Stability analysis of chatter vibration in turning processes

Journal of Sound and Vibration (1985) 102(4), 515-525

STABILITY ANALYSIS OF CHAT-FER VIBRATION IN T U R N I N G PROCESSES M. R.AHMAN

Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge, Singapore-0511 AND

Y. ITo

Department of Mechanical Engineeringfor Production, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo-152, Japan (Received 28 March 1984, and in revisedform 31 October 1984) Most turning operations are carried out with use of three-jaw chucks. The position of the jaws with respect to the direction of the cutting force causes a variation in the mass, damping coefficient and stiffness of the workpiece-chuck-spindle system, the variations being larger when the cutting force is acting against a jaw and smaller when the cutting force is acting along a jaw. These factors are usually considered to be constant when carrying out a stability analysis of self-excited chatter appearing in a turning process and, in consequence, the experimental results so far presented are sometimes found not to agree with the theoretical ones. The variations mentioned above are liable to cause a directional orientation effect in the workpiece-chuck-spindle system and thereby lead to the generation of a type of parametric vibration. Stability analysis carried out with the above variations taken into consideration is found to give a good qualitative agreement with the experimental results. Furthermore, it is found that with the increase of the variation of the mass, damping coefficient and stiffness of the workpiece-chuck-spindle system, the chatter stability of the turning process deteriorates.

1. INTRODUCTION Three-jaw chucks, particularly spiral chucks, are most frequently employed for clamping disc-shaped, ring-shaped and cylindrical workpieces. Statistical enquiries show that for 75% of all machined workpieces which are round and symmetrical in shape the ratio of length to diameter is L / D < 3. Workpieces o f these proportions are normally overhungclamped and machined in three-jaw chucks. Unfortunately, a deviation in shape called out-of-roundness is observed in such workpieces after machining. This out-of-roundness is caused because the position of the jaws with respect to the direction of the cutting force causes a variation in the stiffness of the workpiece-chuck-spindle system, which is larger when the cutting force is acting against a jaw and smaller when the cutting force is acting along a jaw [1-3]. The stiffness is usually considered to be constant when carrying out a stability analysis of self-excited chatter appearing in a turning process [4-8], and, in consequence, the experimental results so far presented are sometimes found not to agree with the theoretical ones. The variation mentioned above is liable to cause directional orientation effect in the stiffness o f workpiece-chuck-spindle system and thereby lead to the generation o f a type of parametric vibration [3]. This effect, mixed with self-excited chatter, causes a significant variation in the stability o f a turning process. Thus, to have 515 0022-460X/85/200515+ 11 $03.00/0 9 1985 Academic Press Inc. (London) Limited

516

M. R A H M A N A N D Y. ITO

a more realistic stability analysis and thereby to determine the actual productivity of a lathe it is necessary to consider the effect of this directional orientation factor. It needs to be mentioned here that among the chatter theories developed so far the one given by Merritt [8] is convenient from the computational point of view, because he has represented a machine-tool-workpiece system in a condition of self-excited chatter by a block diagram with feed-back loop. Analysis of this loop by using feedback control theory yields a straightforward method of calculating the asymptotic and lobed borderlines in a stability chart for a multi-degree of freedom machine-tool-workpiece system, when negligible cutting process dynamics are assumed. His contribution in this area is well appreciated; however, his analysis is inadequate for predicting experimental results fully because he did not consider the variations of mass, damping coefficient and stiffness of the system which are frequently observed in a turning process with a three-jaw chuck. Both theoretical as well as experimental stability investigations have been carried out, as described in what follows, and the experimental results are found to be in quite good qualitative agreement with the theoretical ones. From the investigations it has been found that with the increase of the amount of variation of the mass, damping co-efficient and stiffness of the workpiece-chuck-spindle system the chatter stability of the turning process deteriorates. 2. VARIATION OF MASS, DAMPING COEFFICIENT AND STIFFNESS Frequency analysis of the horizontal deflection of workpieces being turned and held by three-jaw chucks shows that frequency peaks corresponding to the third multiple of the spindle rotation are the most prominent at the onset of chatter [3, 9]; moreover, the deviations from roundness of the same machined workpieces show three peaks and valleys. [3,4,9]. These facts easily lead one to the idea ofsome possibilities of mixing a kind of parametric vibration with self-excited chatter. In the case of turning, such a parametric vibration

I

/

\

-f

\-

/

\

Along jow

Against jow

Figure 1. Variation o f stillness of a workpiece held by a three-jaw chuck. F = directional orientation = 2

(A max-t rain)x 10-2/(,~ max+,~min).

