Process damping in metal cutting

(1)

Process damping in metal cutting

Citation for published version (APA):

Kals, H. J. J. (1971). Process damping in metal cutting. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0278). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1971

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-1-WT-RAPFORT

No.0278

PROCESS DAMPING IN METAL CUTTING

H.J .J. Kals

Dept. of l-rechanical E~gineering Eindhoven University of Technology

Abstract

*

In a previous paper of the author , a new method was presented for the determination of the relevant components of the transfer function

of

the cutting process. Using this method in practice, the specific cutting data obtained, make it possible to explain a series of influences on the dynamic stability of machine tools in cutting conditions. Up to now, these influences could not be explained.

Moreover, the validity of the method is also established to be

true for lower cutting speeds.

(3)

-2-Nomenclature

Angle between the main direction of motion and the

direction of the chip thickness modulation; clearance

angle of the autting tool

a

In-phase component

a • _mln

maximum negative in-phase component

a

AngZe between

k;

and the main direction of motion

J. 13

0 f3

for

a

=

0

b

Width

of

cut~

quadrature component

b

Limit value of width of aut

g

b_cr

Critical width of cut taking into account the phase

equation

c C. J. ... C. J. c _c f o y h h o

Damping coefficient for the machine tool structure

Specific process dwnping coefficient

Resultant specific

process

damping vector

Coefficient for the damping

in the autting process

Natural f:r>equency

Feed f01y:e

Main cutting force

Projection of the dynamic component of the resultant

(Jutting fOl?ce on the main direction of motion

AngZe between

c;

and the main direotion of motion;

rake angZe of the cutting toot

y

for

a

=

0

Chip thiokness modulation

Undeformed chip thickness

Nominal undeformed ahip thickness equal to feed

per revolution

(4)

k k} k2 k. l. -+-k. l. h m n w, W 0 W C w n p q R R n t W 1

Cutting edge angle

Stiffness of the maahine tool struature

Equivalent stiffness of the maahine toot

stpuoture in wopking oonditions

Chip thiokness ooeffioient

Damping ooeffioient

Speoifia ppooeS8 stiffness

Resultant speaifio ppooess stiffness veotop

Cutting edge inalination

Meiss

Rotational speed of wopkpieae

AngulaP fpequenay

AngulaP natupal fpequenay

Angutap fpequenay of puZse pesponse during

outting

AngulaP ohattep fpequenay

AngulaP fpequenoy of pulse pesponse without

autting

AngulaP fpequenay during (Jutting

In-phase aomponent

Quadrature aomponent

In-phase aomponent

Maximum negative in-phase oomponent

Time

-3-Tpansiep funation of the outting ppoaess,

~Fa/~h

(5)

VB Y * y y y*

-4-Transfer funation of the regenerative effeot

Transfer funotion of the maahine tool

st~~oture> ~Y/~F

Cutting speed

Width of the flank wear land of the tool

Instantaneous defleotion of the tooZ

DeZayed ohip thidknesB moduZation

Peak vaZue of

y

Peak value of

y¥

Damping ratio of the struoture

Damping ratio of the struature in working oonditions

(b =:

0)

(6)

-5-1. Introduction

The dynamic behaviour of a cutting process can be represented by a closed-loop model as shown in Fig. 1.

The machine tool is characterized by its transfer function T , in m

.fact the dynamic compliance of the structure, while T describes the c

cutting process transfer function. The loop is closed by the function T

h, introducing the wave on wave chip thickness modulation (Fig.

2.).

The quantity y represents the relative motion between tool and

workpiece i.e. the direct chip modulation. In case of turning, the delayed chip modulation y*, caused by the feed back path of the previous cut, can be written as

It 1

y (t)

=

y (t - -)

n (1)

where t stands for time, and n is the number of revolutions of the

workpiece.

From the assumption of a harmonic motion between tool and workpiece, it follows y .. Y cos wt (2) and

*

y

=

Y

cos wt, (3)

.

*

where a phase shift eXl.sts betwo.en Y and Y .

So, we find for the total chip thickness modulation

*

llh '"

Y - Y

(4)

By introducing llF for the dynamic part of the cutting force, we can _. _ct

(7)

-6-T 1 ' : 1 _ liFo. c Lih (5) and T .. Liy m LiF (6) a

Now, we define the threshold of stability with the aid of the equation

*

y

=

y

So, the following relation is valid

iI-T == 1 -

L

h y

The function Th is diagrammatically shown in Fig. 3.

(7)

(8)

As for the function T

h, the transfer functions Te and Tm are complex too. These functions can be represented by a polar curve, as Fig. 4. shows.

