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
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Published: 01/01/1972
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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 vrijdag19
mei1972 te 16.00
uurdoor
Hubert Jan Jozef Kals
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
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
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
k
1 Chip thiaknesa aoeffiaient k
2 Penetration aoeffiaient k
3 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
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
"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
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.
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
I t 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
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
~
/ 0mot , 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----' . 100a
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.
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.
8
bg
(mm)
---Eindhoven Unlver1lty of Technology
==n~:!:
....
Fig. 1.3. bg (mm) Fig. 1.4. ($ 4 2~\
I I ~I,'
\"
... \\ ,
II
I' I ... \~-
...
_
I I !' \·"'
.I
J",
''f.<
'
~'
'·'
'.,
",
I ~ 1.0 ~ 20 v(mls) The limit width of cut b versus cutting speedg
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
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..
.}_,
~'
"-i
V
!:
'
-"·
---~
''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.
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).
(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).
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
v NominaZ carriage speed 0 w Direction of exeitation x 8t Static defleetion
x
Amplitude of displacementx
Amplitude of displacement at natural frequeney 0x
1 Half peak-to-peak value of displacement X Amplitude of velocity at naturaZ frequency0
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 decrementv Dimensionless frequency
~ Phase angle between force and tool displacement
w Angular frequenay
w
0 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
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 V cos (2 .4)
u == cos a cos Sw (2.5) vw V it follows (ayF + i b F) y
=
a ww u + a (u + u ) + a u + ~vw vw vw wv vv vv + i [ b u + b (u + u ) + b u ] WW WW VW VW ·wv VV VV (2. 6)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• FThe 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
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- k
m U) 2 + i c wk
(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.) 0w'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
(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 Ik
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
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,
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.
kX5
--•
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· 7.20Fig. 2. 3. Resonanoe euYves of a bedstide for severat
vatues of V
0.
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= F
1/(cV0)r'The quantity V0 represents the nomina! sliding
speed, F
1 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
5
t
=
0.1 kx,
tw=
0.4F,
4 F1 k w • -eV0 3tw·aim
0t.
2rmt
Q2 Q4 0.6 1.4 I 1.6Fig. 2.4. Resonance curves of an analogue model for various values of the ratio of motion kw.
I ·~=96N
-
11
=48~ "..,..-/.v•
v·
/
4I
/ .
-~~·
!/
V
tran headstoek ~R 3I
re,-
eod·se -2I
V
i!
-{I
1 ,--..•
kw• 1 kw• 1 ~48N) ~ISN) 0 '( 1 2 3 4 5 6 7Fig. 2.6. The half peak-to-peak-value to force ratio kX1/F1 at natural frequency (v = l) versus
carriage speed V
0 for two values of the
e:xaitation force.
8 9 1 0
it holds for the amplitude of velocity at natural frequency
X
< V • This is confirmed by the experimental results shown0 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· 10twBleekdiagram and analog"tte 1110de1. of the moving ~e inctuding Coulomb f:r.'i.cti.on.
The proof that for k
=
I the relation.X /V=
I is validw 0 0
can be seen from the next derivation. X
0 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 F
1 is covered by Fl k xst 2 l; k xo (2.24) or c { f x m o c x 0 (2.25) Finally it yields Fl x k w 0 c V
v
(2.26) 0 0However, 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
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.
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
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
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 theradial 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 F
1/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
to this Fig. 2.11. represents some experimental results. Summarizing one can conclude that for V
0
=
0 the harmonieforces 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 V
0 on the
transfer funation of Vanherok's too~holder for ~o vaZues of the exoitation forae F
Q4
~
r----
x~~~
--
Vo Fi --_:::_;:.:::._77?7»'7777~ fr -Q2 ~14 Ql Z::_t•Q35 CS4 128 192 255 320 384Fig. 2.11. The amplitude to force ratio of the reZative displacement between tooU1oZder and carriage
l<x
1-x
2)/F1
1f
versus carriage speed for VaPious values0of 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 V
0, the coupling between carriage and frame is periodically rigid. Because V
0 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.
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.
k
1 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
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 + k1) '"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
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 k 1, 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 + kln 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.
(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.
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 m t = 2 1;, m t/(m k ) m Ns/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.
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
k
1 Chip thiekness coeffiaient k2 Penetratien coefficient k 3 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/m2 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
w
0 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
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. 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 thedirection 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, ÀFig. 3.2. F'ig. 3. 3. bg (mm) mot:C45N
w
h0 • 0.072 mm CmtA~~0.15 t.:experimentol volues i'
a I - \\
4À
\\~\.
/
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
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 w
0 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
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 (ç 5 + 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
ç
5-values obtained from the logarithmic decrement. The ki-value
Fig. 3.4. Fig. 3. 5. 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) 3.5The overall damping ratio ç
5 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 ç
5 and the frequency fc of the
