In the early stages of the development of machine tool chatter analysis, the coefficients relating to the variations in chip thickness

SELF-EXCITED VIBRATION

IN TWO-DIMENSIONAL CUTTING PROCESS N. HASSAN

Department of Production Engineering, Technical University, H-1521 Budapest

Received October 30. 1984 Presented by Prof. Dr. M. HORVATH

Summary

Test has been conducted in rough machining to study the self-excited vibration of cutting system. modelled as a "two-degree of freedom system with time lag." The system consists of a work piece with grooves assumed to vibrate relatively to the rigid tool in oblique-cutting.

Introduction

The dynamic behaviour of cutting tool is one of the most important characteristics of assessment with respect to their chatter [1].

Self-excited vibration that occurs violently in metal cutting operations puts limit to the practically possible size of cut referred to as the regenerative chatter. It is one of the various kinds of vibrations that arises in metal cutting [2].

In the early stages of the development of machine tool chatter analysis, the coefficients relating to the variations in chip thickness, ... etc. of the cutting forces have been regarded as constants for a wide range of cutting conditions.

More recently, it has been found that variations in the limit of stability for cutting conditions, particularly at high cutting speeds, can be explained only if the cutting coefficients vary with cutting speed, depth of cut, feed, etc. [3].

Nomenclature

dP x: dynamic cutting force in x-axis dP_{y :} dynamic cutting force in y-axis

Q: angular speed [rad/s]

r: feed rate [mm/s]

N: cutting speed [rpm]

to: dept of cut [mm]

208 ^{V. IfASSAS}

~40 ~20 workpiece

sensor in x-direction headstock

tailstock

tray - center

recorder

Fig. I. The block-diagram for measuring cutting forces and vibrations in two-dimensional system

Test was performed at various depths of cut (0.05 - 0.7 mm), at constant feed of 0.1 mm/rev, with variable cutting speeds (47.5 -750 rpm) and the cutting velocity was 0.05 - 1.57 m/so

Mechanical model

Figure 2 shows the mechanical model of the system with two-degree of freedom.

The following assumptions are considered:

- the vibratory system of the mechanical model is linear,

- the direction of the variable component of the cutting force P is constant,

- the variable component of the cutting force depends only on vibrations in x-axis and in y-axis,

- the value of the variable component of cutting force varies proportion- ally and instantaneously with the two vibrational displacements x and y,

- the regenerative chatter and the mode-coupling principles are taken into consideration.

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A

Fig. 2. Mechanical model of two-degree of freedom system

Theory

Under steady-state cutting conditions the following parameters are used.

V=RQ=2nRN, Q= 2n and

T The mean cutting force is defined by:

P=Cpqxp where

C p: factor depends on the material of the workpiece, cutting conditions and the geometry of the tool

Xp: exponential constant q: chip area

q=toSo 8P -C x ^txp-lsxp

-;;:-t - ^PPO 0

00

(1)

210

So: nominal feed [mm/rev]

V: cutting velocity [m/s]

Ksx :chip thickness coefficient in x-axis [N/m]

Ksy: chip thickness coefficient in y-axis [N/m]

K Ix: dynamic chip thickness coefficient in x-axis [N/m]

K Iy: dynamic chip thickness coefficient in y-axis [N/m]

Kg: angular speed coefficient [N s]

Kv: cutting velocity coefficient [Ns/m]

T: time lag at each revolution [s]

v: displacement coefficient Co: feed rate factor

R: radius of workpiece [mm]

dS o : chip thickness variation dr: feed rate variation

dQ: angular speed variation dc_{o :} depth of cut variation dV: cutting velocity variation

nJ x : mass in x-axis [Ns2/m]

In\, : mass in y-axis [Ns2/m]

C~: damping coefficient in x-axis [Ns/m]

C_{y :} damping coefficient in y-axis [Ns/m]

Sx: stiffness in x-axis [N/m]

S\, : stiffness in y-axis [N/m]

Experimental procedures

Test was performed on a horizontal lathe powered by a 5.5 kw d.e. motor, having center to center length 1500 mm. A workpiece of A42 steel, 350 mm long and 40 mm outside diameter was clamped between the headstock and tailstock centers.

From Fig. 1. it is obvious that the workpiece used had grooves which means that the stiffness of the system was artificially decreased. The section III (Fig. 1) was considered for investigation.

Tool geometry:

overhang L 20 mm

relief angle 'Y.. = 6°

rake angle j = 12°

tip angle e

=

^78°

nose radius r = 0.4 mm size of the tool shank 25 x 22.5 mm.

let

from Eq. (I) and (2)

K<;=-

. o( ?P

⁰

i1P _ to K (;So - So s·

Under steady-state cutting conditions the cutting force is defined as P= /(So, to, V).

Therefore, the cutting force variation becomes

where

V=RD, d V=RdD,

Substituting Eqs (2), (3) and (5) into Eq. (4)

dP= So to KsdSo+Ksdto+KgdD.