CHATTER STABILITY IN TURNING PROCESSES

517

may, in general, be considered to be due to the variation of system parameters: for instance, the stiffness o f a chucked workpiece. In aiming at the clarification of this variation in detail, first o f all an attempt has been made to investigate the magnitude of the variation o f stiffness in a workpiece-chuck-spindle system through static experiments, and it has been found that there is a significant amount o f variation in the stiffness, depending on the direction o f applied load with respect to the position of jaws, as shown in Figure 1. In the next stage, the workpiece-chuck-spindle system was excited by using the set-up shown in Figure 2 and from the mechanical impedence thus derived the variation of stiffness arising from the directional orientation was investigated. It has been found that not only stiffness but also the equivalent mass and the equivalent damping coefficient of the system vary, depending on the direction o f the exciting force with respect to the position of jaws, and the results are shown in Figure 3. The results when arranged in I HydraulicunJt I

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Figure 3. Variation of mass, damping coefficientand stiffness due to directional orientation.

518

M. R A H M A N A N D Y. ITO

terms of the damping ratios become constant, as shown in this figure. Moreover, from the experiments, it has been found that the value of the natural frequency does not vary with the direction of the exciting force with respect to the position of jaws. Thus, it may be assumed that the variations of equivalent mass, equivalent damping coefficient and equivalent stiffness take place in the same ratio. 3. THEORETICAL ANALYSIS As demonstrated by its many previous useful applications, it is very convenient to use the chatter theory proposed by Merritt [8] when computing the stability limit of a self-excited chatter. A parametric vibration may have both linear and non-linear components. So, to obtain a complete solution a non-linear theory must be applied so that proper account may be taken of the interaction between self-excited and parametrically excited vibrations. However, for simplicity of analysis, the non-linear part has not been considered in this work, and so Merritt's idea has been assumed to be applicable. In consequence, the theoretical analysis is a modification of Merritt's chatter theory, in which it is, first of all, necessary to derive the following three equations: the uncut chip-thickness equation, the cutting process equation and the structure equation. In what follows, thus, these three equations are given, and it is assumed that only the structure equation has to be modified in a case of self-excited chatter mixed with parametric vibration, such as that occurring in a turning process in which a three-jaw chuck is used, as mentioned in the preceding section. 3.1. U N C U T C H I P - T H I C K N E S S E Q U A T I O N [ 8 ] Upon referring to Figure 4, the instantaneous uncut chip-thickness can be written as (a list of nomenclature is given in the Appendix) u(t) = Uo(t)-y(t)+tzy(t-

T).

(1)

--//

Jl

l( Tool (rigidly mounfed)

Figure 4. Uncutchip-thicknessvariation. The Laplace transform of equation (1) is u (s) = Uo(S) - y ( s ) + tz e-rSy(s).

(2)

.3.2. cuTrING PROCESS E Q U A T I O N [6] The cutting force is F ( t) = kcu( t).

(3)

CHATTER

STABILITY IN TURNING

PROCESSES

519

The Laplace transform of equation (3) is

F(s) = kcu(s).

(4)

3.3. S T R U C T U R E E Q U A T I O N If the structure could be represented by a lumped parameter model with one degree of freedom, as shown in Figure 5, the equation o f motion, i.e., the structure equation, is given by

F( t) cos ( a - f l ) = mo(d2/dt2)[y( t)/cos ct] + Co(d/dt)[y(t)/cos ct] + ko[y(t)/cos ct].

(5)

Figure 5. Structure model with one degree of freedom. NOW, as already mentioned earlier, the stiffness, damping coefficient and mass o f the workpiece-chuck-spindle system all vary three times in one rotation in the case of turning with three-jaw chuck, depending on the direction o f the resultant cutting force with respect to the position~ofjaws. Here, to make the analysis simpler, and also on the basis o f the discussion at the last part of section 2, these factors are assumed to vary in the same ratio, which will be termed here as F. Then the structure equation, with consideration of the effect o f directional orientation, takes the form

F(t)g=m(1-Fsin3toot)fi(t)+c(1-Fsin3toot)y:(t)+k(1-rsin3toot)y(t),

(6)

where g = cos (or - f l ) cos a, j;(t) = (d2/dt2)[y(t)] and ) ( t ) = (d/dt)[y(t)]. Equation (6) can be transformed as follows:

(1/g)[my(t) + c)(t) + ky(t)] = F(t)[1 - F sin 3tOot] -I.

