Stability Lobes in Milling

Stability Lobes in Milling J. Tlusty (1). W. Zaton, F. Ismail, McMaster University, Hamilton, Ontario

The time domain simulation approach to the nnalvsis of chatter 1 9 applied h e r e to ~ m p r o v e the knowledce of the effert of rlitting speed "lohing" and of snerial riitter~ (nonrinrform pitrh) on stahilitv of milling. In this method the stahilitv lohes arr ohtained under more realistir ronditions whf're all the simplifying assumptions made in the previous analvsas of the+- effertq, esperiallv the restriction that all C u t t e r teeth have the %me dirertional orientation, a r e abolished. The new approach shows that, in general, the gains in stahilitv are milch smaller than previouslv maintained. Fxperimentnl results presented herr verify that the d i a i t a l simulation of rhattrr in milling: compares with renlitv very r l osel v.

1.

Introduction The effect of cutting speed

"Lohing:" on the

chatter in machining: has heen investigated hy manv authors 11-31. All these authors est,ahlished the stability lohes for repulflr milline cutters rising

i)

simplified approach, in which simplifying assumptions were taken.

The most significant o f

these n s s n m p -

tions was that a l l the cutter teeth have the same directional orientation. This would he the r.I(se for a cutter with infinite diameter, or f o r hroaching. Slavicek 121 and Vanherck 131. using this s i m p l i f i ~ d approach, also treated cutters with nonuniform tooth

I'

pitch and they showed, theoretically, that suhstantial increase of stahility c o u l d h e achieved with such c u t t e r s over particular spred ranges depending on t h e desicn parameters of t h e cutter.

In references

14.51.

time-domain digital si.mrila-

tion of the milling process, including chatter. was introduced.

In this method, the different. and vary-

ing directional orientations of the individual teeth are fully respected. Cutting tests presented in the next section show that this new approach t.o chatter in milling expresses the real situation more accur-

FIG. 1 Vihratnrv m o d e l i i s e d in the digital simulntion. The exr.itar tests a l s o showed other modes which w-re too far from these spindle modes and were at least five times stiffer. Therefore, it. was reasonable to establish the vihratory model of the machine a s shown in Fig. 1. I t is R system with two mutuallv perpendicular d e a r e e s of freedom in the X and Y directions, respectively, whose parameters are those

ately. Since experiments show the closeness of the

of the spindle modes given ahove.

digital simulation to reality, t.hjs simlllation is applied here to pstahlish the stahility lobes for a.

simulation in the time domain of the milling proress for the following four crises; up-milling and down-

regular cutter a s well as a cutter with a nonuniform

milling in the X direction (UX, DX), a.nd up-millinc and down-milling in the Y direction (UY, DY). In a l l r . a s e s , the clit.ter was half immersed (the radial depth of cut ~ q l l a l shalf the diameter of the cutter). The

tooth pitch.

This will then demonst.rate the effect

of cutting speed on s t a h i l i t y , and will s h o w the amount of improvement associated with a special cut-

This system was used to carry out the dicitfll

ter, under more realistic. conditions where the simplifying assumptions taken in 11-31 are nholished. The presentation here is concerned with high

critting speed used in the simulation was 1 1 2 mfmin T h e cutting and the feed rate w a s 0 . 1 mm/tooth.

speed milling.

steel; it was taken as r = 2E9 N / m 2 . The limit axial depth of cut was ohtainrd f o r the four cases of mill-

2.

Experiment,al Verification of the

ing mentioned ahove.

Digital Simulation:

Actual cutting t e s t s of 1020 steel, RHN = 12.5, using the same cutting conditions used in the dieital simulation, were carried out.. Fig. 2 shows thr lim-

The exciter tests were used to measure the modal parameters at the end of a face milling cutter 101.6

mm in diameter, having 6 teeth mounted to the spindle of a TOS FA4 milling machine. These tests g a v e the following dat.a f o r the modes assor.iated with the spindle in the two m~rtually perpendicular dirpctions. X and Y, respectively: Damping Modal Stiffness Frrqii~ncy (Hz)

(S)

(N/mm)

X

sia

7.6

1424532

Y

603

8.1

iRinifi

Annals of the ClRP Vol. 32/1/1983

stiffness used in the simulation corresponded to 1020

its of stahility c o m p i r t ~ d from t.he digital simrrlatjon and those ohtained from the cutting test.s. Also shown in this fieiire a r e the limits of stahility rompiited using t.hr simplified approach, a s it was done in the p a s t 1 2 , 3 ] , where a fixed mean directional orientation f a r t o r is taken.