211

(2)

(3)

(4)

(5)

(6) Under the dynamic cutting conditions the cutting force is a function of four independent factors P(So, r, V, to). The cutting force variation for small changes in these factors can be written as

(7) where K ^1, K ^2' K ³ and K ⁴ being the dynamic coefficients which can be de- termined from sophisticated dynamic cutting tests. To determine the dy- namic coefficients, the following conditions are considered:

I -Condition

D=constant, to = constant and So=variable d D=dt o =0 and

Substituting these in Eqs (6) and (7)

(8)

4 Periodica Polytechnica M. 30/2

212 \. illSSn

J J -Colldif iOIl

So = constant, I () = constant and Q = variable d/"= 1n dQ and dSo=dto=O) So from Eqs (6) and (7)

J J J-Condil iOI1

So = constant, Q = constant and to = variable dSo=dQ=dr=O, and similarly from Eqs (6) and (7)

K4=Ks Substituting Eqs (8), (9) and (10) into Eq. (7)

(

10 . ) 2n

d?=K1 dSo + So Ks-KI Qdr+

+[~J -(~:

^{K s -K1})2nR TJdQR+Ksdt o.

(9)

(10)

(11 ) Where, dS o , dl o , dr and dQ are functions of time and are independent from each other, calculated as

dS o =dSox+dSoy=(X-X1')-^v(y-YT) dt o =dl ox + dtoy=x-ry

dr

=

^drx

⁺

^dr\"

=.x -

^[;.i'

Vibration in x-axis

The cutting velocity Vis constant in x-axis, i.e., dV=O. Therefore, from Eq. (11) the dynamic cutting force in x-axis can be written as:

d?x= K 1x[X-X_{1' -} v(Y- YT)J

+

Ks)x-vy)

+

+(~: Ksx-KIX)2~

^{(.x-v.v). (12)}

Where,

(~:

^{Ksx - K}

l.~)

is the feed rate coefficient in x-axis.

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The feed rate factor Co considered by Knight [3] is C _Ks-KI

0 - KI

In x-axis Eg. (13) becomes

Therefore, the dynamic chip thickness coefficients may be defined as

and

to 1

K - - K - -

lX-SO SX1+C_{o '}

Vibration in y-axis

213

(13)

(14)

(15)

The cutting velocity Vis variable and the vibration y brings a change in the angular speed

Similarly, the dynamic cutting force in y-axis was determined from Eg. (11) as follows

dPy

=

K ^{ly[(X -} xT ) -v(y-YT)]

+

Ks,.(.X: - vy)

+ (~:

^{Ksy - K}

I)) ^{Tx +}

(16)

Where

(~:

^{Ksy - K}

I)) ^{(v + 2~oR)}

is the feed rate coefficient in y-axis.

The displacement coefficient v is calculated as

(17)

4*

214 ^{v. HASSA.\}

The stability condition

The equations of motion of the mechanical model with two-degree of freedom system are written as:

mxx+Cxx+Sxx= -dPx my. V

+

Cy.v+Syy= -dPy

(18) (19) Substituting Eq. (12) into Eq. (18) and Eq. (16) into Eq. (19), Eqs (18) and (19) can be represented in matrix form

AX+BX+DX+EX

=0 where,

A, B, D and E are constants and

X=[~]' x=[~], x=[x]

^and

x=[X(t-T)]

y y y y(t-T)

Let us solve the matrix (Eq. 20) by considering the following:

x(t) = lY.e)·1

and

and

x(t - T)

=

^lY.e)·(I-T)

y(t)

=

^f3e)·1

y(t - T)

=

^f3el.(l-T)

(20)

Substituting these in Eq. (20) for trivial solution of matrix equation, if [; ]

=0 and [;] #0.

The characteristic equation is written as

b¹²)·+d¹² +e¹²e-^AT

1-

₀

Q22)·2+b 22),+d22 +e22e-)·T - (21) Equation (21) can be represented in the form

D(),)

=

Re D(),)+i lm D().) (22) where,

),=i'P, i=yC!

and

e -^AT = e -^i'PT = cos

('PT) -

i sin

('PT)

SELF-EXCITED I1BRATlOS JS TWO-DJ.!! E.\'SJOSAL CCTTJ.\{j PROCESS

Therefore, the characteristic Eq_ (21) takes the new form:

D(iP) =

la

^{11 p2}

+

ibl l P +d 11

+

^ell (cos (PT)-i sin (PT))

I ib₂₁ P

+

d ²¹

+

e21 (cos (PT) - i sin (PT))

215

ib21 P +d12 +eI 2(cos ^(PT)-i sin (PT))

I

=0 (23) - an p²

+

ib₂₂ P

+

dn

+

e22 (cos (PT)-i sin (PT))

Let M( P)

=

Re D(iP) real part S(P)= Im D(i'P) imaginary part Eq. (22) can be represented as

D(i P) = M (P)

+

is(P) (24) The solution of Eq. (20) is asymptotically stable, if their roots have negative real parts.