Where T , mostly can be described with the aid of a compound of some

m

single-degree-of-freedom systems, the complex nature of T is not c

well known. In general, T is supposed to introduce a phase shift, but _. _c the deteliUination of the cutting process transfer function is found to be rather difficult.

This phase shift can be introduced by the assumption of an interaction between T and T on account of the relative motion between tool and

c m

workpiece.

Accor.ding to Tobias theory [1] we define:

(8)

-7-The polar curve of the open loop of the feed-back system in Fig. 1. represents (T .T .T

h) and consists of an infinite number c m

of circles, all going through zero, and turning clockwise around zero with increasing value of the angular velocity oo. (Fig.

5.).

The diameter of every circle is completely determined by the value (T .T ). The so called Nyquist-curve intersects the negative part

c m

of the in-phase axis in a great number of '11 "points P., where on the J

threshold of stability the following relation has to be considered

We define and - { T .T .T_{c m} _h } T.T == a + ib c m = 1 00=00' J

According to Fig. 3. the next relation will be valid

(10)

(II)

(12)

2

q = p (2 - p) (13)

Under the circumstances given before, the quadrature component of the function (T .T .T ) is equal zero, so it yields for the in-phase

c h m component

{ T ,Th,T } _c _m ... 2 a (14)

00=00.

J

For the assumption that the force is in-phase with the chip thickness modulation we can derive

(9)

R n a . m~n " " -Te

-8-(15)

The quantity R stands for the minimum part of the in-phase component n

of the transfer function T • The equations (10), (14) and (15) lead

m

to Tlusty's

[2]

relation for the limit value

T cg .,.

-] 2R n (16)

However, for complex numbers of T , this limit value will not occur

c

for R "" R • _n

Contrary to the determination of T , for which reliable methods are

m

available

[3J [4J,

the measurements with respect to the function Tc are complicated.

However, where both quantities are essential for permitting an accurate prediction of machine tool chatter at the design stage and for predicting the stable working conditions for nUmerically controlled rnachines, a good knowledge of T

c fundamentally.

Extensive work in this field is done by a.o. Smith and Tobias [5], Kegg

[6J

and Albrecht

[7J .

. 2. Determination of the transfer function of the cutting process

In the methods used the determina of the process transfer function two main directions can be tinguished, viz.:

1. methods where the regenerative effect is taken into account [8]

2. methods excluding the regenerative effect [6J [9]. In a previous publication from the. author [9J it was shown, that, if

(10)

-9-a m-9-achine tool c-9-an be described with the -9-aid of -9-a single-degree-of-freedom system owing an angular natural frequency w , during cutting for cutting

o

speeds higher than 1 mis, the next relation will hold

I W ... W

n 0 + 21;; , s (17)

where wn stands for chatter frequency and ~ s represents the damping ratio of the complete sys tern on the -threshold of stability.

Equation (17) allows us to obtain values for the specific process stiffness k. as well as for the specific process damping c" Actually,

L L

the word "spe,cific" refers to a width of chip of 1 mm.

In general, the equations (16) and (17) lead to the following relation, which in practice will be valid on the threshold of stability for

l; 2 « ] .

s

r, .

=

(~-

1)

cos a

S

\w

_mt

(18)

The angular frequency w goes with a pulse response of the complete _c system during cutting under limit conditions, avoiding the regenerative effect, whereas w

mt represents the angular frequency of the pulsresponse for a width of cut b ; 0 and a moving carriage. The angle a gives the orientation of the main direction of the vibratory system with respect

to the direction of the chip thickness modulation.

In this way we end up with the following relation for the specific process stiffness

2m

kl.' == -b W c (w _. I1L,t)

C ~lU

g

where b represents the limit width of cut. _g

(11)

-10-It has to be noticed that, with c as the damping constant and k_m

as the spring constant in working conditions, and m as the mass of the structure, the overall damping ratio of the system can be written as

(c + b c.)

's

=

--:==:=!g:,:=~===-2

"m

(k _m+ _{. g}b .k . ) _~

while the damping ratio for b = 0

c l;

=

-mt

2V

mk m (20) equal to (21)

As for the process stiffness, the proces damping is supposed to be proportional with b, which later on will be shown to be very realistic. Starting from the equations (20) en (21), it follows for the specific process damping w 2m c c. "" -b {cos<X - (w - W t) - l; t W t} ~ g w mt c m ill m (22)

During experiments, a special toolholder is used, representing a single-degree-of-freedom system 'vith a dyn.<L'1lic stiffness which is relatively Im-1 compared with the dynamic stiffness of the lathe [10]. This gives us the possibility of decreasing the influence of the dynamic behaviour of the lathe.

