Pergamon Int. J. Mach. Tools Manufact. Vol. 34, No. 4, pp. 489-498, 1994

Copyright (~) 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0890-6955/9456.00 + .00

A S T U D Y O N T H E C H A T T E R C H A R A C T E R I S T I C S O F T H E

T H I N W A L L C Y L I N D R I C A L W O R K P I E C E

J. Y. CHANG,t G. J. LAIRS and M. F. CHEN§

(Received 15 December 1992; in final form 17 March 1993)

Abstract--The purpose of this study was to analyze the chatter characteristics of a thin wall cylindrical workpiece. Initially, the structural dynamics of the workpiece were investigated, then the chatter phenomenon during the cutting experiment were observed. From the results, it was clear that the value of the stiffness coefficient of shell mode decreases rapidly when the ratio between the inner diameter and the wall thickness increases. In the meantime, the value of the damping ratio for shell mode was very small. The high value of dynamic compliance causes the shell mode to be easily excited.

a

Di D,, el e2

ei

fl £12 ,Yl3,f2~ • • L h K Ki Kt2 L m - 1 n

P S V x,

NOMENCLATURE

depth of cut, mm workpiece inner diameter, mm workpiece outer diameter, mm deflection in direction y deflection in direction z total deflection in direction y natural frequency of beam mode (m = 1), kHz natural frequencies of shell mode (m = 1, 2, . .; n = 2, 3 . . . . ), kHz chatter frequency, kHz workpiece thickness, mm static stiffness ( P/e~, N/lxm) stiffness coefficent of beam mode (Kt = P/el) N/p,m stiffness coefficient of shell mode (Kt2 = P/e2) N/l~m overhang length of workpiece the number of nodes in axial mode shape the number of circumferential waves radial static acting force feed rate, mm/rev cutting velocity, m/min the distance from cutting position to the check, mm damping ratio of beam mode (m = 1) damping ratio of shell mode (m = 1, n = 2)

1. INTRODUCTION

OWING to the low stiffness of the thin wall cylindrical workpiece, chatter in the cutting process of these workpieces occurs more easily than in the machining of solid workpieces. Machining of thin wall workpieces becomes one of the most difficult tasks in the factory. Many papers [1-8] have previously discussed the chatter vibration. But, they have only discussed the chatter vibration of solid workpieces. There are no papers which systematically discuss the chatter phenomenon of thin wall workpieces. The purpose of this study is to present the experimental chatter characteristics of thin wall cylindrical workpieces.

The ratio of inner diameter to wall thickness and the ratio of workpiece length to inner diameter have a close relationship with the dynamic characteristics of the thin wall cylindrical workpiece [9-16]. In particular, the ratio of inner diameter to wall

tDepartment of Mechanical Engineering, Tatung Institute of Technology, 40 Chung-Shan N. Road, Sec. 3, Taipei, Taiwan 10451, R. O. C.

~tTo whom correspondence should be addressed. §Mechanical Industry Research Laboratories, Industrial Technology Research Institute, 195-3 Chung-

Hsing Road, Sec. 4, Chutung, Hsinchu, Taiwan 31015, R. O. C.

489

490 J.Y. CnAN6 et al.

thickness considerably influences the dynamic characteristics of the thin wall cylindrical workpiece, for the dynamic characteristics of the workpiece is of beam mode when the ratio of inner diameter to wall thickness is small, and tl~e workpiece can be considered as a thin shell when the ratio of inner diameter to wall thickness is large. The general vibrational behaviour of thin wall cylindrical workpieces consists of the dynamic characteristics of both a beam and a shell, which causes the chatter phenom- enon of the thin wall cylindrical workpiece to be more complicated.

Merritt and others [3-7] have pointed out that the stable cutting depth depends on the value of dynamic compliance of the machine-tool-workpiece system. In the thin wall workpiece cutting system, the dynamic compliance of the workpiece is the dominant factor in the system stability. In the present study, the stiffness coefficient, natural frequency, and damping ratio of thin wall cylindrical workpieces were investigated in detail.

Initially, numerical and experimental methods were used to investigate the dynamic characteristics of the thin wall cylindrical workpiece. Then the cutting tests of the thin wall workpieces and the solid workpieces were carried out in order to compare the chatter phenomena. A number of cutting tests of thin wall workpieces were also carried out to clarify the special chatter phenomenon.