(7)

Expanding the right-hand side of equation (7) and also assuming that higher orders of F sin 3tot may be neglected, one gets F(t)[1 - F sin 3tOot]-1 = F(t)[1 + F sin 3tOot].

(8)

The final form o f the equation thus is 9 (1/g)[mfi(t)+cy:(t)+ky(t)]=F(t)+3F(t)FtOot--~F(t)FtOot

33

81 55 +~F(t)FtOot +...

(9)

Since the Laplace transform of

a[ t"F( t) ] = (-1)"( d " F / ds") = ( - 1 ) " F n ( s ) ,

(10)

one has the following results for the respective Laplace transforms:

a[3FFtot] =-3ro~F'(s),

a[~FF~3t3] = - ~ r 3 F " ( s ) ,

a[~oFFo, s t 5] = - ~oro, s tS F~ ( s ).

(11, 12) (13)

If one carries out the stability analysis of the chatter loop for a step response, assuming that a kind of trigger such as a sudden change of system parameter causes chatter onset,

520

M. R A H M A N A N D Y. ITO

then

F(s) = I/s, F'(s) = - 1 / s 2= - ( 1 / s ) F ( s ) , F~(s)

= -120/$

6=

F"(s) = - 6 / s 4= -(6/s3)F(s) -(120/sS)F(s).

(14)

Hence, if one takes the Laplace transform of equation (9) and arranges the result in the form

g(3Ft~176176176176176 y ( S ) - s ( m s 2 + cs+ k ) 1-t s ----Ts § ----3-s

(15)

then one obtains the dynamic compliance

y(s) 1 F(s)-km[(S2/to2)+(28s/to,)+l]X

(+3FtOo 1 s

27FtO3o 243FtOo5 ) - - -sT - § -----3--s "'" '

(16)

where g / k = 1/km, oJ2,= k / m and 6 = c/2(km) I/2. It is convenient symbolically to write the dynamic compliance by using the normalized compliance G~(s), and thus use equation (16) in the form

y ( s ) / F ( s ) = (1/km)G~(s),

(17)

where Gin(s) coincices with that of Merritt's theory if there are no variations of the mass, damping coefficient and the spring constant in a machine-tool-work system. 3.4. STABILITY ANALYSIS OF CHATTER LOOP AND EXPERIMENTALRESULTS The block diagram of the chatter loop derived by Merritt, which is also applicable for chatter'problems with a parameteric vibration, is shown in Figure 6. The transfer function

.oC>

( [

Cutting process dynamics

~

Structure dynamics

Y()

Primaryfeedback path <

Regenerative
relating to Uo(S) and u(s) can, therefore, be obtained from the block diagram'as follows:

u(s)/uo(S) = 1/[1 + (1 -/~ e-r')(kc/k~)Gm(s)].

(18)

From equation (18), the characteristic equation of the chattel" loop can be written as 1 + ( 1 - / x e-r~)(kc/km)G~(s)=0.

(19)

Substituting s "---j~ointo equation (19) and also rearranging gives

ikc/km)G,,(jto) = - 1 / ( 1 - / z e-~'T).

(20)

C H A T T E R STABILITY IN T U R N I N G

PROCESSES

521

The left side of this equation is the product of the harmonic response functions of the cutting process and the structure: that is, y(jto) F(jto) y(jto) k~ G,,(jto). u ( j t o ) - u(jto) F ( j t o ) - k , ,

(21)

The right hand side of equation (20) can be represented by

G w = - l / ( 1 - i . t e-J'~r),

(22)

whose plot can be considered as a locus of critical points. Substituting equations (21) and (22) into equation (20) gives

y(jco )/ u(jto ) = Gcp.

