On the calculation of stability charts on the base of the damping and the stiffness of the cutting process

On the calculation of stability charts on the base of the

damping and the stiffness of the cutting process

Citation for published version (APA):

Kals, H. J. J. (1970). On the calculation of stability charts on the base of the damping and the stiffness of the cutting process. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0236). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1970

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WT-RAPPORT No.0236

ON 'I'I-IE: c..?'JJ2ULl\TION OF SrABILITY CP.J.i.Rl'S ON THE BASE OF l:mE

DAl'1PIN3 Al\1J) THE STIFFNSSS OF 'l'I-JE CU'Yl'IKS PROCESS.

by

H.J.J.

Kals

Eindhoven, University of Technology, the Netherlands

k',c,-/',,:"Yl to be _{to -the} Genel~a"t _{Assembly of} the _{C.I.R.P . .,}

This rep::>rt deals vlith a new rnethod for calculating stabili·ty charts. Sinple eA'}Jeriments I only m:::asuring frequencies I yield the values necessary to establish the threshold of stability. NC»l, the dynamic cutting coeffici.ent can be determjJ1ed.

A very good reserrblw'1ce betvleen the calculated values and the

experim;:mtal ones is sha;rlD for cutting speeds! higher than 60 m/min.

ZUSN'21E.'NFASSUNG

Dieser Bericht bE~schaftigt sich mit einer neuentvlickelten Methode zur rechnerischen Ermittlung von StabiliciitscUagramnen.

Die zur Best:lmrnung der Stabilitatsgrenze erforderlichen WerLe go.'1en dabei aus einfachen Frequenzmessungen. hervor. Sodann lassen sich die dynarnischen Schill. t:tJr-raftkoeffizie'1 te bestllm12l1. J3t2ispielsweise ,,,,ird schliesslich fur Schnittgescliivindigkeiten tib.er 60 m/min gezeigtl

Class die tibcreinsb.rrc.11lmg zwischen rechnerisch und eh']?erirnentell ennJ:ttelten Werte se.'Lr gut ist.

Ce rapport traite d1une nouvelle methode de calculer des graphiques de stc-milite. Des experiences simples, seule..rnent le mc~urage des fregueDces I dOlTI1ent les valeurs necessaires p::>ur fixer 121 limite de stahili te. Ensui te Ie coefficient dynarnique de cisaillamen'c p::mt etre determine. Une bonne concordcffice e.l1tre les valeurs calculees et

les valeurs ohte.r_{1Ues par les experie..'1ces est arrivee en cas que les} vi tesses de cisailletlP...nt seront plus hautes que 60 m/min.

-1-1. INI'I~')JUcrION

Actually, there are uvo rrain directions in tl1e field of dynamic stabili ty-'cesLs of warninG tools {1}.

-2-First of all there is a methcx::1 characterised by measuring the t-ran.sfer function of the tool.

Thent the crid.cal depth of cut is dcte:t:mined with the E?<luat:ion

The qu.allti ty kd seems to depe;.'1d upon the geometry of the cutting

process

as:

vJell as upon the viorking m-J.terial.

R _n is the ffi3X.Um.lffi _. negative real part of the polar curve. _.

The second method sinply consists of carrying out experimexlts in

order to establish the crit.ical depth of cut for standardized cOrieu-lions. 'lhe progress of the investigations concerning cutting stability is

nainly obstJ:::'ucted by an insu.fficient knc11dedge o~ the kd~value.

The latter value depe.nds on many gu.antities such as feed, cutting speed, tool wear, geonBtry of the tool, and vlorkpiece material.

Especially, the influence of the working material on cut'ling stabili·ty roak.es it difficult to compare results from cutting tests according to both methods.

2. THE INCREXiCNT1\L CO'ITIl\'G S'l'ITFbIESS

Peters and Vanherck {2} assUlre that it is alloNed to take the incre.'tental cutting stiffness k_i for the alrea?-'y ~ntioned kd value. 'lhus I they

calculate the critical depth of cut vli th the relation

b 1·

cr = 2(-R )k.

n 1.