2. THE ANALYSIS OF DYNAMIC CHARACTERISTICS OF THE THIN WALL CYLINDRICAL WORKPIECE

2.1. The measurement and computation of dynamic characteristics

The typical workpiece discussed here is shown in Fig. l(a), where L is the cantilever overhang length, Di is the inner diameter, Do is the outer diameter and h is the thickness of wall. The typical modal shapes for clamped-free thick wall cylindrical workpieces are shown in Fig. l(b), where m - 1 is the number of nodes in axial (X axis) modal shape, The workpiece can be considered as a cantilever beam when the value of Di/h is smaller than 5, and its dynamic characteristics will then be considered as a solid beam.

When the value of Di/h is larger than 60, the workpiece can be considered as a thin shell. The typical radial nodal patterns for clamped-free circular cylindrical shell are shown in Fig. 2, and for each vibration mode the workpiece has a displacement in the y direction of the undistorted cross section. This causes the dynamic characteristics of the thin shell to be more complicated than those of a solid beam. m - 1 means the number of nodes in axial (X axis) modal shape and n means the number of circumfer- ential waves, m and n are the subscript indices of the vibration mode in the following.

Y

_ . _ - £ - . _ - : ) ~ - - _ - _ - E _ _ i D o . . . . . . . . . . . . . .

( a ) Workpiece d imensions

m = l m = 2 m = 3

( b ) Modal shapes

FI6. 1. Workpiece dimensions and modal shapes for a cantilever cylindrical workpiece [9].

Chat ter Characteristics of the Thin Wall Cylindrical Workpiece 491

Nodal orrangerneq : for m---3, q~~-

. - 2 2 1 ": . . - , " ] O / . j . :~ ' .~ ) . .~ Cir c'J mfer eq t ia!

. • - . . . .~__x___ node Axiol node

rn~l m=L m-=3

( a ) Axla! noda! pattern

n = 2 n = 3 n = 4

( b ) Circumferential nodoi pattern

FIG. 2. Nodal patterns for a cantilever cylindrical shell [10, 11].

The dynamic compliance of a thin wall cylindrical workpiece is determined by the stiffness coeffÉcient, natural frequency and damping ratio of each mode. In the thin wall cylindrical workpiece cutting process the main displacement occurs in the radial direction, so the dynamic characteristics of the radial mode are taken as the main ones. The chucking condition of the workpiece affects the dynamic characteristics of the cutting system, so a solid plug was inserted into the chucking end of the thin wall workpiece and the chucking length (50 ram) and chucking force (6.9 kN per jaw) were kept constant, see Fig. 3(a). When the cutting force acts directly against a jaw as shown in Fig. 3(a), the relative stiffness will be the smallest, and the following analysis is produced for this loading condition.

In Fig. 3(a), ei, el and e2 are, respectively, the deflections of points a and b in direction y and the deflection of point b in direction z as the static force P is applied, although the value of ei can be obtained by measuring with a dial gauge and by numerical calculation. The values of el and e 2 must be gained by numerical calculation. The thin wall cylindrical workpiece vibrates in both the beam mode and the shell mode and in this study, e~ can be considered as the deflection for beam mode (m = 1) and e 2 can be considered as the deflection for shell mode (m = 1, n = 2). The static stiffness coefficient K is equal to P/ei. The stiffness coefficient of beam mode K~ is defined as P/e~ and the stiffness coefficient of shell mode K12 is defined as P/e2. The static stiffness coefficient K is determined by all the stiffness coefficients of beam modes and shell modes. When the stiffness coefficients of higher modes are much larger than the values of K1 and K12 , the static stiffness coefficient K can be approximately

1 1 1 determined from K1 and K12, namely K - K1 ÷ K12 "

The numerical calculation of the deflections was carried out by the finite element program (NISA II). The workpiece was divided into 240 3-D shell elements, each element had four nodes and each node had six degrees of freedom (three translations and three rotations). The boundary condition at the chucking jaw position was con- sidered as a fixed end.