309

“I

DX

For

D

Curting

a

SUlUlbtIm

0

~ t ~ n d ~ - ~ t m ~ ~ = n r n t m

M ~ S

(Wl?Jdv&tm:

UY

comparison,

the

stahilitv

lohes

romnrited

ii-lng

t h e simplified approarh a r e presented.

DY

ux

I8

16 14

12 h

E” E

- 8 E

6 4

2

0

FIG.

2

1,lmlt a x i a l d e p t h s of r u t o h t a l n e d from.

C u t t i n g t e s t s , S i m u l a t i o n , Simplified approach. FTG.

-

Stability l o h e s

3

Regular c u t t e r .

Half immersion U p - m i l l ~ n g , f x = f Figure digital

t o t.hosc while

2 shows c l c a r l y t h a t

s i m u l a t i o n approach found

the

in

every

the

case

g a v e v e r y close r e s i i l t.s

experimentally.

agreement

in On

the

other

r e s u l t , s was good

for the

Immersion Up-Milling = f = fin0 H7.:

Half

i)

Case a ) f

hand,

Fig.

IIX

X

Y

shows

3

thr

stahility

a n d DY c a s e s , i i s i n g t . h r m e a n r l i r f x t i n n a l o r i e n t a t . i o n

through t h e d i g i t a l

resulted i n gross errors f o r t h e DX a n d IIY m i l l i n g s . F i i r t h e r d i g i t a l s i m n l a t i o n s and r u t t i n g tests

T h e h o r i z o n t a l a x i s scale i s r a t i o

w e r e a l s o c a r r i e d for e n d m i l l i n g of

z is t h e number o f

T h e end m i l l Iised w a s a 4 - t o o t , h

7075 aliimini.iim.

HSS r,tit.t.rr,

d i a m e t e r a n d 7fi.2 mm i n o v e r h a n g .

2.5 mm

in

T h e modes a s s o c i -

a t e d w i t h t h e e n d m i l l m o d e s h a d p r a c t i c a l l y t h e same

f r e q u e n c i e s of

t h e X and Y d i r p c -

If300 H z i n h o t h

t i o n s , a n d t h e same s t i f f n e s s e s k

= k

X

Y

Nfmm.

= 11000

from t h e a r t u a l c u t t i n e a n d t h e d i g i t a l s i m u l a t i o n o f 7075 A 1 was r e m a r k a h l v good. T h e s e results diffr.rr.d from t h o s e o h t a i n e d

using the simpli-

For i n s t a n c e , h o t h t h e c u t t i n g t e s t s

f i e d approach.

a n d d i g i t a l s i m u l a t i o n g a v e t h e same r a t i o of 0 . 4 the

i n s l o t t i n g to that. ohtained

l i m i t d e p t h of c u t

i n half

immersion.

directional factor,

I n t h e case o f

u s i n g t h e mean

t h i s r a t i o was O . f i .

From t h e s e e x p e r i m e n t a l ohvious t h a t

of

the digital

simulation

it. i s

to t h e

approach

ohtained

teeth.

(hrnken l i n e ) .

where Y = z n / f ,

n is t h e number o f

revolll-

t i o n s p e r second a n d f is t h e n a t . u r a 1 freqi1enc.y.

In

t h e m o d e l p r e s e n t e d h e r e z = 4 a n d f is t a k e n as f = f x = fin0 Hz.

T h e parameter q o n t h e v e r t i c n l a x i s is

t h e ratio h e t w r r n

l i m i t . of

the

s t . a h i l i t y and

the

minimum l i m i t o f s t a h i l i t y o h t . n i n e d i i s i n g t h r d i g i t a l ( N o t e : T h i s minimum s t a h i l i t y w i l l d e p e n d

o n t h e p a r t i c u l a r t u n i n g of t h e t w o modes a s w e l l a s w h e t h e r t h e o p e r a t i o n is h a l f i m m e r s i o n o r s l o t t i n g . )

A s c a n he s e e n i n of

computation.

