Numerical simulation of non-linear chatter vibration in turning

Int. J. Mach. Tools Manufact. Printed in Great Britain

Vo[. 29, No. 2, pp.239-247, 1989.

0890-6955/8953.00 * .00 Pergamon Press pie

NUMERICAL SIMULATION OF NON-LINEAR CHATFER VIBRATION IN TURNING KRZYSZTOF JEMIELNIAK* a n d ADAM WIDOTA*

(Received 5 October 1987; in final form 30 June 1988) Abstract--In this paper, basic relationships and algorithms for numerical simulation of non-linear, self-excited vibrations in single degree-of-freedom cutting systems are presented. Non-linearities due to the tool leaving the cut, as well as interference between the cutting tool clearance face and cutting surface waviness, were taken into consideration. Examples of vibration simulation results are shown.

NOMENCLATURE ao

b ¢

dt

F~h h hx

n~ hex kxa

K~ LSS m r/

el WR

T Vo X, X T

xh ~ok Oto

~q

average uncut chip thickness width of cut stiffness of the machine tool system iteration period incremental feed force damping component of force Fxd damping of the machine tool system dynamic cutting damping coefficient damping of cutting process LSS damping factor dynamic cutting stiffness coefficient stiffness of cutting process Low Speed Stability factor mass of the machine tool system number of iteration step workpiece rotational speed period of the workpiece revolution nominal cutting speed relative tool-workpiece displacement in present and previous workpiece revolution velocity in the half of iteration period instantaneous clearance angle nominal clearance angle angle between nominal and instantaneous cutting speed vectors

1. INTRODUCTION

SELF-EXCITEDvibration during cutting, called chatter, has been investigated in a number of studies aimed at describing and predicting the limit of stability against chatter and at- determining the correlation between this limit and the cutting conditions. The conventional stability analysis is based on linear differential equations of machine tools and cutting forces. The majority of authors support the opinion that one of two main causes of chatter is the regenerative effect, which can occur in single degree-of-freedom systems. For this reason, and also for simplicity as well as for clarity of interpretation of simulation results, it is assumed in this paper that the machine tool system is represented by a single degree-of-freedom system in the feed direction (x-coordinate). Such an assumption has also been made in many previous works. It is possible to extend the presented relationships to systems with many degrees-of-freedom, thus obtaining more complicated relations. *Technical University of Warsaw, Institute of Production Engineering, ul. Narbutta 86, 02-524 Warsaw, Poland. 239

240

.K. JEMIELNIAKand A. WIDOTA 2. BASIC RELATIONSHIPS

The equation of motion of mass m for single degree-of-freedom systems has the following form

(1)

m Y + h± + c x = F~a

where Fxd =

-K,,a

(x -

xr) - H~;

K~d=bkxa; O ~ = b h ~ ;

(2)

xr=x(t-T);

T = l/nwR.

After the limit of stability has been exceeded, the chatter amplitude according to the theoretical solution of equation (1) increases up to infinity. It is observed and well known, however, that chatter amplitude does not increase indefinitely but stabilizes at the finite amplitude of vibration. This stabilization is caused by non-linearity of the machine tool components and cutting process. For this reason the linear equations (1) and (2) cannot be used for analyzing the system behaviour and chatter course after exceeding the stability limit. Such an analysis, however, is necessary for the investigation of relations between chatter frequency and amplitude and also of the influence of online cutting parameter changes on the cutting system during chatter [1]. The strategy of automatic chatter identification and suppression should be based on the results of that analysis. 3.

CUTTING SYSTEM NON-LINEARITIES

There are two basic cutting system non-linearities counteracting the infinitely growing chatter amplitude: the tool leaving the cut [4,5], and the interference of the cutting tool clearance face and the cutting surface waviness [3]. The first is due to the fact that with the increasing chatter amplitude the tool starts to move outside of the Cut for a part of a cycle. This effect can be introduced into the model by a more precise description of Xr in equation (2). That means that more than one preceding workpiece revolution should be taken into account (Fig. 1). Thus we obtain XT = minimum I x ( t - T ) , ao + x ( t - 2 T ) , 2 ao + x ( t - 3 T ) . . . . ].