The modal natural frequency and damping ratio can be obtained from the mentioned numerical method and also by hammer testing. Figure 3(b) shows the schematic diagram

HTH 34:4-D

492 J . Y . CHANG et al.

~ L=lOOmm ---~

Xi

Diel gouge 1

Load cell

e2

L~ Lo~cl ceLl

( o ) Set-up for measuring the stiffness

coefficients of workpiece

Acc~rome~cer

Hemmer kl±

AnotyzerJ

( b ) Hammer testing set -up

FIG. 3. The set-up for measuring the dynamic characteristics of the workpiece.

of hammer testing. Here the hammering direction is against the jaw and assists with the measurement of the modal stiffness coefficient. The modal natural frequency and damping ratio are obtained from averaging ten sets of data.

In the following section the overhang length is set at L = 100 mm and the effect of Di/h on the dynamic characteristics is discussed in detail.

2.2. The effect of Di/h on dynamic characteristics In Fig. 3(a), the value of the static stiffness coefficient is smallest at the free end.

Hence, the effect of Di/h on the value of modal stiffness coefficients and the static stiffness coefficient was first investigated for the free end. To obtain an extensive range of values of Di/h, the inner diameter Di was set at 30.5 mm, and five values of the wall thickness h were used (h = 3.0, 2.5, 2.0, 1.5 and 1.0 mm).

In Fig. 4(a), the abscissa represents the value of Di/h and the ordinate represents the modal stiffness coefficients and static stiffness coefficient. Symbol + represents the measured value of K and symbols II, • and • represent the computed value of K, K~ and K12, respectively. The measured and computed value of K are very close to each other which shows the accuracy of the numerical calculation.

Chatter Characteristics of the Thin Wall Cylindrical Workpiece 493

_- 2t~,

E x i

K 1 ÷ : measured

"L 1

J 24 \\

t/3, 5 I0 15 ~ ~ !~ ]5

(o) Di/h

o Z

\ \ \

N

i i l l l

(b) Di/h

r~ ~O

. t ~O

o

E o

f ~

°f F

0.~!

amL 512

i

I i i J t i

(c) Di/h

(D~=30.5mm, L = 100mm, F16.4. The effect of Di/h on workpiece dynamic characteristics Xi = 100 mm).

When D#h = 10.17 (namely, h = 3.0 mm), the value of K12 is much larger than that of K1 and K, therefore, the thin wall cylindrical workpiece has beam characteristics. When the wall thickness decreases, namely as the value of D#h increases, K1 decreases and K~2 decreases much more rapidly to reduce the value of K, and hence the shell mode gradually affects the workpiece characteristics. When Di/h = 30.5 (namely, h = 1.0 mm), the values of K1 and K12 are very close, and, therefore, the workpiece will have the vibration characteristics of both beam and shell.

Figure 4(b) shows the result of the hammer test, where f~ indicates the natural frequency of beam mode (m = 1) and f12 indicates the natural frequency of shell mode (m = 1, n = 2). The value of f12 decreases with an increase of Di/h. This is caused by the sudden decrease of stiffness coefficient of shell mode Kt2. The value of f~ changes only slightly at the same time.

In Fig. 4(c), the damping ratio 812 (shell mode) is smaller than that of ~l (beam mode).

From the above analyses it will be seen that the value of stiffness coefficient of shell mode K12 decreases rapidly initially with an increase of Di/h. The natural frequency of shell mode also decreases and the damping ratio of shell mode is smaller than that of beam mode. This means that the dynamic compliance of shell mode increases and the occurrence of chatter vibration of shell mode is more likely as the value of Di/h increases.

2.3. The effect of varying thickness on the modal stiffness coefficients, the static stiffness coefficient and modal natural frequency

Figure 4 shows that the value of Di/h will dominate the dynamic characteristics of the workpiece when the load acts on the free end of a thin wall workpiece. But in the actual cutting process, the cutting point and the wall thickness will be changing continu- ously. So it is necessary to investigate the effect of varying wall thickness and cutting force loading point on the modal stiffness coefficients and static stiffness coefficient in the cutting process.