(23)

Hence, intersections of the plot of y/u with points on the critical locus give harmonic solutions of the characteristic equation which define the border-line of stability. The actual calculation procedure for equation (23), with consideration of the effects of directional orientation, ig very difficult when using an analytical method. Thus, the following stability charts have been produced by using a mini-computer and X-Y plotter. One can now consider a structure, the dynamic compliance of which is given by equation (16), and for which it is assumed that f , = to,/2~r = 300 Hz and 8 = 0.05. Then

y(s) u(s)

kc/km =

2

(s/2~300) +[2(0.05)/27r300]+1

[ x/l-

k

3rtoo_F27rtog 243rtog '~ s-------g---- t-. 9 .]. s 3~

If the factor depending on directional orientation is not included in this equation i.e' r = o, then the equation turns to be the original equation proposed by Merritt and the gain-phase plot also takes on the shape ascertained by him; however, for different values of F the gain-phase plot will also take on ditterent shapes as can be seen in the following. Since an overlap factor of unity is the most critical value from the viewpoint of the commencement of chatter, the critical locus was plotted only for this value. First of all, the gain-phase diagram of y/u (in the absence of the directional orientation factor) and critical locus are shown in Figure 7(a) for easy comparison of the results when the factor F is present, as shown in Figure 7(b). In the case of Figure 7(b) the values o f f , and 6 are kept the same as those for Figure 7(a) and the value of k+/k,, is maintained at 0.105: i.e., the value at which the first intersection occured in Figure 7(a). In addition, the value of the directional orientation factor, F, is varied from 0 to 0.3. In Figure 7(b), it is interesting to note that with the increase of the value of F the points of intersections of critical locus and the gain-phase diagram move leftwards, i.e., the structure characteristics move towards instability, whereas, as shown in Figure 7(a), the loci of (kc/k,,)G,,(jto) moves only in the upward direction on the gain-phase diagram when the value of F is zero. One more interesting aspect of the behaviour to be noted here is that for a slight increase of the value of F the intersections start occurring at values of ~, even less than 0.5. In the absence of F, however, the intersections never take place at values of v less than 0.5 in this case of a single degree of freedom system. Here, v is the phase factor given by toT = 2rrfT = 27r(n + ~,) (n an integer). In Tables 1 and 2, the intersection points and critical speeds obtained from the above mentioned figures are shown, respectively. Stability charts drawn for different values of F are shown in Figure 8. From this figure, it is once again vividly clear that with the incre'ase of the value o f F the stability of the system decreases. In this context, it needs to be mentioned here that in the case of a four-jaw chuck frequency analysis of the radial component of cutting force did not reveal the fourth

M. RAHMAN AND Y. I T O

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(b) k c / k m =0.!05, F a s

parameter.

multiple of the spindle speed [3, 9]. In this case, therefore, it c a n be assumed that the influence of the directional orientation is missing. Moreover, the experimental results obtained through the cutting tests revealed that the limiting depth of cut, as shown in "Table 3, in the case of a four-jaw chuck is much higher than that of a three-jaw chuck, even under nearly the same cutting conditions. This phenomenon may also be considered to verify qualitatively the facts mentioned above, though there are some possibilities of

CHATI'ER STABILITY IN TURNING PROCESSES

523

TABLE 1

Tabulation o f intersection points and critical speeds at I ~ = 0 kc/k,,,

f(Hz)

v

n=0

n=l

0.105 0.02

315 308 326.2 305.6 336"5 304"5 351 302"4 423"5 300 520

0"758 0.843 0"67 0"884 0"63 0.916 0"6 0-968 0.543 0.99 0.526

424 365 487 345 534 332 588 312 777 301 988

181 167 195 162 206.5 159 220 153"5 274 .

0.15 0"20 0"50 1"0

n=2 115 108"5 122 106 128 104"3 135 -.

.

n=3

n24

842 80 88"9 78.6 92.6 77.7 97"5 --

665 63"6 69.8 62.5 72.8 62.0 76"3 --

.

TABLE 2

Tabzdation o f intersection points and critical speeds at F = 0.001

kdk,,,

f(Hz)

v

n=0

n=l

n=2

n=3

n=4

0.08 0-10 0.15

310"9 324.1 301"4 340.0 303-3 354"6 300 380

0-721 0.589 0.895 0-53 0"923 0.52 0"949 0"48

431.2 550.3 336.8 641"5 328.6 681.9 316.1 791.6

180.7 204.0 159"1 222.2 157-7 233.3 153.9 256.8

114"3 125.2 104.1 134.4 103"8 140.7 101"7 153-2

83.6 79"5 77"4 96"3 77-3 100.7 75"9 109.2

65"9 63"0 61-5 75"1 61 "6 78"5 60"6 84.8

0-20 0.30

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524

M. RAHMAb/ AND V. ITO TABLE 3

Limiting depth of cut for different numbers of jaws Type of chuck used

Cutting speed (m/min)

Feed rate (mm/rev)

Limiting depth of cut (mm)

Three-jaw Four-jaw

100 100

0.1 0.1

3"2 5-0

changing other characteristics of a machine-tool workpiece system when changing the type of chuck.