T'ne tlU .. T118rical values of k. are obtains'<l by stad.c cutting tests. Fig. 1

. J.

sho,7s I for orthogonal cut·ting I the qllange lIF of th.e resul·tant cutting force due to em increase I~h of the . The incremental cutting stiffness ID-.::l.y be defined as

where

f3 represents the·cmgle between the vect.or !§! and the direction of

rrot.:i.on of the tool.

-3-Peters and Varlherck evaluated the calculated b -values with ~"<P2rim2nt:al cr

data from a special tool holder. A fairly good recernblaYlce beu'leen

both results Vlas found. HCNlever, exper:ilnents carried out in 'the LaboratOJ:Y of Production Engineering at the Eindhoven Uni versi ty of Teclmology I

using the sarre tool holder I did not confirm the usefulness of the methcx1 in the sarre extent.

We measured the cutting forces wi t11' a three-component dynarrometer which has its first natural fregue..nc.y at approxLmately 1.5 kHz. Our

:i::'esul ts are, sha/Jn in Fig. 2. In general, the calculated b cr-values I are considerable snaller thcu~ the exper.irrPJltal ones.

Among other things ,Fig. 3 ShOllS our ClJXves for k

i according to the methoJ,

of Peters and Vanherck, belonging to the calculated stability chart of Fig. 2.

It should be no-aced that all our tests are carried out for the follOlJing condi tions:

- orthogonal cu-tting

- workpiece TIaterial: C45 N

- tool: standard thrCNl-away type carbide tool tip P30

geometry: a

=

5°, y ~ 6°, K ~ 90°, Kl ~ 30°,

r ~ 0.4 Inn, A ~ 0°

, E:,

- nominal feed: 0.072 rnm/rev.

3. 'I'FIE DN1PlliG OF 'l'HE CIJITIl'-."G PROCEss

3 . 1. General

Many investi.gators in this field have already perceived the existe..nce of danping in the cutting process . In this context the Trost well-knolm relcttion l'>F

=

kl Llh+k

2 LlX-I-k3 II I') is giverl by Tobias {3}.

Hailever! e>"-p2r:irn,,"nts "in onjpr to

se

i - DlJlnerical valur",s for t.~is cl.."'..TT'f'i!!'J' phenomenon arc stipfOsed to be difficult. Consc.quently, reliable values

for the process o.rJI(1ping caused by the Vlo:ckpiGce rna.tc:cial rue not available at this rrorrent.

Tne test rig which is used in cOJperation "lork of the C.I.R.P. Ha-Group investigations of 8usceptiliili ty to chat·ter materials gave us the Ix>ssibili ty to carry out e.xperinr211ts in order to get nu:rnerical values of the darrQing ratio during tun1ing operations.

3.2. Process damping and iI~}ent~iffness as basic quantities for stability: charts

Tlusty, Polaceck a.o. {4} derived

where T represents the transfer funct.ion of the cutting process _{c .} while R is the real part of the traI1~ferfunction Tm of the machine

tool.

-4-When k

i is supposed to be indqjendent of the depth of cut b the dynarric

cutting force can be writtell as

Hence ,. it fo1101vs on the threshold of stability

1

b_cr k} .. ::; _{. 2} C _-I~ )

For a single-degree-of-freec1om system we can "derive

where k is the stiffness of the machine tool, ~\ the damping constant,

l; the damping ratio, and

Wo

the angular velocity at natural frequency.

NOW,I'le can write

be necessary to .J-1,'-• .L'C'-'-.::>c, b ord3r to beCOln8 instaf..lili·ly, or

If vIe excite the tool by a

to measuxe the

NOd I vle can calculate the the aid of the ,.,.,.."""",,,"",

r-"I..W_iC'", during cutting 1 i t is possjJ)le rC13p()DSe ~Ierore regeneration occurs.

da"T[)ing ratio of the system vlith

where n is the nunber of

The value

wd

is characterised by

Thus I the anpli tude ratio yields

or

_ 'lTnps

wm

_o

2/Tn

ratio

ill is the rna.S5.