As mentioned previously, when Di/h = 10.17 the workpiece will resemble a beam, and at D#h = 30.5 the workpiece will possess the characteristics of a beam and a shell simultaneously. These two cases are considered in investigating the effects of a changing loading point and varying wall thickness. Consider the finishing process and let the cutting depth be the recommended value of a = 0.5 ram. The values of K, K1 and K12 at the cutting point will be obtained from the numerical computation. The results are shown in Fig. 5(a) and (b).

In Fig. 5(a) and (b) the ordinate axis shows the log scale of the model stiffness coefficients and the static stiffness coefficient while the abscissa axis Xi is the distance

494 J .Y. CHANG et al.

E looo

z

lOO

10

q~ ¢-

~ .ql..=- L=IO0 mm ,,.]

- , - , • , . ,

%N K12 ,

I I i I I i I i i I

20 40 60 BO 1 O0 20 40 60 80 100

X i mm X ; mm

( o ) D i / h = 1 0 . 1 7 ( b ) O i / h = 50.5

Fro. 5. The effect of varying wall thickness and cutting force loading point on stiffnesses K, K~, and K~2 (computed results).

between the cutting force loading point and the chuck. Comparing Fig. 5(a) with (b) it will be seen that K12 reduces greatly as the value of Di/h increases. During the cutting process K12 maintains a constant level when Xi-> 30 mm. Both KI and K increase continuously during the cutting process, while K, KI and K12 increase greatly when Xi < 30 mm. From the above results, it is seen that g l 2 is mainly affected by the wall thickness, that K1 is mainly influenced by XI, and that K is affected by the wall thickness and Xi. These changes in the stiffness coefficients will affect the dynamic characteristics of the workpiece system considerably during the machining process.

Due to the varying wall thickness during the cutting process, the modal natural frequency will also change. The computed results are shown in Fig. 6. At the start of

.=~

O"

ID

1 • ' I I j , , , ~

f12

/ Ol/n~lO,]7

O r ' ~ - 3 0 , b

Z . _ _ _ "

I

80 !00 J [ I i t i

20 ~0 60 ),'~ mm

flo. 6. The effect of varying wall thickness on natural frequencies (computed results; D~ = 30.5 mm, L = 100 mm).

Chatter Characteristics of the Thin Wall Cylindrical Workpiece 495

cutting, namely the uncut situation, the values of fl for the cases of Di/h = 10.17 and D#h = 30.5 are very close, and do not vary very much during the cutting process. In addition, f12 for these two cases of Di/h show a great decrease during the cutting process. From the above results, the effect of the varying wall thickness on the shell mode frequency during the cutting process can be appreciated.

3. THE CHATTER PHENOMENON OF THE THIN WALL CYLINDRICAL WORKPIECE

3.1. Experiment set-up and method

In order to discuss the relationship between the dynamic characteristics and the chatter phenomenon, a series of cutting tests was performed. The experiment set-up and measurement system are shown in Fig. 7. A righthand offset tool-holder was used. To improve the stiffness of the tool, the overhang length was limited and set as 28 ram. A replaceable carbide insert tool with side cutting edge angle 45 ° was used to produce the radial displacement.

Each edge of the throw-away tip was used only once to avoid the effect of tip wear on the cutting process. The cutting forces were detected by a piezoelectric type of dynamometer, and the frequency components of the cutting force signal were analysed by a FFT analyser.

The experimental results and the dynamic characteristics of workpiece discussed in Section 2 were combined to clarify the chatter characteristics of the thin wall cylindrical workpiece.

Engine Lathe Center- Dlst&nce,BOOmm Swing over BeQ:540mm

\ Three-Jaw '~ Scrott-Chuck - - /

I ~////~-

/ Dynamometer

Work Length : 150 mm Chucking Length , 50 mm

L

__-~---- 1

~ _ 1 Throw-Away Tip Feed Rate S=O.I mm/rev

Cha~ge I Cutting Speed V=125 m/rain AmpliFier Depth of" Cut a:O.5 mm

' r l ! Recorder _J

I

L ~requency ~ A~pi;tude Spectrum I Spectrum o? I]ynamlc Component |An.Lyzer j

FIG. 7. Experimental set-up for cutting test.