It also should be assumed that force Fxa is described by formula (2) only when ao+xr-x>O. \ •

\

/ ,~-~\

~//

/ \, ',. Z'

/

\

\

fx\

-./

\\

FIG. 1. Traces remaining on the cutting surface when the tool leaves the cut.

(3)

Simulation of Chatter Vibration in Turning

~

~

~

~

,

<

241

0

Fro. 2. Dependence of instantaneous clearance angle on the relative vibration of the tool and workpiece.

Vo

/
In the opposite case it would mean that the tool jumps out of the cut and the force F.a = 0. The second non-linearity is caused by changes of instantaneous clearance angle ~ o k connected with the relative tool-workpiece vibration (Fig. 2) etok = et o + "q ~ et o + k / v o .

(4)

When this angle approaches zero, the damping of the cutting process [equation (2)] increases non-linearly. Because the instantaneous clearance angle cannot be negative "qmin ~ -- Oto ,

one can assume that the additional damping force caused by this effect increases to infinity when cto, approaches zero. This effect causes a considerable increase of the stability limit at the low cutting speeds and is known as the L o w S p e e d S t a b i l i t y (LSS) [3]. Thus we can assume that this additional damping force does not occur for the vibration speeds k>0. The same effect can be obtained mathematically by replacing the damping component of force F x a from equation (2) F~h = --/-L~ by the form (Fig. 3) F~,, = - h , ~ Hx.t

(5)

242

K. JEMIELNIAK and A. WIDOTA

where h,,, = f l - LSS

ao 3C/Vo + YC/Vo for,~ < 0

1

for3c ~>0

(6)

where LSS > 0. The LSS factor in equation (6) expresses the importance of this damping effect and should be obtained experimentally. Having no adequate experimental results it has been assumed here that LSS = 1. Finally the incremental feed force should be described by the form

--Kxd(X XT) hex nx.fC

for ao + x r - x > 0

0

forao + x r - x ~< 0

- -

- -

(7)

where Xr is expressed in the form (3) and hex in form (6). The equation (1) with the cutting force expressed in form (7) and equation (3) are taken as a base for chatter simulation model. 4.

ALGORITHM OF NON-LINEAR CHATI'ER NUMERICAL SIMULATION

The numerical simulation of chatter is based on the determination of displacements, velocities and accelerations of the system in succeeding small periods--iteration steps, dr. The basic relationships allowing this simulation were given by Tlusty and Ismail [5] 2(n)

= [F~d -- h:c(n) - cx(n)]/m 2(n) dt

(8)

,i:(n+i) =~(n) +

x ( n + l ) - = x(n) + ,~(n+ 1) dr. As one can see from the equations (8), the authors have made the following simplifications --during the period from the moment n to the moment n + 1, the acceleration is constant and equal to that at the beginning of the period dr; --velocity during the period dt is constant and equal to that at the end of the period dr. These assumptions cause an inaccuracy of the simulated values in comparison to values calculated from the analytical solution of differential equations (without nonlinearities) for succeeding workpiece revolutions. It is possible to obtain a considerable increase in simulation accuracy by using relations presented below. At first, velocity ±n(n) in half of the current iteration period, dr, is obtained on the base of velocity :th(n-1) in half of the previous period and acceleration )~(n-1)at the end of the previous period (Fig. 4)

.~h(n) - :oh(n- 1) + 2 ( n - 1) dr.

(9)

Then the displacement x(n) at the end of the current period dt can be obtained as x(n) = x ( n - 1 ) + ±h(n) dr.

(10)

The velocity at that moment can be expressed as ±(n) = J:h(n) + 2 ( n - 1 ) d t - 2 ( n - 2 ) dt/2.

(11)

Simulation of Chatter Vibration in Turning

243

I I

i

I

lJ

oI i

i

I previous I I I I I n-2 J. I I I~(n-Z) I I ~ t ( n - 2 ) i t ~r(n-ZlO't ~ =lllillllllL~(n-1)

I I (n-l) dt I I I I i dt current I 1.