It has to be noticed that the overall datping ratio

Z;:S can be written as Consequently Ps Z;:S :::: -:\f J = = = 2vm (k+bk.) ]. Ps := 21;;

~

m (k+bk.) s ].

NO.1, on the threshold of stability the follOitling

b . k. == Ps Wo :::: 2 1;;

·4

k2

+

b k k.

cr 1 S 1

system

ber ki ::: _{2 l;s .} 2 _k { b k.

--

2 k { z.;

+

cr 1 s

Finally I "\lith k

=

ill w 2 We find

o b k

~2mw2

r ~. l; c l O S 1. -I-

f

+

I ... )

₌

\ 2 /;;s z.;s 2 ;-

...

., "

.

}

Once nore, we "Till aSs1..lI1:D th.at k

i will not be influenced by b. If the stability chart under certain condi lions is kncwn l i t is

p:>ssible to calculate the values of k. with the aid of the z.; -values

, 1 s

obtained by means of the logarithmic decrement. 'l'his k. --value will 1

,be unique <md not be influenced by the dynamic behaviom:- of the tool. 'Ine obtained k;- ru'Jd S'" -values can be used for predicting' stability

~ ~ .

-6--charts for every machine tool of which the transfer function is knO'.\711. It should be noticed that where the direction of 6P is LUlknOlln, j.t is difficlllt to extrapolate k.to an other c3irection than the main

1

-one of the rig.

The assumption that the cutting process "\lill add dmTlping to the vibratory sY5te:m is confirmed by the results in Fig. 4 and Fig. 5.

rrhese results are obtained from experim2l1ts measuring t.he puLse response of the, test rig during cut-ting. Fig. 6 5hO'I1S a typical example

of such a response.

Considering the system on the threshold of stability, it follotls

In general, the follCJt.ling relation exists

where lJk is the aITDunt of

to

be in order to tJ1G

natt!ral o£ at H

""n'

-7-notion will be

00 _{c . n} =00

_{\j.t -}

.~.~

Z:;S

If b == 0 and the carriage of the lathe rroves I the next relation vlill also be valid

It has to be noticed tl').at in this case w _am renresents the anm'\lar _{'" ';)~} veloci ty at natural frequency of the tool while tl1e carriage Iroves. The corres};X)nding damping J;at..io is t;mt'

Consequently

2

k == m 00

'in am

It is es:tablished that the transfer function of the machine tool depends u};X)n the velocity of the ccLrriage. The m3.gni:tude of th.e change in conpliance will be influ8D.ced by ~t and in salle extent also by the carriage speed (5) .

. 2 2

Assurn:i.ng ~ .« 1 and ~ t « 1 , it };X)ssilile to obtain approxinlL"'!.tely

s m

the process stiffness with the equation

Thus I writing for the overall damping in pract..ice

on tl1e threshold of stability the next: relation will be valid

We consider the feed as a paran~ter for this relation.

Consequently

w

Thi~ yields - W mt 2 -- 2 oj

r,--..2...=:A+V 1 +1\2 w_mt · . Assurl1ing 1\2 «1, it fol1oHs We 2 ---=:1\+l+~A

+

w mt •• -. •• -·l+A we w_mt '" 1

+~---2

'---2

1 1 - [ ,\111-r ' 8 ' ?mt or in practice

If the experim::mtal results satisfy tl1is equr:ltion at the tlrreshold of steIDili ty, the validity of t118 theory in the preceding pages has

b2en proved.

Fig. 7 shaNs tlle stability chart for I;mt =: 0.080. '1'11e curve for I; s'

made vlitl1 the aid of tl1e logarithm:Lc decrem2nt1 for the corresponding

cutting-data of the stability chart of Fig. 7, is shoNn in Fig. 8.