496 J . Y . CHANG et al.

TABLE 1. COMPARISON OF CHATTER PHENOMENA BETWEEN THIN WALL AND SOLID CYLINDRICAL WORKPIECE HAVING THE SAME MOMENT OF INERTIA I (WORKPIECE MATERIAL IS $ 4 5 C )

D i m e n s i o n s o f C u t t i n g c o n d i t i o n C u t t i n g d e p t h S t ab l e o r C h a t t e r v i b r a t i o n C h a t t e r f r e q u e n c y w o r k p i e c e u n s t a b l e m o d e

D~ = 3 0 . 5 m m S = 0 . 1 m m / r e v a = 0 . 8 m m u n s t a b l e m = 1, n = 2 fc = 6.7--->5.8 k H z h = 3 .0 m m V = 125 m / m i n L = 100 m m

Do = 30 .9 m m u n s t a b l e m = 1 L = 100 m m

S = 0.1 m m / r e v a = 8 .0 m m V = 125 m / m i n

fc = 1.0---~1.2 k H z

3.2. The general characteristics of chatter of thin wall cylindrical workpiece From Refs [3-7] it is known that the larger the value of the static stiffness coefficient,

the larger the cutting depth for stable cutting can be. Also, from beam theory [17], it is clear that the static stiffness coefficient is a function of the moment of inertia (I) of its cross section. In order to investigate the chatter phenomenon of the thin wall workpiece, the chatter phenomenon of a solid workpiece and a thin wall workpiece with the same inertia moment (I) must be compared. The workpiece dimensions and cutting conditions for this investigation together with the experimental results are shown in Table 1.

In these experiments the cutting starts from the free end. When the chatter occurs it will begin from the free end, and it will continue for a period of time till the dynamic stiffness increases to a higher level. The chatter phenomenon was observed during this period.

Table 1 shows the shell mode (m = 1, n --- 2) of the thin wall workpiece was excited. The chatter frequency fc decreased from 6.7 to 5.8 kHz. The beam mode (fundamental mode, m = 1) of the solid workpiece was excited for the cutting depth a = 8.0 mm, and the chatter frequency fc increased from 1.0 to 1.2 kHz. In order to investigate this difference, the dynamic compliance of each workpiece was measured by the hammer test and the results plotted by a curve fitting technology and shown in Fig. 8.

Figure 8 indicates that in the case of the thin wall cylindrical workpiece before cutting, the dynamic compliance of the second vibration mode (m = 1, n = 2) was larger than that of the first vibration mode (m = 1). It was for this reason that the

Z

E = 10 ~

T_~-

i t t ~

o E I0" o

.o E 10-' o

ld

- - : s ta r t o f cut t ing

. . . . . . : end o f cut t ing

/ , /

Thin

. . . . . . . . l . . . . . . . .

i . . . . . . i . . . . . . . . i . . . . . . . .

F r e q u e n c y f Hz

FIG. 8. T h e v a r i a t i o n o f d y n a m i c c o m p l i a n c e d u r i n g the c u t t i n g p rocess .

Chatter Characteristics of the Thin Wall Cylindrical Workpiece 497

5

4

3

2

f l 2

f12

S = 0 . 1 m m / r e v

V = 1 2 5 m / m i n

a : O . S m m

\

/ Xx t~k" ~ ~ . . ~ J

• : D i = .30.5 m m

. : D i = 5 9 . 5 m m

• : Di = 7.5.5 mnn

L = 1 0 0 m m

f23

f13

f12 . . . . . . . --------:-Z

, I r I , I L i , ~ , I , 1 , 10 20 30 40 50 60 70 80

D i / h

FIG. 9. The effect of D/h on the chatter frequency when cutting the thin wall cylindrical workpiece.

damping ratio of the second mode (shell mode) was lower than that of the first mode (beam mode) and during the cutting process, the stiffness coefficient of the shell mode decreased. This caused the shell mode to be more easily excited.

The first vibration mode (fundamental mode, m = 1) of the solid workpiece was easily excited because the dynamic compliance of the first vibration mode was larger than that of the other modes. The solid workpiece was a Timoshenko beam, and since during the cutting process its mass will be decreased, the natural frequency will increase during the cutting process.

