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CHATTER STABILITY OF A SLENDER CUTTING TOOL IN TURNING WITH TOOL WEAR EFFECT RICHARD Y. CHIOU†§ and STEVEN Y. LIANG‡ (Received 30 June 1997)

Abstract—An analysis of the chatter behavior for a slender cutting tool in turning in the presence of wear flat on the tool flank is presented in this research. The mechanism of a self-excited vibration development process with tool wear effect is studied. The components contributing to the forcing function in the turning vibration dynamics are analyzed in the context of cutting force and contact force. A comparison of the chatter stability for a fresh cutting tool and a worn cutting tool is provided. Stability plots are presented to relate width of cut to cutting velocity in the determination of chatter stability. Machining experiments at various conditions were conducted to identify the characteristic parameters involved in the vibration system and to identify the analytical stability limits. The theoretical result of chatter stability agrees qualitatively with the experimental result concerning the development of chatter stability model with tool wear effect. 1998 Elsevier Science Ltd. All rights reserved

NOMENCLATURE AS BS Ct c ⌬T F Fx fq fu Hf Hr I Kf Ksp kc kfd km lc lw m mf r sl Tp t t0 t1 U V v w woc n ⍀ x

a coefficient in the equation of chatter frequency a coefficient in the equation of chatter frequency a cutting pressure constant structural damping coefficient of the workpiece vibration system time interval cutting force perpendicular to the primary cutting edge on x-z plane contact force normal to the primary cutting edge oscillation frequency of tool contact force normal to the cutting edge without considering tool retrieval ratio of penetration rate contact to the directional stiffness stiffness ratio integer number penetration rate constant specific contact force static directional cutting stiffness 1 2 w H , a parameter 2 n f structural stiffness of the workpiece vibration system clearance length tool wear length equivalent mass of the vibration system kfd, a parameter radius of the workpiece slope of the cutter trajectory on the workpiece surface workpiece periodic of the workpiece time the instant when the workpiece begins to move down off the cutting tool the instant when the workpiece begins to move up against the cutting tool amplitude of the chatter vibration total volume of displaced material at the wear flat contact cutting speed chatter frequency, 1/Tp width of cut natural frequency rotational speed of the workpiece current surface displacement of wave modulation

†Assistant Professor, Mechanical Engineering Department, Temple University, Philadelphia, PA 19122, U.S.A. ‡Associate Professor, The Georgia W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. §Author to whom correspondence should be addressed. 315

316 x· ⌿(x) ␥c

R. Y. Chiou and S. Y. Liang current surface velocity of wave modulation variation of workpiece displacement perpendicular to the cutting edge clearance angle damping ratio of the workpiece, √km/m