Actually, the values of k. and c. obtained in this way are projections

1.

l-of the original vectorial quantities

k7

and c;. In case of orthogonal

. ~ l

. . + +

cuttlng ki and ci will act in the plane through the direction of the cutting force F and the feed force F

f .

v

Thus, in order to determine k7' and c; we need two toolholders with a

1, 1,'

(12)

-11-of k~ and

l.

-+

c .•

1 These quantities are supposed to be independent of the

vibration direction of the structure within a wide range.

Now, we assume on the threshold of stability the delayed chip thickness modulation to have no influence on the process damping. So, we

assume this modulation to have no other influence on the transfer function than its geometrical contribution to the chip thickness modulation.

n

ij

The relation between the relevant component of the dynamic cutting force and the quantities derived previously in this chapter can be expressed with the aid of equation (9) as

(23)

In this relation we assume once more the process stiffness and the process damping to be proportional with b. This assumption is justified up to a great extent by the experimental results in Fig. 7. (See eq. (28». For an angle 8, respectively y, betwee.n the main direction of a given.

-+ -+

structure and ki' respectively 0i' it follows

Keeping in mind that h.y ... _ 1 6h it yields finally T

=

b (k~ cos 8 + 1 W k

it.

cos y ) C l. 1 (24) (25) (26)

(13)

-12-3. Experiments

For the measurements special toolholders are used, starting from a model similar to the C.l.R.P. cutting stand accepted for cooperative work in the Ma Group

[10].

To obtain all the information necessary

• -+ -+ • •

for compos1ng k. and c., measurements are carried out for two d1rect10ns

1 1

of the vibrating system. Where a characterizes the angle between the main direction of the toolholder and the direction of the nominal chip

thickness modulation, perpendicular to the cut surface of the workpiece,

o 0 ' I ,

two values are used for this quantity viz. 0 and 32 30. (See Fig. 8.), In accordance with the conditions on which equation (18) will be valid,

the measurements are carried out for cutting data on the threshold of stability. So, before starting measurements the experimental stability charts are needed. In order to avoid the regenerative effect data are only taken l"ithin the time for one revolution of the workpiece, while during the time preceeding the pulse the toolholder is locked in order to eliminate any contribution to the delayed chip thickness modulation.

In spite of the precautions mentioned above at low cutting speeds this method does not satify quite well. This is due to an increasing dynamic

cutting force. By decreasing the width of cut only to a small extent we can evade this problem. The values WI measured in this way have to be extrapolated to . W • Equation (28) sho\vS that for practical purpose,

c

within a small range, the angular frequency can be taken proportional with

b k.

b, when 0.08 <

T

<0.4.

m

From the relation

n;-;

b k.

w "'1./~ ~

1 , m it follows

(14)

-13-+ ••• ) (28)

To avoid lobes in the stability charts the experiments are carried wmt

out for ---

>

30. This implicates b

=

b • The pulse response of the

n cr g

toolholder is measured with the aid of strain gauges mounted on springs, and the data are lead to a high speed 8 bit x 1024 word core memory to capture and hold the digital equivalent of the analogue signal as a function of time. The values for ;,: and ware computed with regressive technics. It points out that the computed results for the frequencies show considerable less scatter than the computed results for ~. The measurements during cutting concerning the results presented in this paper were carried out three times. The results have been corrected

with respect to the change in damping of the toolholder due to the static deflections which changes the oil slit. In order to eliminate the

influence of the nose-radius of the cutting tool all experiments have been carried out on tube-shaped workpiaces. Standard throw-away type carbide tool tips (P 30) have been used.

The following tool geometry has been applied: 900 60 :

K.'" ; a lOt ,

Using SKF 1550 steel for working material, the experiments have been

carried out for two values of the feed viz. 0.072 rom/rev. and 0.208 mm/rev.) within the range of 0.5 - 2,3 m/s for the cutting speed. Tool wear was

kept smaller than 0,15 tn.'U VB while the ttlmperature of the workpiece

during cutting remained within the range of 35 - 400C.

4.

Results

,

For the direction of 32030 the results of the measurements on w , w t

c m

and ~mt as well as the computed values of ~ , and w - using k. and c. . are s c 1 1

(15)

-14-plotted in Fig.9 •• Fig. 10. shows the computed values for the specific process stiffness k., according to eq. (19), for a feed of 0.072 rom/rev.

~

and 0.208 rom/rev. In this figure the curves for SKF 1550 steel show a course which is equal to that of the graph for C 45 N steel presented previously by the author

[9].

Concerning Fig. 11., presenting the computed values (eq. (22» of the specific process damping, the results show a process damping versus cutting speed which varies with.in a wide range and passes through a minimum for about the same cutting speed yielding the minimum bg-value

(see Fig. 20.).