-8-1

A second curve in the diagram of Fig. 8 sha-tls tlle calculated values. For practici3l reasons f during expedJl1E>nts, the values for the parameter

b cr are. replaced by values \/lhich are a shade smaller. The third curve shalls that I;mt slightly depeno.s upon the cutting speed. Fig. 9 gives the curve shoding tile fequenC"j versus cutting sr-eed at b , "ldhile

cr

another cu.-r:ve sha:.'ls the fre'JuenC"j for b ::: O. A very good resemblance behleen the calculated values a.nd tl1e' e.xoerirrental ones can be establishe . '

-for cutting speeds higher than 60 m/mLn.

So, it points out tllat the right slope of tlle stability chart is not only def:Lned by the compliance of the tcol ane: an incremental cutting stiffness k

i 1 but also by a process da-rnping.

Altl10ugh, at la:.v cutting speeds, the calculated values of 1; do not s

agree well "'lith the fO>Vyv·"y· ones, it is doubt:~less that a process

dan-ping will also have great influence i.n this range of cutting speeds, Up to nad f a fm:-mer explanation carulot be given.

FrOm the foregoing 1 it is clear t.hat only for cutting speeds larger

than 60 m/minl ne .. l method gives a gcxyJ. re.semblc:mce with the eA"}?erim:::ntal results accorrung to

Conse<:JU.ent1y I the new method will give the best k{-values for cutting speeds higher than the mentioned ones.

The preceeding theOlj1 sh~¥s

or b k. er 1 We ::::: 2 1;; m (I) W '" 2 ( - - - 1) 'm (l) W s c o r n , w mt e om

with the approxixnation w -'- '" w the incremental cutting stiffness

ffiL. om' • . will be 2 m W (w - (() .) e e me k. ::::: _'1 ---~---_b er

Fig. 3 sha.vs the calculated Ie. -values. 1

A considerable difference is to seen bet.ween the latter values and those calculated according to Pet.ers-method.

.

.

'I'he difference bebrleen both k. -values SID.:l.1l at 80 m/min. As Fig. 5

1

shalls a very 10\11 value for 1; , at the ment.i.oned cutting , this

e

difference can be expected.

3.5. 'me inf),u.ence of thE:u:~ear Qf:... tlle .. 'teol ~e J2rocess da~ on tlle inc:~S:'l.~n~l clJ!. ... t:Ln~lftne§~

-:1-Fig. 10 shods the curves of tile overall darnping ratio and of the frcc.fU.ency taken from the pulse-resp:mce, versus tool 'dear the cutting S0-~~

v

= 76 l1'/mL>1 a'1d b == 1.25 m:n. As frQll 1.'esu) ·ts f the of the

two or three till1'2S its ini t,ial value.

~is influence is taken into accouryt during all expcrim2l1ts nentiOIlE:;d in this pc1l::>er by kE:sping the wear oJ: the tool vii t.hin the range of O. 1 -:- O. 2 mm.

Where the change in frego,ency is only SITl;211, and this c..hange

can be eX1£)lained by tJle increasing dc'1lr0in9 in a great manner ( 'I:Je can conclude that the incI:eTllC!llt:al'cutting st:iffness calculated according to the 112'",1 1nethod will not be influenced by the'ldear of the tool.

Conclusio:'1S

-..-..---For cutting speeds higher than 60 ,m/m:tn the met.hod proposed in . this paF~r gives, to nad f the bsst "mal ytical approxilnation

for tJle e:xpedJnental resul·ts. Only frecIuencies have to be neasuren in order to get all -the information necessary for

calculation of the incr(O'.mental cutting stiffness and the process d-::rrnping. A furmer advan'tage of ,this memod is I tJlat mese

experirrents f i. e. measuring frequencies I can be easily done in each labJratOl:Y.

FurmeDrore I tllis m:::thod gives the r::ossiliili ty to ge't real values

-10'"

of tile c1amp.i,ng ar1d the of roaterials. r.r;l-'l8Se values v1ill be unique and mey vJili not be in:Elu_2Tlced by the dynamic data of the

tool. Na,'" we can derive me c1ynmnic cutting-C02fficient, as we can compare rfl.rJ.terials on stLsceptibili ty to chattex.