1. INTRODUCTION

The problem investigated in this paper involves the determination of the chatter stability model for a slender cutting tool in the presence of cutter flank wear in machining operation. The phenomenon of machine tool chatter has been a matter of concern to industry because of its detrimental effect on part quality, cutter life, and machine utilization. An analytical understanding of the chatter behavior can extend experimental knowledge, which assists in the design of cutters as well as in the planning of cutting processes. Several studies of chatter vibration in machining have been documented in the literature, for example Refs [1–4]. This study clarifies the basic mechanism of self-excited vibration and describes the critical limit of the chatter. It shows that the criteria for chatter stability could be reduced to the context of cutting stiffness, the dynamic characteristics of the structure around the cutting point, and the regenerative effect. In the area of chatter vibration in relation to tool wear, the general characteristics of chatter vibration occurring with worn turning tools were experimentally investigated [1, 5–7]. The cutting process in the presence of flank wear was found to consist of pure cutting by the leading edge, shearing and compression by the trailing, and compression by the clearance face. The resulting surface contour is greatly influenced by cutting velocity, since it depends on the ratio of the wear flat length to the surface waviness length. Since the latter is proportional to the average cutting the trailing edge of the flat cutter will remove or deform proportionately more at a lower cutting velocity, thereby varying the magnitude of contact force and the vibration energy. Several important facts regarding the tool wear effect on cutting tool chatter vibration have been reported [1–3]. They indicated that the frictional force acting when the flank surface of a tool contacts the workpiece during chatter vibration plays an essential role in the supply of vibratory energy. The turning tools used from above authors are regarded as one or two degrees of freedom systems which are able to deflect in the main cutting force direction or in both directions. As the tool wear flat of cutting tool increases, the chatter stability decreases. However, some other points must be investigated with respect to this wear mechanism and the characteristics of cutting tool chatter vibration. In the present study, the chatter model which plays a role in the absorbing of energy is somewhat different from the above mentioned chatter models. Tlusty [9] pointed out that the wear flat on the tool flank plays an important role of positive damping in the occurrence of the chatter vibration. This is shown by the strong effect of tool wear on the positive imaginary inner direct coefficient of dynamic cutting forces. From another perspective, the cutting force was shown to depend on the velocity of tool penetration of the workpiece, and the damping force varies with the cutting velocity due to the elastoplasticity of the workpiece in contact with the wear flat. As chatter occurs greater tool wear manifests itself as increased damping in the cutting process resulting from the interference between the tool wear flat and the waviness on the workpiece [10]. It was also shown that the magnitude of the contact forces normal to the contact surface between the tool nose region and the work material depends on the volume of material displaced by the tool penetration [11, 12]. Therefore the chatter model in this study deals with the selfexcited vibration only due to deflection of the cutting tool in the feed direction. The aim of this investigation is to propose a method to determine the chatter stability very precisely through the observation of the chatter vibration of the cutting tool in the feed direction. In this paper, a dynamic turning model for cutting an ideal rigid workpiece with an flexible cutting tool is derived. This approach represents the chatter stability of the cutting tool as a function of both cutting force on the cutting edge and contact force on the flank wear flat. In the present investigation, the workpiece is cut so as to observe the mechanism

Chatter stability of a slender cutting tool in turning with tool wear effect

317

of the cutting tool chatter stability corresponding to the continuous variation of width of cut and cutting speed. A specially designed flexible cutting tool was made to vibrate in the feed direction only and to investigate the cutting tool vibration system. This study allows more insight into the impact of flank wear on the cutting system, including chatter stability, cutting tool stiffness, etc. The validity of the analytical results is examined through experimental study in the identification of characteristic parameters and the estimation of stability limits. 2. SLENDER WORN TOOL DYNAMICS

To gain fundamental understanding of the fairly complicated dynamics of the cutting process, models consisting of vibration motion with a single degree of freedom are often discussed, such as in [1, 9]. This analysis focuses primarily on the horizontal chatter motion of a slender cutting tool which can only vibrate in feed direction. Fig. 1 illustrates such a system in which a flexible tool, vibrating parallel to the feed and perpendicular to the cutting velocity direction, has a layer turned from the surface of a workpiece. Following from the consideration of a cutting force on the x–z plane in proportion to the chip thickness [13, 14], the equation of motion in the feed direction as shown in Fig. 1 without tool wear can be given as m x¨ + cx¨ + kmx = kc(x)

(1)

and (x) is the variation of chip thickness in the direction normal to the cutting edge,

冉 冉 冊冊

(x) = [x t ⫺

1 ⍀

⫺ x(t)]

(2)

Substituting Eqn (2) into Eqn (1) and repeating the expressions

n =

冪 m , = 2√k m , H = k km

c

kc

r

m

m

yields the following

Fig. 1. Model scheme of the vibration system and deformation zone in the chip formation process.