As for the c.-values near the minimum, they are negative~while for

l.

increasing cutting speeds the process damping c may rise to high _. _c positive values, exceeding the structural damping of the Vanherck toolholder,even for ~mt

=

0.17.

In this range of cutting speeds the values of the process damping may' be positive and negative as well. This may be the reason why some investigators', found the dynamic cutting force leading the direct chip thickness modulation, while others became aware of a delayed dynamic force, since cutting conditions, especially the cutting speed, were very

different.

0 U .

As it is already shown by the eqs. (19) and (22), comparing the results from Figs. 10. and 11. it can also be seen that the graphs concerning the process damping and the process stiffness show a very similar course. This is confirmed once more by Kegg's results (see Fig. 18.). [6].

Evidently, both quantities come In Fig. 12., the quantity

of the cutting speed.

c.

l,

is

-the same physical mechanism. di,2grammatically shown as a function

(16)

-]5-solution method for the limit value

hg,

which has been derived from Gurney's method.

DJ.

Using the values from Fig. 12., theoretical values for the stability charts of Fig. 14. have been computed in this way. A very good resemblance with the experimental results is to be seen in

the same figure.

In fact, the experiments started from the experimental stability charts of Fig. 14.

Although experiments carried out previously,

[9]

using measurements based on the consecutive amplitude ratio of the pulsresponse as a reference, did not show a good reliability of the method for cutting speeds le.ss than 1

mIs,

the results from Fig. 14. prove the method to be properly within the whole range of cutting speeds mentioned in this paper.

k.

In Fig. 13., for a feed of 0.208 rom/rev., the vectorial quantities 1

-+

and c. for the several cutting speeds are shown. As can be seen from the

l.

figure, the directions of both quantities depend strongly upon the cutting velocity.

With respect to the process damping, this is contradictory to assumptions made by Andrew

[12]

and by Kegg

[13J.

At last, the final results will be

shown in Table 1., yielding the numerical values for the length of the vectors mentioned and the angle with respect to the direction perpendicular

to the cut surface.

A remark has to be made for the reproducibility of the method. As,

for instance for a feed of 0.072 rom/rev. and for all cutting speeds used, with respect to the measured values for w , scatter remained within a

c

range of 1.9%, the reliability of the method proves to be very good.

5.

Process damping and its influence on the threshold of stability

Ignoring the process damping eq. (16) results in:

(17)

-]6-If we take this damping into account, we may write:

c + c

T .. ___ c_

cg k

=

(c + c ) c (.oJ 0 (31)

where c represents the projection of the process damping vector at the

c

main direction of the vibratory system.

c .. b c. c ~

From eq. (31) it follows:

or: b c. b k.:= 2 { 1; + g ~ 8 l } k

2~

2 z:;, k b g ... k. 1. c. (1 - ...2:. w ) k. 0 l lIence, b g is a function of 2 k. 1; k and ___ . 1. (32) (33) (34) (35) c. (.oJ 2 z:;, k

15. shows b_gversus

\.,0

for several values of k. . From

I w ~

Fig.

this graph it is very clear that: for ...;;;;--o -+ 1, the influence of the

II". •

l-process damping at the threshold of stability increases strongly.

c. w

For ~ 0 ~ 1 there will be unconditional stability for every b-value

k. 1

as far as regeneration is concerned.

From Fig. 15. it _{is also clear that the influence on bg of a variation}

c_i Wo 2 k

of increases by increasing the -value. Of course we can also k.

l.

get this result by differentiating Eq. (35) with respect to

c. (.oJ

1 0

k.

(18)

2 ~ k k. 1. c. 2 (1 - -.!.. w ) k. 0 1.

-17-(36)

With respect to the shape of stability charts of machine tools some important conclusions are to be made with the aid of the foregoing:

(1)

(2)

(3)

c. w

for 1. 0 «1 there will be no significant influence

k.

1.

of the processdamping on the threshold of stability, for

c. W

1 _k.0 _~I _{uncon 1.t1.ona sta}d" 1 b'l' _1. _{1.ty is to be expected} 1.

from the standpoint of regeneration, if

c. W

1. 0

k.

1.

+ 1, we have to make a distinction between - machine tools with a high sk-value:

process damping will have great influence, - machine tools with a low sk-value:

process damping will have small influence. For different dynamic structures, some remarkable differences in the course of the bg'-value as a function of cutting speed can be explained now.

Up to now, the explanation of these differences was beyond the scope of the existing knowledge about machine tool chatter.