2, = 1 w h i c h

3, t h e z o n e s of

Fig.

t h e same f o r h o t h

s t a h i l i t y are p r a c t i c a l l y

The s t a h i l i t y

is v e r y

c o r r e s p o n d s to a r o t a t i o n a l w a v e of

Produces one f u l l

undulation on

high

around

speed which the surface

For u l t . r a h i g h s p e e d s . o n l y

a wave e x i s t s h e t w e e n s u h s e q i i e n t

a f r a c t i o n of

high

methods

In t h e present case it

hetween suhsequent t.eeth. w o u l d h e n = 151) r e v / s e c .

verificat.ions,

lohes

s i m u l a t i o n ( s o l i d l i n e ) and t h o s e

computed u s i n g t h e s i m p l i f i e d approach

simulation. Agai n t h e a g r e e m e n t h e t w e e n t h r resiil ts o h t a i n e d

significantly

= fi00 Hz.

V

t,eeth

and t h e s t a h i l i t y w i l l cnntiniir t o i n c r e a s e with f u r -

c h a t t e r prohlem i n m i l l i n a expresses r e a l i t v very

ther

closelv.

s t a h i l i t y ahove V = 2 d i f f e r s s i g n i f i c a n t l y hetween

In view of

this

fact,

the digital

simiila-

t i o n is used i n t h e f o l l o w i n g t w o s e r t i o n s t o o h t a i n .

incrpasps o f

speed.

Case h ) f x = fiO0.

lohes f o r a r e g u l a r c u t t e r a s w e l l as a c u t t e r w i t h

The s t a h i l i t y Fig.

S t a b i l i t y Lobes f o r a Regular C u t t e r The v i h r a t o r y

lation

here

model

iised

The d a t a a s s o c i a t b d w i t h n e s s e s kx = k

Y

digital

simri-

Case

Fie.

t . u n i n g of

=!,

x the two

= fin0 HZ; case fY = finn, f y = 833 H z ; a n d c a s e c ) f fY =

x

=

HZ.

In puted

this

for a

section,

the

stahility

r e g u l a r c u t t e r where

equally spaced,

and pa.rallrl

lobes are

~1.1. t h e

teeth

to e a c h o t h e r .

com-

are The

rutter h a s 4 st.raight t e e t h ( n o n h e l i c a l ) . In e v e r y c a s e . r o m p u t a t i o n s are c a r r i e d out f o r h o t h h a l f immrrsion up-milling

is s l i g h t l y

case o f t h e d i g i t a l s i m u l a t i o n .

stiff-

and damping r a t i o s f

case a ) f

= fi33 Hz:

T h e l o h e , h o w e v e r , a r o u n d Y = fl.5

1.

Fig.

t h e t w o modes are:

= .4ER N/m,

m o d e s are c o n s i d e r e d : h ) f x = firm.

the

Y

n a r r o w e r a n d t h e s t a b i l i t y i n c r e a s e is s m a l l e r i n t h e

depicted i n

T h r e e r a s e s of d i f f e r e n t

= 0.03.

goo

in

to t h a t

is s i m i l a r

f

lohes i n t h i s case a r e shown i n A o t h m e t h o d s o f c o m p u t a t i o n g i v e c l o s e re-

4.

sults. 3.

increase i n

t h e m e t h o d s of c o m p u t a t i o n .

through computations, a t r u e pictiirr of t h e s t a h i l i t y nonuniform t o o t h p i t c h .

T h e r a t e of

and s l o t t i n g i n t h e X-direction.

C )

t h i s case.

f

=

finn,

f

J

5 is t h e p l o t

= 900

of

m:

the stability lohes in

R o t h m e t h o d s of c o m p u t a t , i o n g i v e v e r y

close r e s u l t s .

I t is i n t e r e s t i n g t o n o t i c e t h a t t h e

r e s u l t s o h t a i n e d h e r e a r e i n good a g r e e m e n t . w i t h t h e r e s u l t s ohtained

model used h a s of 518 a n d fin3

i n s e c t i o n 2 , w h e r e t.he v i h r a t o r y two widely-separated frequencies

the HR.

FIG. 4

Stahility lohes

-

Regular Cutter.

FIG. 7 Stahility lohes - Regular Cutter. Slotting, f x = 6 0 0 , f = 633 Hz. Y

Half immersion Up-millinp, f x = 6 0 0 . fv = 633 Hz. - ~-

------r

!

-

1 -

r

2

q t

8 .

3;

I

6 -

4 1

2 -

t

0

FIG. 5

Stahility lohas

-

L A

FIG.

Regillar Cutter.