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冉 冉 冊冊

1 x¨(t) + 2wnx·(t) + 2nx(t) = 2nHr[x t ⫺ ⍀

⫺ x(t)]

(3)

With the existence of flank wear, the dynamic cutting force is affected by the actual contact between the workpiece and tool nose region, including its adjacent flank face [1, 2, 4, 7]. To determine the variation of dynamic cutting force induced by such a contact at the wear flat, the behavior of the workpiece material in the vicinity of the contact area has to be understood. Because of the existence of the flank wear, the material approaching tool cutting edges may be deformed in two ways. In the upper portion, the material is deformed and removed as a part of the chip. In the lower portion, the material cannot move upward to become part of the chip; instead it is extruded and pressed under the tool [4, 11]. This extruded material flows under the flank wear region and eventually departs from under the tool flank face. During this extrusion process, the material is displaced downward by different distances, depending upon the instantaneous relative position of the workpiece surface to the cutter. The relative position is a function of not only the cutting velocity tangential to the work surface but also the ‘plowing’ velocity perpendicular to the surface resulting from the cutting tool compliance. Therefore this volume of material displaced at any instant of time thus depends upon both the cutting speed and the relative tool flank motion. This material must be restrained by its surrounding material [10], and an elastic-stress field is established under the tool flank face as shown in Fig. 2. The stress field is in equilibrium with a normal force exerted on the contact surface where the separation of material around the tool flank face occurs. The normal force is referred to as the contact force, and its direction varies in accordance with both the cutting velocity and the cutting tool vibration. The effect of the contact force is to change the cutting force on the x–z plane, thereby modifying the forcing function in the equation of motion (1). The cutting system model as shown in Fig. 1 in the presence of tool wear flat can be represented as: mx¨ + cx· + kmx = kc(x) + Fx

(4)

where Fx is the contact force in x direction. It has been proposed that this force is in proportion to the total volume of displaced material V, i.e. [11] Fx = KspV

(5)

Fig. 2. Deformation zone in the chip formation process.

Chatter stability of a slender cutting tool in turning with tool wear effect

319

In estimating the displaced volume, this study uses a linearization scheme to approximate the cases of small inner and outer modulations with vibration amplitudes. Fig. 3 shows such a case, in which the cut surface can be treated straight in the small displaced triangular area ABC under the wear flat. For a chatter vibration with an amplitude of U at a frequency of , the displacement of the cutting tool x(t) perpendicular to the primary cutting edge at time t relative to cutter can be described as x(t) = U sinwt

(6)

In this expression the clearance angle of the cutting tool is assumed to be large enough to avoid interference between the workpiece and the clearance face beyond wear flat. The trajectory of the cutter tip, as it slides across the machined surface, begins at point C and ends at point A. The slope of this trajectory between C and A is sl =

⫺ x· sin v

(7)

The instantaneously displaced area ABC can then be expressed as: area(ABC) =

1 1 [(lw + lc)lctan␥ ⫺ l2c tan␥c] = l2w 2 2

sl

冉 冊

sl 1⫺ tan␥c

(8)

Therefore the displaced volume then can be represented as V=

1 w l2 2 oc w

sl

冉 冊

sl 1⫺ tan␥c

(9)

For a small amplitude vibration, as assumed in this analysis, the displaced volume can be simplified as V=

1 ⫺ wocl2wx· wocl2wsl = 2 2v

(10)

The contact force fx is thus f x = kspV = ⫺ kfx· where kf =

wocl2wksp 2v

Fig. 3. Schematic diagram of the displaced volume underneath the tool.

(11)

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The general characteristics of the contact force formulated in Eqn (11) agree with those proved by Wallace and Andrew [15]. In their work, a series of experiments in wave cutting was carried out with workpiece surface wavelength being the only geometrical variable. The amplitude of the contact force due to flank contact was found to be inversely proportional to the wavelength of the cutting wave. This observation matches with the expression for the contact force in this analysis, since the amplitude of the forces is proportional to the cut surface slope resulting from the wear flat. Wear on the tool flank brings about a zero or sometimes negative clearance angle depending on the relative tool motion with respect to the workpiece. It is understood from [8] that the zero clearance angle enlarges the positive direct complex inner cutting coefficients. Expression (11) also supports this argument in that the contact force is negatively proportional to the vibration velocity; therefore the wear flat creates a positive damping effect in the vibration system. The dynamic relationship between the contact force and the tool-work relative displacement is complicated by the directionality of the displacement as shown in Fig. 3. The contact force is generated as the cutting tool moves down against the workpiece, and it becomes zero as the cutting tool moves up off the workpiece. Since the contact force is always compressive and assumes only non-negative values, the beginning half of the upward motion and the finishing half of the downward motion are associated with a zero contact force. This aspect is graphically illustrated in Fig. 4 where x = 0 as t = 0 is assumed. Modification of Eqn (11) results in the following Fx(t) =