For instance, comparing stability charts of machine tools and those established for the Vanherck-Peters toolholder, especially for high s-values for the toolholder, shows a remarkable increasing of the limit-value with increasing cutting speed for the latter. (Fig. 19.). This cannot easily be brought in agreement with Knight's theory about a small increase of the level of stability with increasing cutting speed basedl on the be.haviour of the mean shear angle. [14] •

Peters

US}

found out that ~-values for conventional machine tools seldom exceed 0.03, while in case of single-elements, values of 0.002 are common practice. In this context, investigators often use special tools for

(19)

-18-chatter research which have low z;-value.s. Two further examples will be mentioned here: Peters boring bar [l6] with a 1';-value of 0.008 and a

special toolholder used by VUOSO

D7J

for chatter research with 1';= 0.01. Not much information is available concerning values for the stiffness of a middle-type machine tool. In case of a less stiff machine tool as the radial drilling machine, Landberg

08]

mostly found ~pring constants of about 1.5 x 107 N/m at a distance of 1 m from the column. The natural frequencies for middle- type machine tools mostly are in the range of

100 ":' 200 Hz.

D

7J

Considering the mass," values will seldom exceed 100 kg, a spring constant · 2 7

of 50 x(2~.150) • 5 x 10 N/m will represent a reasonable medium value for this type of machine tools.

For the Vanherck-Peters toolho1der, for standard conditions as proposed for cooperative work in the C.I.R.P. Me-Group [19J and under working conditions [20J it follows k "" 1.5 x 107 N/m, and 1;; ::< 0.15. Thus, it

yields 1';k· 0.15 x 1.5 x 107 • 2.25 x 106 N/m.

Comparing data, for Peters boring bar we find

~k

=

0.008 x 1.6 x 107 " 1.3 x 105 N/m, and, on the average, for a middle-type machine tool

Z;k. 0.02 x 5.107

=

106 N/m.

Consequently, using SKi 1550 steel, the limit values of Peters boring bar (f

=

168 Hz) hardly will be influenced by the process damping, as shown

o .

in Fig. 16., which has been confirmed by experimental resul ts • [I6J. In case of Vanherck's toolholder (f

=

157 Hz), according to Fig. 16.,

o .

even with respect to a middle-type machine tool, a considerable influence on the threshold of stability is to be expected, which has also been confirmed by many investigation using the same toolholder and applying several working materials.

[i

9J •

Actually, a process damping will be responsible for the increase of

(20)

-19-A comparison as made here will only be valid for about the same behaviour of the quantity

steel, supposing a natural c. w

~ 0

k. • For orthogonal cutting of SKF ]550

1.

frequency of the tool of about 150 Hz, the value of this dimensionless quantity will increase up to 0.5. Taking into account the change of k. according to Fig. ]0., with respect to

1.

Vanherck's toolholder and for the conditions mentioned before, a working area under limit conditions can be given as indicated in Fig. 17.

As, according to Kegg's results

[6J,

for frequencies higher than 150 Hz, T decreases slightly with increasing frequency while the out-of-phase

c

o

angle increases up to 75 for f

=

450 Hz (See Fig. 18.), this can only occur due to a considerable decrement of k. and a, at the same time

1.

practically unchanging, out-of-phase component. It should be noticed that, where the results from Figs. 10. and 11. show a very similar course for both, the process damping and the process stiffness, also the increase of the quadrature component with increasing frequency will be considerable less than proportional. This is confirmed by Kegg's results.

With respect to the VUOSO toolholder, the next conclusion can be made now. Although the ~k-value of the VUOSO toolholder (f

=

520 Hz) is low,

o

5 2~k c. w

r;k

=

5.10 N/m, the

k. -

value is high, while the .

\.0

-value \"i11

]. . 1.

easily exceed 1. So, starting from a certain value of c., only a small

1.

rise will cause quite high values for the limit width of cut, and finally will lead to unconditional stability of the cutting process as far as regeneration is concerned. Fig. 16. shows the influence of a process damping on the limit value, predicted for SKF 1550 steel, taking into account a change of k. according to Fig. 18. Since results for

J.

several kinds of mild steel are about the same, we assume this extra-polation to be allowed

[I9] .

(21)

-20-This clears up why experiments, carried out under standard conditions in different laboratories, with the aid of the VUOSO toolholder, did not reveal comparable results with respect to the limit values.

D7].

A relevant conclusion needs to be made here: both toolholders mentioned are not suited to carry out comparative tests on susceptibility to chatter of materials, based on comparing stability charts.

It is important to notice that for decreasing values of k. the influence

J.

of process damping increases strongly.

Continuing, it is also possible now to explain the considerable differences between the influence of tool wear on the threshold of stability of the

VUOSO toolholder, and of a normal machine tool, as is shown in Fig. 19.

D7].

Where in contrary to the machine tool (f • 130 Hz), using the same

o

c. W

3. 0

k. are rather high,

material, for the toolholder the values for

J.