_

n-1

i

I

i

I J I I I I, L

'

D

(time]

I

I I I

I

-

'

Y(n) dt I

I

I

I [

i

I

I

I

}

I

I I

n-2

•

s~p of it=r~ n

:°-Y |

dt next

l

n-1

n

time)

~'(n-1)

FIG. 4. Diagram of the applied method of numerical simulation.

On the base of x(n), ~(n), X(n) is evaluated

xr(n) and the presented feed force model (7), acceleration

.~(n) = (Fxd [x(n),±(n), xT(n)] - c x(n) - h ~ (n))/m.

(12)

244

K. JEMIELNIAKand A. WIDOTA

If, at the end of the considered time period, the tool is in the material, then it can be expressed as x(n) - x:r(n) < ao

(13)

a n d the traces on the cutting surface are made xr+l(n) = x(n).

(14)

In the opposite case the "old" trace remains on the cutting surface, magnified by the

average nominal chip thickness . x r + l ( n ) = xr(n) + ao.

(15)

The value xr+l(n) will be used in the next workpiece revolution (pass) as xr(n). On the basis of the presented relationships, a computer programme SDS1 in Turbo ~'PROCEDURE SIMULATION )

-! In: = O;

sumt:= 0 ]

--I n := ~h:= x := :=

n + ~h+ x + ~h+

1; ~ $ dr; ~hz dr; ~ $ dt - ~bI dr/2

~b:= 0

- xT[n] >= ao

I xT[n]:=xT[n]+a° I

I N

-

I

I Fxd: =-Kxd$ (X-XT[nl)-Hx*~ xT[n]:=x

~-~

Fxd:= Fxd + LSSIHxSx

I

* (~Ivo)l(QCo+~/v o)

!

I ~ Force with random component ?

~- - - ~

FxcI:=Fxd + FRNI) ]

1

Ix

!

1~:=(-clx - h$~ + Fxd)/m

I PROCEDURE SYNTHESIS ]~

Waveform Y ~ori ntout ? /

~

I

Wavmform in succe~ing revolutions

I

I_

I N

<

n>T/dt > End of Dass ?

YI

(N.t

:)

FIG. 5. Flow diagram of the main simulation procedure.

] l

Simulation of Chatter Vibration in Turning

245

Pascal language has been worked out which enables chatter simulation for input machining parameters; m, h, c, kxd, hx, LSS, Vo, ao and nwR. As a primary excitation, the velocity impulse [:t(1) = 50 mm s -1, x(1) = 0] can be selected at the.beginning of the cutting or random primary cutting surface. The cutting force can be "ideal" according to equation (7), or with additional random components representing cutting noise, known from experimental measurements. As a result of numerical simulation it is possible to obtain and to draw positions of the vibrating tool in subsequent workpiece revolutions. A second programme option enables an evaluation of instantaneous amplitude, frequency (calculated as an inverse of last vibration period) and phase angle between present vibration and cutting surface undulation. Flow diagrams of main calculating procedures of this programme are shown in Figs 5 and 6. The simulation accuracy for various machining parameters was examined using a special testing programme. Examples of obtained results are shown in Fig. 7. There are simulated and analytically computed vibration courses during two workpiece revolutions after velocity impulse plotted one on another. Differences between them are plotted in a ten times greater scale on the right side of these courses. Simulation error calculated in this way depends on the dynamic system of machine tool parameters and iteration step values and does not exceed some per cent of instantaneous amplitude. The planned application of the simulation programme is thus satisfactorily taken into account. 5. APPLICATION OF SIMULATION To illustrate the possibilities of the presented programme, the results of chatter simulation for two selected combinations of cutting conditions are shown here; related data are given in upper parts of the presented figures. Properties of machine tool [stiffness c (N ram-l), damping h (Ns mm -'1) and mass (kg)] are represented by the

(0cE00Re P ) SYNTHESIS

{Rb<=O and ~h>O

Xoim --~

Amplitude printout

I ~Y ( xT[n]>=O and XTb
tT:=XTbSdt/(XTb-XT[n]) I + sumt

IN

1

t:= XbSdt/(Xb-X)+sumt f := 1 / ( t - t b) tb:= t Frequency f printout

i

fi: = f$(t-tT)/(tT-tTB)~/~ 1360

I

tTB:=tTB Phase a n g l i p r i n t ° u t

I J

t

sumt:=sumt+dt IXTb:mXT[n]

Is that the f i r s t

\

\workpiece r e v o l u t i o n ? /

"

y

I

J

1

l~b: x Xb:= Xh)

FIG. 6. Flow diagram of the estimated instantaneous amplitude (Xo), frequency (f) and phase angle (f,) between current vibration and wave on the cutting surface.