再

fx(t) = ⫺ kf x· t1 + nTp ⱕ t ⱕ t0 + nTp n = 0,1,2,3,$ 0

(12)

otherwise 3. CHATTER STABILITY ANALYSIS

The chatter stability analysis herein is based on the study of characteristic roots in the Laplace domain. Denote the transforms of fx(t) and Fx(t) as fx(s) and Fx(s), respectively, Fx(s) = fx(s)

e ⫺ st1 ⫺ e ⫺ st0

(13)

1 ⫺ e ⫺ sTp

Since fx(s) = -kf s x(s), therefore Laplace transform of the contact force is

Fx(s) = ⫺ kfs

1 ⫺ e⫺s

3

e ⫺ s 2w ⫺ e ⫺ s 2w 2 w

x(s) = ⫺ kfs

e ⫺ s 2w

1 + e⫺sw

x(s)

Based on a first order approximation of,

Fig. 4. The resultant contact force under free vibration.

(14)

Chatter stability of a slender cutting tool in turning with tool wear effect

e ⫺ sTp ⬵

321

1 , 1 + sTp

冉

Fx(s) ⬵ ⫺ kfs 1 ⫺

冊

kf 2 s x(s) ⬵ ⫺ kfs x(s) + s x(s) 4w 4 w

(15)

The Laplace transform of the equations of motion of the turning process incorporating the effects of tool wear becomes (s2 + 2wns + 2n) = 2n Hr[e ⫺ TpS ⫺ 1] ⫺ kfds +

mf 2 s w

(16)

冊

(17)

where kfd = w2nHf, mf =

w2nHf kf kf = H , and Hf = 4 km kc r

Substituting s = jw into Eqn (16) gives

冉

w2n ⫺ w2 + mfw + j(2wn + kfd)w = w2nHr cos

w w ⫺ 1 ⫺ jw2nHr sin ⍀ ⍀

Equating the real part and the imaginary part of Eqn (17), the characteristic roots of the equation of motion can be solved by w4 ⫺ 2mfw3 + (A2s B2s ⫺ 2As + m2f

(18)

⫺ 2w2n)w2 + (2Asmf + 2mfw2n)w + (2Asw2n + w4n) = 0 where, As = w2nHr, and Bs =

(2wn + kfd) w2nHr 4. EXPERIMENTAL INVESTIGATION

A series of experiments have been conducted to further examine the validity of the chatter stability model. The schematic diagram of the set-up is shown in Fig. 5. A specifi-

Fig. 5. Experimental set-up.