. an increase of process dampi.ng caused by increasing tool wear,

(9]

which grows with time, easily leads to real different values for the limit width of cut.

Experiments, carried out with Vanherck's toolholder did also reveal

an important increase of the level of stability with increasing tool wear. [21J • In this case the high ,k-value of the toolholder will be responsible for this important influence of process damping.

With the foregoing, the pres~nce of damping in the cutting process enables us a pronouncement for a flat course of the limit value versus cutting speed of the Vanherck-Peters toolholder as the damping declines until small values (See Big. 20.). In this figure it draws attention that

the differences with respect to the values of the left side of the border-line are not as great as those with respect to the right flank. This is confirmed by the results of k. and c. presented in Figs. ]0. and 11.

l. J.

(22)

-21-the influence of this damping remains more important here, due to low values for k .• This will also be the reason why, even in case of machine

l.

tool structures with low ~k-values, always a considerable increasing of the limit value of width of cut at lower cutting speeds can be seen. It is evident now that the specific process stiffness k. primary will

l.

be responsible for "low speed stabilitytl. Where the curves from Fig. 20. represent experimental stability charts, the black marks indicate the predicted limit values. For ~mt ~ 0.055 and

=

0.17 these values are determined with the aid of similar graphs as shown in Fig. 16., taking into account the corrections on ~mt for the varying oil-slit of the damper as well as for the carriage speed.

Bearing in mind a great derivative of the limit value b for the higher g

and the lower cutting speeds, a process damping and a process stiffness which are not quite proportional with b, especially for high values of b , may cause great deviations between both results. In this way the _g accordance between the predicted results with the experimental ones is very good.

6. Conclusions

Process damping in metal cutting is proved to be essential to describe cutting dynamics with respect to the dynamic stability of a machine tool in cutting conditions, the influence of process damping increases strongly according as

only a small

C. til

l. 0 + 1. In case of a high damped stiff machine tool,

k. l.

process damping will lead to considerable increase of the threshold of stability, particularly for low values of the specific cutting stiffness. It should be noticed that the influence for negative damping will be considerably smaller than for positive damping. The results presented in this paper are confirmed up to a great extent by

(23)

-22-the fact that some important influences on -22-the limit width of cut, which up to now could not or insufficient be explained with the existing

theories, do have an explanation now.

Taking into account the results of Fig. 21. concerning the damping during cutting of C45N steel

[9J ,

which show even a decreasing damping for very low cutting speeds, once more an affirmation will be gained for the evidence made before, that the specific process stiffness primary "lill be responsible for "low speed stability".

In this way we have to reconsider Tobias theory

OJ

concerning the increase of stability at lower cutting speeds as well as Kegg's "low speed stabilitylt solution [13].

Acknowledgements

The experiments presented in this paper have been carried out in the Laboratory for Production Engineering of the Eindhoven University of Technology. Thanks are due to Professor A.C.H. van der Wolf and

Mr. J.A.W. Hijink for their stimulating discussions and to Mr. A. van Sorgen for assisting with the experiments. For the calculations required

(24)

-23-References

[1]

Tobias, S.A., Machine tool vibration. Blackie & Son, Glasgow (1965).

[2]

Tlusty, J., Pola~ek, M., Danek, 0., Spa~ek, L., Selbsterregte

Schwingungen an Werkzeugmaschinen. V.E.B. Verlag Technik, Berlin (1962). Koenigsberger, F., Peters, J. and Opitz, H., C.I.R.P. Ann. 14(1966)96. Van der Wolf, A.C.H., The development of a hydraulic exciter for the investigations of machine tools. Doctor's thesis, University of Technology, Eindhoven (1968).

[5] Smith, J.n. and Tobias, S.A., Int. J. Mach. Tool Des. Res. 1(1961)283. [6] Kegg, R.I.., A.S.M.E. paper No. 64 - WA/Prod.-l1 (1965)464.

[7J Albrecht, P., A.S.M.E. paper No. 64 - WA/Prod.-ll(1965)429.

[8J Van Brussel, H. and Vanherck, P., A new method for the determination of the dynamic cutting coefficient. 11th. Int. M.T.D.R. Conference, Manchester (1970) •

..

~

Kal s, H. J . J ., C. 1. R. P. Ann. 19 ( 1971) 297.

[to]

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. R~port pr~sented to Group Ma, University of Louvain, 19 sept. (1967).

~

IJ

Vanherck, P., C.I.R.P. Ann. 17(1969)499.

[t2J Andrew, C., Proc. lnst. Mech. Engrs. 179(1965)877.

(13]

Kegg, R.L., C.I.R.P. Ann. 17(1969)97.

[i4] Knight,

v1.A.~

Int. J. Mach. Tool. Des. Res. 8(1968)1.