The

Slotting Figs.

6,

7 and

R

show the stahility lobes for

the r e g u l a r cutter when slott,ing in the X-direction As can he

I R

2

_-

i---

4

u

Stahility lohes - Regillar Cutt.er. Slotting fx = 600, fy = PO0 Hz.

Half immersion lfp-milling, f x = 600, fV = 900 Hz. ii)

\

large

discrepancy

between

the

results

nhtained from the two methods of computation is attrihut.ed mainly to the following: in the simplified

tion, the simplified approach and the dieitxi1 s i m r i -

a.pproach. the directional orientation fnctor associated with the mode i n the Y direction is taken as u Y = 0. which amounts to neglecting altogether the p a r ticipation of this mode. Therefore, i i s i n g the mean

lat.ion. is very l a r g e . In the cases a) n n d h) the lohes aroiind V = 0.5 and V = 1 n r e narrower and the

directional factors in the case of slotting reduces the vihratory model depicted in F i g . 1 to a single

stahility increases are smaller f o r the digital simu-

degree of freedom system in the X direction o n l y , rather than two d e g r e e s of freedoms i n X and Y. For

for the c a s e s a), h) and c ) , respectivelv.

seen i n these figures, the discrepancy hetween the results ohtained f r o m t . h e two methods o f computa-

lation compared to those ohtained from the simplified anproach. A l s o , the r a t e of increase in stahility at the ultrahigh s r ~ e e d range is much lower. These differences are more pronounced for the case c ) where the two freqrienries of the system are far apart.

such a system the mode coupling mechanism of chatter hecomrs inactive.

Ohviously this i s an oversimpli-

fication which leads t,o gross e r r o r s , as can he seen in F i g s . 6 to 8. 4.

Stability Lohes for a Cutter with Nonuniform Pitch The cutter f o r which the stability lohes are

computed in this section is shown i n Fig. 9. It. has 4 straieht teeth Inonhelical ). The spacings between these teeth are not equal. They follow an arithmetic series (linear variation) with the minimum pitch p =

and the pitch increment p = lRo. Therefore, the ratj.o of the increment in pitch to the averaee pitch

4,

63'

2

P is: p/p, = 1R/90 = I f 5 This cutter, with such coarse variat.ion in pitch, is supposed to he efficient in terms of providing high stahility against chatter for long waves of u n d u l a t.ion on the surface hetween suhsequent teeth. These long waves can he ohtained n t the high and ultrahi~h

FIG. 6 Stahility lohes - Remllnr C u t t e r . Slotting, P x = fy = fi00 Hz.

cutting speed ranges which are the main concern i n this work.

31 1

Q 0 -&

99 O FIG.

The %me

Cutter with nonuniform Di tch.

9

vibratory model

used

in the previous

section is used here to compute the Stability lohes of this cutter. FIG. 12

Stahility lohes - Nonliniform Pitch.

Half immersion Up-milling, fx = 600, f ii)

Y

= 900 Az.

Slotting Figs. 13 to 15 show tho stahility lohps f o r the

h) and c), resppctivaly. As in the c n s e of a regular c u t t e r , the differences in r ~ s u l t sohtained from the two methods of computation in s l o t t i n g are suhstantinl. They are attributed again to the s a m e reason mentioned in the previous section. cases a ) ,

Regular Cutter vs R Cutter with Nonuniform Pitch Comparing the results nhta.ined in the previnlls two sections hy t.he digital simulation onlv, we may 5.

FIG.

10

Stahility lobes

-

Nonuniform Pitch. f x = fy = 600 Hz.

Half immersion Up-millinE.

conclude the following: a) In terms of thp ratio of stahility incroase to cuttar the minimum horderline stahility, t h ~ with linearly-varying pitch does not offrr a n apprpciahle advnntage over the regular cutter.

Half Immersion Up-Mil 1 i n g case a ) f = fy = 6on HA: Fig. 10 shows the st,ahilit,y lohes ohtained from

1)

the two methods of computation.

This is hpcause, while the nonuniformity of the pitch disturhs the reaeneration of the waviness on the cut siirface, thP mode colipling mechanism

The difference in

still remains active. This is especially t r u e when the two frnqiiencips of the system are close and riltimntely when they are equal.