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R. Y. Chiou and S. Y. Liang

cally designed cutting tool shank mechanically locked in front of a tool post on a conventional lathe was used to cut the rigidly mounted 6061-T6 aluminum workpiece of 50 mm (1.95 in.). The cross-sectional configuration of the tool shank is a rectangle with a height 31.75 mm (1.25 in.) and a width of 6.35 mm (0.25 in.). The turning tool used is regarded as one degree of freedom vibration system which is able to deflect only in the thrust cutting force direction. The vibration of the turning tool is detected by a accelerometer attached to the back of the shank. The acceleration signals were amplified by a charge amplifier (Kistler 5004) prior to being digitized with an emulated digital oscilloscope. The acceleration signals were used to observe the sudden change of the vibration amplitude for this chatter study. The cutting tool was high speed steel with a tool geometry of rake angle 7° and clearance angle 15°. The width of cut was increased until chatter vibration was observed. The characteristic parameters, namely the cutting stiffness kc, the structural stiffness km, and the natural frequency and damping ratio of the tool and tool holder system, involved in the dynamic cutting model were all determined experimentally in this study. Impact testing was performed to identify the structural response of the machine tool system. The natural frequency wn and damping ratio associated with the flexible cutting tool were found to be 403 Hz and 0.0567 respectively. The direct proportionality between steady state cutting force and uncut chip thickness is a basic approximation of the cutting stiffness made to the cutting process in this analysis. The experimentally measured thrust cutting force as a function of the feed, or uncut chip thickness, over a range of cutting velocity with different width of cut are shown in Fig. 6. The force versus feed curves were essentially linear for feeds from 0.0127 (0.0005 in./rev) to 0.05 mm/rev (0.002 in./rev) over the range of cutting velocity from 48 (157 fpm) to 139 mpm (454 fpm). Based on these cutting force data the value of kc can be approximated

Fig. 6. Machining force measurements for 6061-T6 aluminum workpiece.

Chatter stability of a slender cutting tool in turning with tool wear effect

kc = Ctwoc

323

(19)

where Ct = 206 N/mm2 (29,850 lb/in.2) To estimate km of the tool shank, an experiment was performed to exert static forces on the tool holder from the tool post. Set-up was designed for simultaneous measurements of the force by dynamometer and the displacement by dial gauge. The values of km were obtained by calculating the ratio of the force to the displacement measurement; it was estimated to be 492 N/mm (2,806 lb/in.). The specific contact force ksp, is defined as the normal contact force per unit volume of material displaced by the penetrating tool. This contact parameter is assumed to be independent of the tangential load. Since the volumetric strain produced by the tangential load is negligibly small compared with that induced by the normal load in cutting, the penetration process can be considered equivalent to an indentation process of an elastic work material by frictionless tool-shaped indenter. Although some theoretical work has been done on indentation process [11, 12], experimental results for tool indentation on an aluminum workpiece in the axial direction during turning has not been available. In this study, the experimental set-up as shown in Fig. 7 was used to determine ksp from the simultaneous measurements of normal contact force and displaced volume as the tool was pressed into a stationary workpiece. Fig. 8 shows experimental results of the specific

Fig. 7. Experimental set-up for the determination of specific cutting coefficient ksp. 120

contact force (Ibµ)

100

80

60 Ksp = 17,670 N/mm3 (6.5×107 Ib/in3)

40 89 N 20

3.3×10–3mm3) 0

0

20

40

60

80

100

120

140

160

180

200

displace volume 10–2(in2)

Fig. 8. Normal contact force measurements for 6061-T6 aluminum workpiece.

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R. Y. Chiou and S. Y. Liang

contact force ksp with different wear lands and displaced volumes. From the ratio of the force to displaced volume ksp is averaged and found to be 17,670 N/mm3 (6.5 × 107 lb/in.3) in this case. 5. CHATTER STABILITY NUMERICAL SIMULATION

To plot the test data as a function of kc/km (where km is the directional static structural stiffness), as the width of cut was varied, kc was expressed in terms of the width of cut. From the cutting force data for the 6061-T6 aluminum workpiece in Fig. 6, the value of kc at a feed of 0.001 in/rev is a function of width of cut. Based on these cutting force the value of kc can be approximated kc = Ct woc. Computing the ratio of kc/km for this series of tests as a function of width of cut yields kc 29850 = w = 10.6woc km 2806 oc

(20)

It is noted that the forgoing ratio is computed for a feed of 0.025 mm/rev (0.001 in./rev); however, owing to the linearity of the cutting data, this ratio is good for a wide range of feed. In the case of a sharp tool, the contact force between the workpiece and tool flank face is essentially negligible with Fx = 0. The stability limit can be derived from Eqns (16) and (17)). The critical condition and the region where the system is stable can be obtained as Hr ⱕ 2(1 + )