It can be proved [4J that the system is stable if

L (-

m 1)K + I sign S ^(P K) = - 2 (25)

K= I

A computer program was made with the aid of the algorithm resulted from the theorem of G. Stepim for computation zeroes of the real part of Eq. (23) and to check whether Eq. (25) is true or not.

The program has been written in WANG 2200 computer.

The results show that the characteristic equation (21) depends on a large number of parameters. The dynamic cutting force in x-axis dP ^x in Eq. (12) depends on the change in chip thickness dS_{o ,} on the change in the feed rate dr, and on the change in the depth of cut dt_{o .} Therefore, the dynamic instability can occur as a result of dS_o and dt o variations.

Discussion

Figure 3 shows a theoretical and experimental stability chart of cutting process in two-dimensional system. The asymptotic borderline of stability is the principal borderline, since it defines the maximum depth of cut (0.155 mm) which will result in stable cutting at all speeds. The prediction indicates that the stable region increases when the cutting speed increases from a minimum to around 320 rpm.

It has been observed that the limit of stability of a system is greatly affected by the inequality in rotating speed of the workpiece, depth of cut, chip thickness coefficients Ksx and K sy , feed rate coefficients, dynamic chip thickness coefficients K Ix and K Iy , cutting velocity coefficient K Vy and time lag

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Culling speed N (rpm)

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SELF· EXCITED VIBRATION IN TWO·DHfENSIOSAL CUTTING PROCESS 217

T. Figure 4 concluded that the maximum vibration amplitude is directly proportional to the width of cut and the system becomes unstable if it exceeds the critical equivalent width of cut.

Figure 5 shows the linear relation between the cutting velocity and the average main cutting force for each depth of cut in order to determine the cutting velocity coefficient. Similar curves have been got at different depths of cut. The average main cutting force and average thrust force as function of the depth of cut for each cutting speed are measured.

]

~ .2

'5. E o .2 c

0..5 265 (rpm)

'2 0.25

.0 .s:

o.~---r---~---r---T~

o 0..25 O.SO 0..75

to.

Widlh of cuI (mm)

Fig. 4

dPy I cl:

Kvy=~= an Vy

10=o..2mm to: depth of cut R : correbtion factor

o.~----~---r---~~

0..33 0..66 0..99 1.32

Cutting velocity v (m/s) Fig. 5. The cutting velocity coefficient Kv\.

218 S.IIASSA.'i

A typical graph of these forces is shown in Fig. 6. All the curves follow this expected form. The forces versus the depth of cut are linear.

The dynamic chip thickness coefficients K I}' and K Ix in Eq. (14) and (15) are more affected by

(~:)

than the chip thickness coefficients Ksx and Ksy' Therefore, the ratio of 10 and So have a high effect on the stability condition of the cutting process.

250 N=530(rpm) KS -tonC£5y=~

3 ^{y- dto}

o..X 200 R=0.9803

"0 0 c a..>

j

¹⁵⁰

I

g'

~ _u

~ 100

I

I

t ^I

~

I

I I I I

I

Q25 0.5 0.7'5

J

O.-pth of cut to(mm) Fiy. 6. The thickness coefficients

Conclusions and directions for further research Conclusions could be summarized as follows:

1. The level of stability is increased as the depth of cut is decreased because the

(~:)

reduces the values of the dynamic chip thickness coefficients K Ix and K_{I }, .}

2. The predicted borderline of stability shows good qualitative agreement with experimental results and indicates stability in low speed range due to the damping effects of the cutting process.

3. The dynamic cutting coefficients are properly evaluated from the steady- state cutting parameters. The feed rate variation dr has a stabilizing effect.

SU.r UCIlLIJ IIHU.-1'//(J.\ IS J'1I'(}·IJI.\f/;'.\·SIOSAI. C11rJV(i I'IWC/;SS 219

4. The following factors are noticed to influence the cutting stability: The cutting velocity coefficient, stiffness, chip thickness coefficients, feed rate factor, damping coefficients, dynamic chip thickness coefficients, displace- ment coefficient and time lag.

5. In future an on-line data processing system could be made by connecting the control system ofCNC or NC-Iathe with computer to use the stability chart and to stabilize the system by changing the cutting conditions (feed, cutting speed and depth of cut) automatically.

Acknowledgement

This work has been carried out in the Department of Production Engineering, Budapest Technical University. Great thanks are due to my supervisor Prof Mcitycis Horvcith, Head of the Department for his continued advice and encouragement.

References

I. WEeK, M.: Assessing the chatter behaviour of machine tools, Annals of the CIR P, 1978 2. HOSHI, T.: Study for practical application of fluctuating cutting speed for regenerative chatter

control, Annals of the CIRP, 1977

3. K~IGHT, W. k: Chatter in turning: some effects of tool geometry and cutting condition, 1nl. 1.

Mach. Tool Des. and Res. 1972

4. ST(:PA~, G.: Stability of machine tool vibrations under regenerative cutting conditions, Period.

Polytechn. 24, 169 (1980)

Nassir HASSAN H-lS21 Budapest