While the simplithe results is very significant. f i e d approach shows a suhstant,ial increase in stahility around V = 1 , the actual incrense ohtained from the digital simulation is merely two times. Also, nt. the ultrahigh s p e e d , di.gital simulation s h o w s the

h)

In the case of a r r a u l ~ rciitt.er, for

R

cutting

speed corresponding to V = 1 , a full wave of

rin-

d u l a t i o n on the surface hetween snhseqoent teet.h

rate of increase in stahility to he much lower.

exists and consequently the inrrease i n stahil-

Case b ) and case c): The stahility lohes for these cases are shown in

ity is very hiah a r o u n d Y = 1. Such a condition is virtually impossihle in the c a s e of a rrrttrr

Fig. 11 and Fig. 1 2 , respectively.

with nonuniform pitch, which explains why the increase in st.ahi1ity is not significant a r o u n d V = 1 for t.his c u t t e r .

Both approaches

yield very close results.

I 6~

I

!

i

I

I

4:

0L-i

...

I

.....-

2

. ..

--..

3

.

.. 4

.

....

;

v

FIG. 13 FIG. 1 1

Stahility lohes - Nonuniform Pitch.

Half immersion U p m i l l i n g , fx = 600, f y

312

=

633 Hz.

Stahilitv lohes - Nonuniform Pitrh. Slotting, fx = fy = fin0 Hz.

References 1.

Tohi ns , S . A. Fk

6,

! i

!

i

"The

3.

Vanhr'rck,

P.,

drirtivity

hy

Yilling",

6th

MTDR

19fi5.

"Increasing Use

cheSiter, 4.

Stahilitv lohes

-

Rth

WTDR

hlachine

Pro-

Non-Constant

Conference,

'Ian-

1967.

Tlustv, J.,

and

Ismail,

i n Machining C h a t t e r " ,

Nonuniform P i t c h .

Uilllne

of C i r t t e r s w i t h

C u t t i n g - F d g ~ Pi t r h " ,

t 14

of

I

i1

/

2 -

FIG.

R1 ~ c k i

lrrearilar T o o t h

Effect of

Stahllitv

Conference, Hnnchester,

i i

!

on

V ih r a t i on",

19fiS.

Slavicrk, J., Pitch

i

4 -

2.

i

, "Llnchine-Tool

Son, Claseow,

F.,

"Basic Nonlinearitv

llnnals of

t h e CIRP,

Vol.

30/1/1F)Rl.

S l o t t i n g , f x = 600, f v = 633 Hz. .-

5.

q

8 .

Tlusty, ChRtter

J., and I s m a i l , in

Acoustics,

Vol.

0

I.

-.-.

PIG.

.

.->

15

..

- L

t

3

2

St.ahility lohes Slotting, fx =

-

L.

..

l

v

..

Milling".

F.,

"Special Aspects of

Joiirnal

of

Vihration.

S t r e s s and R e l i n h i l i t y i n Design,

105, January, 1983.

-1

Nonuniform P i t c h .

fino,

f

Y

= 900

HZ.

T h e v a l i i e s of t h e m i n i m r l r n h o r d e r l i n e s t a h i l i t i e s ohtained

f r o m t h r d i g i t a l s i m i i l a t i o n s for t h e t w o c u t t e r s f o r t h e d i f f e r e n t casf's are l i s t ~ d i n T a h l a

1.

-

Tahle 1

Ciitter

Minimlim L i m i t o f S t a h i l i t y h m i n (mm)

finn-finn

fion-m?

fion-900

1 / 2 imm.

Slot

Regular

4.5

1.5

3.5

2.0

fi.5

5.5

Nonuni form

fi.O

2.5

4.5

2.5

8.0

fi.n

1/2 i m m . S l o t

1/2 I m m .

Slot

PI t c h

For a l l cases listed i n T a b l e 1 , t h e c r i t t e r w i t h nonuniform p i t c h r e s u l t s

in higher l i m i t axinl

depth

of c u t . c o m p a r e d t.o t h a t . o h t i t i n e d w i t h t , h n r e g u l a r cutter.

T h i s i s e v i d e n t l y d u e to t h e d i s t i i r h i n e

m e c h a n i s m of

t h o r e g e n e r a t i o n of

undrilations

on t h e

cut surface a s s o c i a t e d w i t h t h e n o n u n i f o r m p i t c h . This might

he

the h a s i c a d v a n t a g e of a cirtt,er w i t h

n o n u n i f o r m p i t , c h o v e r a r e ~ u l a rc i i t t e r .

313