It is :f=X)ssible to l..1se me results for predicting me dynamic behaviour of machine tools during cutting the directions of motion of the tool and the ·test rig are the samet or if the direction of me dyncuuic force is kna:tln.

At lad (.'Ut-ting I Vle calIDot calculate tile stability chart vlell only \vi t11. tile values for the dan1[xLng and those for the cutting stiffness.

'lhe k. -values v1ill not be influenced by the Hear of me to:)l, while

1-the da'rnping rray increase to high values.

NOTI:::

yield well reproductive values. It turned out that this was on account of. the ternpera'ture of the vJorkpiece I which increases during cutting.

The influence has been eliminated by stancklroizing the· experiIr.ental conditions.

REE'EREl\CES :

{l } VANIIRRCI\ I P.

AperG!u General des gtudes du Coefficie..'Tt Dynamigue de Coupe.

Annals ·of the C.I.R.P. Vol. XVII. (1969)

· {2} PEl'ERS I J. and W)}mm~CK, P.

Hachine Tool Stability T~sts and the Incremental. Stiffness.

Annals of the C.I.R.P. 17 (1969)3 p. 225-232

· {3} TOBIAS, S.A.

MaQhine Tool Vibration. Blackie & Son, Glasgo;, 1965.

· {4} TllJSTY I J, I POU."CKrC I 1-1. i DN\rEK rOt and SPACEK t L.

Selbsterregte Sch~vinsnmgen an \\ierkzeugmaschinen. V.E.B. Verlag Technik, Berlin 1962.

· {5} Kll~LS, H. J • J. and HCXX3El\:f8CX)i'1, A . J .

The influence of the carriage speed on t11e compliance of tile- tool-holder.

Re:port NI' 0227 -- Eindhoven, University of Technology. Note presented to C,LR.P. r1a-group - 1970.

"-..

Fs

Fig. 1. Determination of the inoremental. cutting stiffness

according to the method of Peters and Vanherck •

8

critical depth mat: C 45N

of cut (m m) _{h: 0.072 mrn'rev}

t :

0.097 mt

\

~\

- - experimental values -'-.'- theoretical values .6

~

4

~

..

._-_ ... A 6.

~\~

0

,

R,

/"

'.

'\

V

\

V

.

_~

\

17 x f---.

\.

._.x-\

' /

...

"-

\Lx_.-

,,/')(. 1"-... . ··X~._

.-2

o

o

20

40

60 80 100 1"20

cutting speed (m/min)

Fig. 2. The experimental- and the calculated stahiUty chart (Peters method). The quantity ~mt represents the average value of the damping ratio of the tool- l;,hen the cOJ'riage is moving.

~~I--~I--~~-~-~-~--~l

- - r

incremental cutting / , , ' \ mat: C 45 N

stiffness (109 N/m2.) /

l'l...

t ·

0080

. / \

I

my'

. V

~TI-Q050

mm/rev.

/

\

, 4 ---4---+---4---~---~~---~~--__;

~

3

I---+--.~

, " , - " ,

~

A - A }valUeS according " '

-~

'\7-'-'\1 to Peters-method " -0 - -0 corrected values v

-~---+---+-~h'O.100~O,071 mm/r~v,

2

I - - - t - - - t - - - , J - - - - -r - - - i - ! - - - - r - - - t

'-

.

~',

'v

/D)/V

"

,-I-I-~t---I--! ~_

_{-\0"""'- "\. '}

V"'I'",

h::: 0.072 mm/rev ...

0__·

\

o

1 l---I---j-~_+--\) o 20 40

60

' ( " 0

"<

80 100 120 140

cutt ing speed (m/min)

Pig. 3. The incrementaZ cutting stiffness versus cutting speed according to the method from Peters and Vanherck (I) and according to the new method (II).