246

K. JEMIELNIAK and A. WIDOTA

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~c(J.)= Ir Sin

mJO mee~'s,

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•

Coap

P •

t

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L....J

Je~IS.0,hnJ..4AD,o=UNJgO,J4bcd-31NNB,Hx=49.SO,aS:=0.30

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P •

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• •

Co~p SO

on

S.O

L.--.J

on

¢1~~" ¢" ) iq~c

=

-

1,, 341, o n

FIG. 7. Results of the accuracy test of simulation•

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FIG. 8. Simulation results for low cutting speed. R a n d o m p r i m a r y cutting surface.

data of the test rig with a single degree-of-freedom, built in the Institute of Production Engineering of the Technical University of Warsaw. Stiffness kxd (N mm -2) and damping hx (Ns mm -2) of cutting process are obtained on the basis of an analytical--experimental model of cutting coefficients [2]. These coefficients were obtained for a tool holder, hR 171.36.3825, with an indexible insert TNMX 2404-05 SECOTIC TP, workpiece material C30. Workpiece rotational speed n W R is given in rpm, cutting speed vo in m min -~, uncut chip thickness ao and cutting width b in mm.

Simulation of C h a t t e r Vibration in T u r n i n g ~ M R = S M ob = ; l . • , ~ : : O . O 0 ok . , ~ l . = U m O o h X : ~ . M , v O = . I . 4 0 , 0 = I I I 0 0 0 , l r o l , . o e w / r i b ravWtoA o o m p o ~ e v t t , 8 e e d = i . O ,

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FIG. 9. Simulation results. Cutting force with r a n d o m c o m p o n e n t .

The right part of each figure presents chatter waveforms in succeeding workpiece revolutions (tool passes). The revolution "0" denotes primary cutting surface. Chatter marks in the following passes are plotted successively side by side with distances equal to average uncut chip thickness. The same scale is used here for tool displacement.x, except when amplitudes are very small, in which case they are multiplied by ten and indicated by "*" In Fig. 8, chatter for low cutting speed (Vo=30 m min -1) is simulated. The relatively high workpiece rotational speed ( n W R = 630 rpm) was assumed here to obtain clearer figures. Chatter is initialized by the random primary surface visible as pass 0 and increases without further disturbances. Here the stabilization of chatter amplitude can be seen on a relatively low level (about 0.035 ram). A non-linear increase of damping force, i.e. the low speed stability effect mentioned above, is responsible for this. In practice, chatter is initialized and disturbed by random factors of the cutting force. This is simulated in Fig. 9, where Vo = 140 m min-L Here, the non-linear damping has no influence on cutting force and chatter course. But starting from the 13th revolution, the tool temporarily leaves the cutting material with forces that stabilize the chatter amplitude. In conclusion, one may say that the presented method of non-linear chatter simulation is convenient and effective for an analysis of processes connected with chatter regeneration. The programme is very useful, particularly for the investigation of development of chatter frequency caused by various disturbances and its influence on the chatter build-up process. REFERENCES K. JEMIELNIAKand A. W[DOTA,Int. J. Mach. Tool Des. Res. 3, 207 (1984). [2] K. JEM1ELN1AK,Arch. Prod. Engng, Polish Academyof Science, p. 417 (1987). [3] R. L. KEGG,Ann. CIRP, p. 97 (1969). [4] Y. KONDO,K. OSAUUand H. SATO,Trans. ASME, p. 325 (1981). [5] J. TLUSTYand F. ISMAIL,Ann. CIRP, p. 299 (1981). [1]