(21)

As the tool flank wears and the contact force has to be incorporated in consideration of cutting system dynamics, the boundary of stability can be solved from the same equations for a given wear flat lw. Fig. 8 shows the comparison of the stability boundary without flank wear agree with those proposed in [1, 16–18]. In their studies the minimum dynamic stiffness of the structure is related directly to stability criterion and is proposed as an index of chatter performance. 6. EXPERIMENTAL RESULTS

In the machining tests, the cutter was fed by the carriage of the lathe in the direction perpendicular to the spindle axis during cutting. The width of cut was increased until chatter occurred. The feed was 0.025 mm/rev (0.001 in./rev), and the cutting velocity ranged between 48 (157), 82 (268), 109 (357), 139 (454), and 170 mpm (560 fpm). Under these selected cutting conditions a continuous chip was observed without the presence of a built-up edge on the tool. Fig. 9 is the simulation of stability boundary comparison for a sharp tool and a worn tool of lw = 0.05 mm (0.002"). The asymptotic borderline of stability for all cutting speeds is shown in this Figure at width of cut = 0.56 mm (0.022"). According to theory, the region below the asymptotic borderline is absolute stable for all cutting speeds as shown in this Figure at width of cut = 0.56 mm (0.022"). In other words, the asymptotic borderline of stability is the principal borderline since it defines the maximum width of cut which results in stable operation at all speeds. Fig. 9 shows the higher stability accounting for the contact force in worn tools. Also shown in Fig. 10 is a curve which is drawn through the points of intersection of the stability lobes. Stability in the region bounded by these two curves is dependent upon cutting speed; while above the upper curve the system is absolutely unstable regardless of cutting speed. The circles represent the experimental unstable cuts and triangles represent the experimental stable cuts. The interface between those circles and triangles represents the experimental values of the critical width of cut (onset of chatter) for all cutting speeds. All stable as well as unstable cuts verify the theory. Chatter vibration is fundamentally caused by a lack of adequate dynamic stiffness associated with the machine structure. This behavior can be traced to the lack of damping

Chatter stability of a slender cutting tool in turning with tool wear effect

325

Fig. 9. Stability chart with a sharp tool (solid curve) and a worn tool of lw = 0.05 mm (0.002") (dotted curve).

Fig. 10. Comparison between theoretical prediction and tested stability limit for fresh tool cuts. Circles are for chatter and triangles are for stable cutting.

inherent in the structure. When the contact damping effect on the tool flank is considered the region of stability enlarges in comparison to that with a sharp tool. Fig. 11 shows the predicted and tested stability chart accounting for tool wear compared with the critical limit width of cut for a sharp tool at 0.56 mm (0.022"). It is evident that the chatter stability was increased as a result of flank wear. In other words, the stability against chatter improves as flank wear increases. This is in agreement with the results of Ref. [8], which argued that the damping coefficients of the imaginary part of static stiffness increase almost in direct proportion with flank wear and the positive damping effect and higher stability are associated with an increased wear flat.

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Fig. 11. Comparison between theoretical prediction and tested stability limit for worn tool cuts with tool flank wear of 0.05 mm (0.002"). Circles are for chatter and triangles are for stable cutting.

Reasonable agreement between the results was obtained at cutting velocities of 82, 109, 139, and 170 m min-1 (268, 357, 454 and 560 ft min-1). At velocity of 48 m min-1 (157 ft min-1) the prediction of stability was difficult in the vicinity of the narrow stability peaks. Sisson and Kegg [10] reported that the finite radius of the tool nose introduces a higher stability at low cutting speed, which may in part provide a basis for the deviation of experimental data form the theoretical at 48 m min-1 (157 ft min-1). 7. CONCLUSIONS