Q5J Peters, J., Proc. of the 6th Int. M.T.n.R. Conference, Manchester (1965)23.

(25)

~6]

Peters, J., Vanherck, P., Industrie-Anzeiger No. 11(1963)168 and

No. 19(1963).

v7]

Tlusty, J., Peters, J., Matthias, E., Report on cutting tests of

stability against chatter. C.I.R.P.,Group Ma, Doc. No. 3/67, University of Louvain, 20 april (1967).

vs]

Landberg, P., Metaalbewerking No. 9(1959)173.

D~ Peters,

J.,

Vanherck, P., Program of chatter-susceptibility tests. C.I.R.P., Group Ma, Doc. No. 69 Ps, University of Louvain,

15 aug. (1969).

[~,

Kala, H.J.J., Hoogenboom, S.A., The influence of the carriage speed on the compliance of the toolholder. C.I.R.P. Report presented to Group Ma, Report \?r 0227, University of Technology, Eindhoven (1970).

[2U

Tlusty,

J.,

Koenigsberger, F., Specifications and tests of metal cutting machine tools. Proe. of the Conference, 19th and 20th Febr., The University of Manchester Institute of Science and Technology (1970).

(26)

TABLE 1

s

=

0.072 rom/rev. s • 0.208 rom/rev. v

[m/sJ

/k;1

[N/m

2 ]

eo

[0J

I e;l

[N.s/m

2 ]

_Yo

[0J

I

k;1

[N/m~

SO

[0J

le;1

~.s/m~

. Yo

[0]

0.500

-

1.56 x 109 73 1.03 x 106 112 0.667 1.08 x 109 4] 0.45 x 106 334 1.80 x 109 74 1.43 x 106 118 0.833 1.59 x 109 81 1.17 x 106 100 1.41 x 109 26 0.19 x 106 274 1.000 I 1.45 x 109 82

0.66

x 106 138 2.26 x 109

66

1 ~35 x 106 91

I

J ~ 167 I

-

1. 98 x 109 72 1.26 x 106 92 1.250 1. 01 x 109 79 0.69 x 106 211

-

-1.333 2.36 x 109 85

I

1.19 x 106 121

-

-1.667 2.16 x 109 69 0.75 x 106 78

-

-2.000

I

2.57 x 109 67 1.25 x 106 71

-

-2.333 1.69 x 109 59 0.67 x 106 50

-

-SKF 1550 - steel

(27)

SUBSCRIPTIONS FOR THE FIGURES

FIG. 1.

FIG. 2.

FIG. 3.

FIG. 4.

FIG. 5.

FIG. 6.

FIG.

f.

FIG. 8.

FIG. 9.

Block-diagram representing the dynamic autting process

DiagrammaticaZ presentation of the chip-thickness modulation

Polar plot of the transfer function Th on the threshold

of stabiUty

Nyquist-curve of the dynamic system

The composition of

k~

and

c~

1-

1-Check on linearity of the process damping

and the process

stiffness with respect to the width of chip

The

~o

performanoes of the special toolholder

The plots of the experimental results on

w _c~ ~ _m

t

and

w

_mt~

and the oomputed values for

~s'

The computed values of Wc

are obtained from the results on k

_i

and

ci~

using the

dynamic data of the toolholder

,

FIG. 10. The vaZues of the specific

process

stiffness versus cutting

speed

FIG.

11.

The values of the specific

pl~ooess

damping vel'SUS autting

speed

c.

FIG.

12.

The course of the quantity

i:

versus cutting speed

1-FIG. 13.

FIG. 14.

FIG. 15.

-+ -+

DiagrammaticaZ presentation of the vectors k

_i

and Ci for

severaZ cutting speeds

The oomputed b g -values

~:n

C!omparison with the experimental

stabiUty charts

1._ • r;k

C!iwo

T,~

1-nfluence of

k:

and

~~

as the leading numbers of

1-

(28)

FIG.

16.

The infZuenoe of the prooess damping on the limit values for

some experimental toolhoZders mentioned in literature

FIG.

17.

The working area under Umit oonditions for the

Vanherok-Peters toolhoZder in oase of maohining SKF 1550 steeZ

FIG. 18. The influenoe of frequenoy on the outting stiffness

and

the

prooess damping acoording to Kegg's results

FIG.

19.

The influenoe of tooZ wear on the threshold of stabiZity

for different maohine tool struotures

FIG. 20. The influenoe of prooess damping on the threshold of stability

for different vaZues of the structural damping

FIG.

21.

The overall damping and the frequenoy of the pulse response

(29)

±

y*-y

Te' Tm

j

-y-y*

y l

Th

-fig 1

(30)

- - ; ; »

y

--,,>

F

fig 2

(31)

(32)

400Hz

ho

h:ho•

y*-y

b,wk·Cj

/

Rn

fig .4 out-?f -phase aXIs Tc in-phase

axis

I

·387 Hz

I .