-14-175 ----~---;----~ .. ~--+_----_4---_4_-- I frequency mat: C 45 N l f

h: 0.072

mm/rev V: 140

ml

min

0.20

overall dampi

l!9

t

ratio OD5r---4~·---~-~---_r---;_---t---;---~

0.5

1.0 1,5

2.0

2.5 3.0 3,5 depth of cut (mm)

llig. tJ. The ovex'a

n

dallTping ratio and the frequency of the pu Z8

170 f 165 160 155 150 i <, 0.)5 ~ . 0,10 r.:: QO~ I I

-t

frequency of the mat: C 45 N

pulsresponce (Hz) h: 0,072 mm/rev.

b:

1.5 mm

\

.

.... --.~-~m( 0.080 " , >

r\

/ ...,. , \ ( ,

\

A A J ~ V A

-...

6 .

.".---.

ft----'"

~

vA ..

j Is. . . _, j

~

, A A A , . " -I---- -damping ratio _. . . &

-vi

v ~ ~v~ t;;::---v'- _~..:---i l }

-~"

--~ / '

. v /v

~v--v .. 20

40

60 80 100 120 i40

cutting speed Cm/min)

Fig. 5. The dm«ping ratio and the frequency of the overall system versus cutting speed.

-16-s

100 11

m

50

o

) reference: 1000.0 Hz . 1111 (lIl1l1I1l11I111I1l111I11111ll11111I111

V:

90 m/min 'b: 1.5 m m ~mf 0.080

h:

0.072 mm/rev

Fig. 6. ExampZe of a puZse-response.

The mass of the tool holder

=

14)35 kg.

-18-8

critical dept h mat: C 45 N

of cut (mm) h: 0.072 mm/rev ~mt=o.080

-.

.

·6

-~

4

\

2

\

/'

~~

A

r

~~

A~

p./

.

A~

- \

_/

o

20 40

60

80 .100 120 140

cutting speed (m/min)

-19-!

0.25

r - - · · - - - - t l - - -- - - t - - - / - - - - + -~-"-_tl---__tl----_+

overall damping mat: C 45 N

. t

rat io \ h:

0.072

mm/rev.

0.

20

1---+--\\

"-"experimental values for

b~bcr

\~ (~

0 - 0 calculated values for b:: b

_cr

~

v:--vvalues for b, 0

0.15

1 1 -'(~ ~

~. ~ ~\

0.1 0 t--~···---j---"'T~''''''---... - . _ - -~,,-.o-/·--. ~-I-~-_-_--'Ng"';---l---·~~···-t--··--··--t

--v---r---~_\~~~(D~~--~----~F---A~

20

40 60

80

100 120 140

cutting speed Cm/min)

Fig. 8. The OVel'a U dOlitping ratio at the thres ho Zd of s tabi 1,1.: ty ., and the dOlT/ping ratio of the moving toot (b=O) versus cutting speed.

18

0

f

5

r- r 17 0 16

5

. 16

₀

15

5

15

0

14

5

o

.

I

reguency

_{mat:C 45 N}

(Hz) h :0.072

mm/rev

A

\~

6-L:::. b:: b

_cr

(\~

'\1-'\1 b=O " 6

.,.,..

,

\

,

/ 6

6/,

Jo

~'

'~/6

L::;:.

•

t.~-7

.

-\:

A 6

I

.J

A

\.,

_~J

.~ " t -"~ v _v

-'\J

20

40

60

BO

100 120 140

cutting

speed

(m/min)

Fig.

9.

The frequency at the threshoZd of srabiZity>and the

frequenc;y for

b::.::o

versus cutting speed.

190 f 170

50

o.

3

o.

2

o.

1

o

frequency

.<H?-)

- -6-> overall damping ratio

o-r-\?

V' v ._--\ ~ t::. 6.-.'

.

"V V ._--- .

f

"\l

"

j

y

o

0.2

0.4

Fig. 10. The overall dmnping-ratio versu.s too l wea1~.

"

....

-21--'

-

I---""

'"

mat: C 45 N

h: 0.072

mm/rev

~mt'0.060 /l~

V:

76m/min b; 1.25 mm

V

V

,

L

.~

V

V,

0.6

---0.8 (mm) tool wear (VB)