The chatter stability analysis in turning with this tool wear effect makes clear the behavior of the tool oscillation during chatter, and chatter stability can be explained qualitatively. It is based on the decomposition of the forcing function in dynamic machining systems in two contributing components: the cutting, and contact forces. The cutting force directly results from the chip load through the cutting stiffness. The contact force induced by the material displacement at the wear flat has been found to be related to both the cutting velocity and the plowing velocity of the tool at the workpiece surface in the case of small amplitude vibration. A comprehensive expression of the equation of motion for the dynamic cutting system incorporating the effects of these contributing forces is established. Machining experiments were conducted on a conventional lathe with the use of a specially designed flexible tool which can only vibrate parallel to the feed and perpendicular to the cutting velocity direction. The machine characteristics and static stiffness were identified and the chatter stability were observed in verification of the analytical solutions over a range of cutting velocities and width of cuts. Among these cutting conditions, flank wear has been shown to have a significant effect on the chatter stability. REFERENCES [1] Nathan, H., Cook, Self-excited vibrations in metal cutting. Trans. ASME J. Engng for Ind., 1959, 8, 183. [2] Merrit, H. E., Theory of self-excited machine-tool chatter: contribution to machine-tool chatter research1. Trans. ASME J. of Engng for Ind., 1965, B87(4), 447. [3] Marui, E., Ema, S. and Kato, S., Chatter vibration of lathe tools. part 1: general characteristics of chatter vibration. Trans. ASME J. of Engng for Ind., 1983, 105, 100. [4] Marui, E., Ema, S. and Kato, S., Chatter vibration of lathe tools. part 2: on the mechanism of exciting energy supply. Trans. ASME J. of Engng for Ind., 1983, 105, 107. [5] Arnold, R. N., The mechanism of tool vibration in the cutting of steel. Inst. Mech. Engng J. Proc., 1946, 154, 261.

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[6] Rao, S. B., Tool wear monitoring through the dynamics of stable turning. Trans. ASME J. Engng Ind., 1986, 108, 183. [7] Chiou, Y. S., Chung, E. S. and Liang, S. Y., Analysis of tool wear effect on chatter stability in turning. Int. J. Mech. Sci., 1995, 37(4), 391. [8] Tlusty, J., Analysis of the state of research in cutting dynamics. CIRP Annuals, 1978, 27(2), 583. [9] K Yoshitaka, K., Osamu and S. Hisayoshi, Behavior of self-excited chatter due to multiple regenerative effect. Trans. ASME J. of Engng for Ind., 1981, 103, 324. [10] Sisson, T. R. and Kegg, R. L., An explanation of low speed chatter effects. Trans ASME, 1969, 91(4), 951. [11] Shaw, M. C. and DeSalvo, G. J., On the plastic flow beneath a blunt axis symmetric indenter. Trans ASME J. of Engng for Ind., 1970, 480. [12] Wu, D. W., Application of a comprehensive dynamic cutting force model to orthogonal wave-generating processes. Int. J. Mech. Sci., 1988, 30(8), 581. [13] Koenigsberger, F. and Sabberwal, A. J., An investigation into the cutting force pulsations during milling operations. Int. J. Machine Tool Design Res., 1961, 1, 15. [14] Tlusty, J. and MacNeil, P., Dynamics of cutting forces in end milling. CIRP Annals, 1975, 24, 21. [15] W Wallace, P. and Andrew, C., Machining forces: some effects of tool vibration. J. Mech. Engng Sci., 1965, 7(2), 152. [16] Tlusty, J. and MacNeil, P., Dynamics of cutting force in end milling. CIRP Annals, 1975, 24, 21. [17] Lemon, J. R. and Ackermann, P. G., Theory of self-excited machine-tool chatter: contribution to machinetool chatter research-4. Trans. ASME J. of Engng for Ind., 1965, B87(4), 471. [18] Long, G. W. and Lemon, J. R., Theory of self-excited machine-tool chatter: contribution to machine-tool chatter research-2. Trans. ASME J. of Engng for Ind., 1965, B87(4), 455.