5. C O N C L U S I O N S

The variation of mass, damping coefficient and stiffness of a workpiece-chuck-spindle system, depending on the position o f the jaws of the chuck with respect to the direction o f the cutting force, is responsible for the generation o f directional orientation and this effect of directional orientation can be clearly observed when turning is done with a three-jaw chuck. The effect o f directional orientation is found to give rise to a parametric vibration and it plays a significant role in the stability of a turning process. From the investigations carried out in this regard, the following conclusions can be drawn. (1) In the analysis o f the stability o f a turning process, mainly in the case of a three-jaw chuck, not only the self-excited chatter but also the influence o f a parametric vibration, as mentioned above, needs to be considered. This factor is found to play a significant role in the range of stability and the parametric vibration appears in a mixed condition with self-excited chatter. As three-jaw chucks are most widely used in turning, consideration of both self-excited chatter and parametric vibration would appear to be necessary in obtaining a better solution to the actual problem. (2) With the increase of the effect of the directional orientation factor the stability limit decreases. To get rid of this effect, a chuck with four or more jaws could be used; otherwise this effect should be considered when investigating vibration problems in turning machines. ACKNOWLEDGMENT The authors thank Miss A. M. Kumari, Department of Mechanical and Production Engineering, National University of Singapore, for typing this manuscript. REFERENCES 1. M. RAHMAN 1979 Dr. Eng. Dissertation, Tokyo Institute of Technology. A study on dynamic performance test for lathes. 2. G. PAHLITZSCH and W. HELLWIG 1967 Proceedings of the 8th International MTDR Conference 97-118. The clamping accuracy of three-jaw chucks. 3. M. DOI, M. MASUKO and Y. ITO 1982 Proceedings of the lOth NAMRC Conference 409-416. Re-Observation of The Chatter Vibration in Chuck Works. 4. R. N. ARNOLD 1946 Proceedings of the Institution of Mechanical Engineers 154, 261-284. The mechanism of tool vibration in cutting of steel. 5. F. KOENIGSBERGER and J. TLUSTY 1971 Machine ToolStructure, Volume 1. London: Pergamon Press. 6. S. A. TonIAs and W. FISHWICK 1958 Transactions of the American Society of Mechanical Engineers, 1079-1088. The chatter of lathe tools under orthogonal cutting conditions.

CHA'VFER STABILITY IN T U R N I N G PROCESSES

525

7. S. DOI and S. KATO 1956 Transactions of the American Society of Mechanical Engineers 78, 1127-1134. Chatter vibration of lathe tools. 8. H. E. MERRITF 1965 Transactions of the American Society of Mechanical Engineers 87, 447-454. Theory o f self-excited machine-tool chatter. 9. M. RAHMAN and Y. I x o 1978 Proceedings of the 19th International MTDR Conference, Manchester, 191-196. A method to determine the chatter threshold. APPENDIX: N O M E N C L A T U R E

C, Co F f

fo g

k~ k~ ~ko ~1, n l 0

N S

T t I,I 12o W

Y ot

8 F to r n

damping coefficients of structure resultant cutting force or vector force exciting the structure chatter frequency along lobed borderline of stability frequency of variation of system parameters directional factor normalized dynamic compliance of structure static directional cutting stiffness (slope of curve of F versus u) static directional stiffness of structure (slope of curve of F versus y) spring constants of structure equivalent mass of structure spindle speed Laplace operator delay time, 1 / N time instantaneous uncut chip-thickness average or steady state uncut chip-thickness width of cut (measured along cutting edge of tool) relative displacement between tool and workpiece normal to machined surface angle of mode of vibration of structure relative to a line normal to machined surface angle between resultant Cutting force and a line normal to machined surface damping ratio o f mode of vibration of structure factor of directional orientation overlap factor = 2r radian frequency undamped natural radian frequency of mode of vibration of structure