revolution of the workpiece.
APPENDIX lil
Graphical solution method for the limit width of cut bcr
For this method of solution we need the harmonie response
locus of the machine tool, as mentioned in Chapter II.
Sealing this response locus. to the specific response locus
for F1 = I N, every vector OP will give the specific displace-r
ment Ys for the wj chosen. (See Fig. III.a.)
114
0
Fig. III.a. GraphicaZ salution methad for the criticaZ width
of cut bcr·
A relation between the relevant component of the cutting
force fiFa' the specific stiffness k: and the specific process
damping c7 is given in equation (4.23). With
... * ... \l y - y (1II.l)
where \l is the overlap coefficient giving the effect of the
delayed chip thickness modulation (for orthogonal cutting \l 1),
and
c. ].
lc71 cos y
it yields for the amplitude of the dynamic cutting force
With an amplitude of I~~ = I N and e~ as the in-phas·e a
unit vector eq. (111.4) results in
-ei (dYS) - \l y* + y ... +
s s k. dt ].
(liL 2)
(III.3)
(III.4)
(1II.5)
115
The definition of the threshold of stability is
(III. 6)
Using the specific response locus of the structure, of the
eqs. (III.S) and (III.6) the length and the direction of Y+ the s' the direction of -er/(b k.) are known.
~
_,.* length of ~y , as well as
s -The vector (ci/ki)(dY
8/dt) can be written as iw.(c./k.)~+; both
J ~ ~ s its length and its direction can be computed for the wj-value
chosen. The graph of Fig. III.a. can be completed now and the . . +* +/( k ) d~rect~on of ~Y8 and the length of -el b i can be read.
+ It should be noticed that the phase angle ej between Y
8 and
+* ~Y8 bas to satisfy the phase equation
1f d e = -- w. - 2 11 p V J
(III. 7)
This comes forth from the ratio between the hypothetic chatter
frequency and the rotational speed of the workpiece. To calculate
the number of periods p, counted in one revolution of the work
piece (diameter d), we compute the point of the response locus
where b will reach its minimum value b • The value of the real g
component in this point is defined as R and the corresponding g
angular frequency is wg.
p entier ( d w ) 2v g (III.8)
Knowing p, we look for the point (in the neighbourhood of R ) g
where the angle ej satisfies eq. III.7. The value of the width of
cut b which results from this solution, represents the true cri
tical valtie b er Because the quantities ki and ei depend upon the cutting
speed v, we have to repeat this computation for a number of
values of v.
A flow chart of the computer program directed towards this
methad of salution is given in Fig. III,b.
116
READ:
1 • number of points the locus m = of cutting
X5(j) =X/F
~ y ./F J
compute point z where X5 (j) reaches its minimum index z•r, R•R
Fig. III.b. FlJwahart of the computer program.
compute point z where b reaches its minimum b (R = R )
g g
check phase equation
compote bcr' e, wk
PRINT: k, v, q, "i. p,
wg' "'k' bg, bcr' e
false
PLOT: b and b
g er w and
117
SAMENVA TIING
verspanende bewerkingen van materialen met behulp van
gereedschapswerktuigen treden trillingen op welke leiden tot min of
meer afwijkingen in de nominaal ingestelde afstand tussen
snijgereedschap en werkstuk.
De aard van de snijkrachtsvariatie die door de wordt
gegenereerd enerzijds en de dynamische eigenschappen van het gereed
schapswerktuig anderzijds bepalen de grootte van de relatieve ver
plaatsing tussen gereedschap en werkstuk. Deze relatieve beweging
kan onder bepaalde omstandigheden instabiel van karakter worden.
Hierbij wordt tenminste de oppervlaktekwaliteit van het produkt
in ernstige mate geschaad.
Het belangrijkste onderwerp van dit proefschrift is de bepaling
van de dynamische eigenschappen van het werkstukmateriaal in hun
betrekking tot de stabiliteit van het verspaningsproces.
Hiertoe wordt in hoofdstuk III aan de hand van een eenvoudig
model, opgesteld in hoofdstuk II, een methode ontwikkeld voor de
bepaling van de dynamisch relevante materiaalgrootheden.
Met de verkregen resultaten kan de invloed van de materiaal
eigenschappen op de stabiliteit van het verspaningsproces worden na
gegaan. In hoofdstuk IV wordt deze invloed bezien in 'samenhang met
de dynamische eigenschappen van het gereedschapswerktuig. De geldig
heid van het model wordt voor een aantal representatieve verspa
ningskondities aangetoond, terwijl een aantal opmerkelijke gedra
gingen met betrekking tot de stabiliteit van het verspaningsproces,
die in de praktijk worden waargenomen, kan worden verklaard.
In hoofdstuk V wordt een praktische toepassing gevonden in het
draaien van slanke assen. De overeenkomst tussen de experimentele
en de theoretische resultaten illustreert de betrouwbaarheid van de
gemeten waarden voor de dynamische materiaalgrootheden met betrek
king tot de technische toepasbaarheid.
Tenslotte wordt in het laatste hoofdstuk het gekozen model be
sproken tegen de achtergrond van recente bevindingen in andere labo
ratoria.
118
CURRICULUM VITAE
De schrijver van dit proefschrift werd in 1940 te Heerlen
geboren. Na de middelbare schoolopleiding bezocht hij van
1958 tot 1962 de Hogere Technische School te Heerlen, alwaar
hij voor het afsluitend examen van de afdel
kunde met lof slaagde.
Werktuigbouw-
Van 1962 tot 1964 vervulde hij zijn militaire dienstplicht.
Tijdens deze periode ontving hij een opleiding tot genieofficier.
In 1964 begon hij zijn studie aan de Technische Hogeschool
te Eindhoven; in 1969 verwierf hij daar met lof het diploma van
werktuigkundig
Sinds 1966 is hij als medewerker verbonden aan het labora
torium voor Mechanische Technologie en Werkplaatstechniek van
genoemde Hogeschool. In de vermelde funktie vond de gele
genheid het onderhavige proefschrift te bewerken.
1!9