I

,}'-390 Hz . I _,

(33)

fig 5

out-of-phase axis

in-phase axis

(34)

~~;;;;;:;::;;;=----p----p--r~-:2:;n:;:drm--:ai

n

di

rect ion kil fig 6

of

movement

1

st main direction

of

movement

"i

(35)

5 17

fc

(Hz )

17 0

16

5

160

15

5

150

0.20 hs

0.1

5

0.10 ~

o

I

mat: C 45 N l~

110:

0.072

mm

v:

2.33 m/s

,0. ./

V'~

/1.

~ /.

~/

-"/

_t.

~/J.)'

~~.,/'

I

.~

Il./i.~

~

/

'<.,

----~

~-

-~, -~&'

-"1

r

~

_ _

l

S--,~

~i1 ~

n--0.5

1.0

1.5 ·2.0

2.5

3.0

3.5

b (mm) fig 7 ! :

(36)

o

I

0.,=3230

1 r=-

-0

l ]

Q2=Q

-I

I

fig 8

(37)

---_. -""""··~---I'0.18

"

...

t:-

_~L- ~---l---i ~

S - !

~mt

We

1260

Wmt i _ /

;{-~

0 ,

.15

I

' A / - I l---+--+---f---lT--

/L"< -

-~

. •

-X

1200~--~--~-~-(rod/s) 1140

_0.12

~-I--~~"":::;'~.

-

CJ-

~

-'

~e-[-"'---[.:~--~t"'-1080

- -

-.

=Xl .

m

--J~J_-

.09

. ,

h_0072{O:We

(Bl1

COm

p

uted

t

!lI ". • values) 8 , Wmt f - -

It

h

_::08{:~~:~4:::ed

1 0

201---~---t---+---...--

:\',,---

CJ

values)

A: W

mt

06

,!I mm ' A:ts A: tmt

_ ---8-

-8---.

----l...-m-~ __ I,'-- ' I _ _ _ ~,~cc ... '"'1..,.---.-- +,---«

.03

I I gOb+---~----~--~--~----~--~----~--~--~--~Q

o

0.5 1 .. 0

1.5

2.0

2.5

v (mf 5) fig 9

(38)

" II

I - - + - - -

- + - - - - ~-+----

-Q5

_to

1.5

2.0

2.5

v (m/s) fig 10

(39)

o.

Q5 1.0

1.5

fig 11 j ;

i'

iI

j

2.0

- " - - - i ,

i

2.5 v

(m/s)

(40)

-5

2.5

(41)

mot.~ SKF 1550 ho:: 0.208 mm 'K. ::

90°

6

0 a:: 0 y:;

50

A

=

0 0.67

mJs

I

J

I

1.17+

I

m!s

1.00 m/s

fig v 13

0.83 m/s

-I/iOO' C i: lcm

=

0.1xl0

6 Ns/m2 -1> ki: lcm=o.2x109 N/m2

0.50

~

m/s

0.67 mJs

1.17 mls

(42)

'O~--~----~---~----~---,---~----~ , mot: SKF 1550 bg 0

o

I

6nm

~mt~

0.1 I - - - + - - - r -

+---+---r

.---8 r - - - t + \ -4r----+

----ri

--- ---_ ..

~

2mm

8

II

II, AI., •

,+

~ computed values

m

2.0

25 v

(m/s)

(43)

12r----.----.----.----~----~---'"-~~~~.,r·~~rr bg

(mw

_b

g

=

2~'lk

c.

1- __

I Wo

ki

O+----+----~----~---+----+---~----~----+-~-+~~

-0.6

-0.4

-0.2

o

0.2

0.4

fig 15

0.6

0.8

1 CjWO kj

1.2

(44)

b 14

mat!

SKF 1550

(m~'1'1)

ho:

~.07~mm

0 j - - - l a. : 32 30; 'M.: gO

12! mean values

for

kj: 1.5x 109 NJ m 2 (fo~150 Hz) 10 0.0 9 l( 10 9 Nlm2 (f

z

52,.=-O_H:=z)'----lt--_-+--:-::--:-+_-:-::---f---I---I--t---t I Vanherck's

8L- -r __

+ ____

~~

__

~--

__

~

__ - + _ _

t-o-O~lh-o-ld-e-r_+----~--~

6.t---:

_I

~--t--~I___+_+_.I'_l_~----l

i

4

f - - - ! - - - - f - - - / . - - - f - - - f . - - - - I - - - " " - - r - - - ' - - + - - I - - 4 - - - - H "-