3.3. The effect of Di/h on the chatter vibration of the thin wall cylindrical workpiece In this section, in order to investigate the effect of Di/h on the chatter vibration of

the thin wall cylindrical workpiece, the machining of workpieces with a more extensive range of Oi/h was carried out. Three values of inner diameter Di (Di = 30.5, 39.5 and 73.5 mm) and five values of wall thickness h (h = 3.0, 2.5, 2.0, 1,5, and 1.0 mm) were used. The cutting conditions and the result are shown in Fig. 9, where the ordinate axis shows the frequency of chatter fc, and the abscissa axis shows the value of D#h. The symbols, A, x, and, O, represent the three levels of diameter D~ = 30.5, 39.5 and 73.5 mm, respectively. The dash line shows the natural frequency obtained by hammer testing. The vertical solid line shows the range of frequencies descends during the cutting process.

Figure 9 shows that the chatter phenomenon is determined by the value of Di/h and always occurs in one of its natural vibration modes. The chatter frequency of the shell mode decreases as Di/h increases. For any fixed ratio of Di/h the chatter frequency gradually decreases during that cutting process. Because the dynamic compliance of the shell mode increases as Di/h increases, the vibration mode of a higher frequency is often excited in the cutting process when DJh exceeds a particular value.

4. CONCLUSIONS

From the above numerical and experimental analyses on the chatter characteristics of the thin wall cylindrical workpiece, the following conclusions can be drawn:

(1) The chatter phenomenon is mainly decided by the value of Di/h and always occurs in one of its natural vibration modes.

(2) Even if a thin wall and a solid cylindrical workpiece have the same value of

498 J.Y. CHANG et al.

m o m e n t o f inertia, there is a great difference between the cutting depths for stable cutting for the two cases.

(3) Dur ing the cutt ing process the stiffness coefficient of the shell mode becomes smaller, and the chat ter f requency decreases gradually.

(4) W h e n the D i / h ratio increases, the stiffness coefficient of the shell mode decreases and the damping ratio of the shell mode becomes very small. This causes the dynamic compl iance o f shell mode to increase quickly, so the shell mode of the thin wall cylindrical workpiece is easily excited.

(5) Since the value of dynamic compl iance changes during one cutting process, the chat ter vibrat ion mode can jump f rom a lower modal natural f requency to a higher one.

Acknowledgements--The authors would like to acknowledge the financial support from the National Science Council of the Republic of China under Grant No. NSC-80-0422-EO36-03, and the Tatung Company.

REFERENCES

[1] J. D. SMITH and S. A. TOaIAS, Int. J. MTDR 1, 283 (1961). [2] S. A. TOaIAS and W. FISHWICK, Trans. A S M E 1079 (1958). [3] H. E. MERRITr, Trans. A S M E 87, 447 (1965). [4] J. R. LEMON and P. C. AeKERMANN, Trans. A S M E 87, 471 (1965). [5] F. KOENIGSBERGER and J. TLUSTY, Machine Tool Structure, Vol. 1. Pergamon Press, London (1971). [6] M. RAHMAN and Y. ITO, Int. J. MTDR 21(1), 1 (1981). [7] M. RAHMAN and Y. ITO, J. Sound Vibr. 102(4), 515 (1985). [8] G. J. LA,, Y. SAITO and Y. ITo, NSC-DFG JOINT Symposium, p. 243 (1990). [9] L. MEIROVITCH, Analytical Methods in Vibrations. Macmillan, Collier-Macmillan, London (1967).

[10] W. LEISSA, Vibration o f Shells. NASA SP-288 (1973). [11] K. FORSBERG, NASA CR-613 (1966). [12] H. KRAUS, Thin Elastic Shells. John Wiley, New York (1967). [13] V. I. WEINGARTEN, J. A I A A 2(4), 717 (1964). [14] C. B. SHARMA and D. J. JOHNS, J. Sound Vibr. 14(4), 459 (1971). [15] V. RAMAMUTRI and J. PATrABIRAMAN, J. Sound Vibr. 48(1), 137 (1976). [16] C. H. J. Fox and D. J. W. HARDIE, J Sound Vibr. 101(4), 495 (1985). [17] S. P. TIMOSHENKO and J. M. GERE, Mechanics o f Materials, 3rd edition. PWS-Kent, Boston (1972).