A study of bifurcation behaviour in oblique turning operation

ARTICLE IN PRESS International Journal of Machine Tools & Manufacture 49 (2009) 1042–1047

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International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool

Short Communication

A study of bifurcation behaviour in oblique turning operation J. Srinivas a,, K. Rama Kotaiah b a b

Department of Mechanical Engineering, Chaitanya Engineering College, 1-11-27, Laxminagar, Visakhapatnam 530 017, India Industrial Production Department, KL University, Vaddeswaram, Vijayawada 520 132, India

a r t i c l e in f o

a b s t r a c t

Article history: Received 4 March 2009 Received in revised form 8 June 2009 Accepted 9 June 2009 Available online 18 June 2009

This paper presents results of linear stability analysis in turning using nonlinear force–feed dynamic model by considering three-dimensional cutting tool geometry. The modified analytical equations for cutting insert with three-dimensional tool geometry are derived by including relative displacements of the tool with respect to a two-degree of freedom work-piece model. The critical stability points obtained as a function of feed are validated against time-domain solutions. Simulation results are shown in the form of limit cycles and bifurcation diagram. Influence of static-feed term on cutting dynamics over a cantilever work-piece is illustrated along with the experimental results. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Cutting dynamics Tool geometry Linear stability Force–feed Transfer function

1. Introduction Prediction of unstable operating conditions is of vital importance in machining practice. The regenerative instability due to self-excited vibrations (known as chatter) has been studied [1–3] for a long time. Often, stability lobe diagrams are developed to select the conservative process parameters. Especially, the analytical prediction of stability in turning and boring is wellestablished with flexible cutting tool models. In many cases [4–6], two-degrees of freedom models of cutting tool and work-piece are employed to realize the practical machining conditions. The more general model requires inclusion of oblique cutting conditions and multidimensional dynamics for obtaining accuracy. Recently some authors [7,8] illustrated a multi-dimensional model that accounts for a relative tool motion in both feed and radial directions. However, these models have not included the effect of static feed on regeneration phenomenon. On the other hand, the effect of force–feed nonlinearity in turning chatter had been studied independently [9,10] with one-dimensional models. In few works [11,12], two-degree of freedom oscillator model of cutting tool was selected to include the nonlinear force–feed. But these nonlinear force–feed models cannot be used in oblique cutting conditions. In the present paper analytical multidimensional stability equations are formulated by considering nonlinear force–feed effect in turning using three-dimensional tool-insert geometry. Feed and radial forces are included in the model appropriately and

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E-mail address: [email protected] (J. Srinivas). 0890-6955/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2009.06.004

dynamic equations are derived. The revised expression for stable depth of cut as a function of chatter frequency is obtained. Effect of static feed on regenerative stability is validated with timedomain solutions as well as using an experimental analysis.

2. Dynamic cutting force model The basic parameters in turning process are time varying chip thickness h(t), the depth of cut b and the cutting angles. The side cutting edge angle (c) and the inclination angle (i) are measured on the rake face of the tool. The relation between the forces and chip thickness is obtained by considering the coordinate systems x and y as shown in Fig. 1. Here the stiffness and damping coefficients of cutting tool along x and y directions are indicated by lower case letters kx, cx, ky and cy, while that of work-piece are denoted by upper case letters Kx, Cx, Ky and Cy, respectively. When the dynamic displacements in cutting directions are not affecting the dynamic chip thickness, the problem (reduced into 2D model) can be written in terms of modulated chip thickness h(t) defined according to: hðtÞ ¼ f cos c þ Dx cos c  Dy sin c ¼ hs þ Dx cos c  Dy sin c

(1)

with

Dx ¼ fxc ðtÞ  xc ðt  tÞg  fxw ðtÞ  xw ðt  tÞg

(2)

Dy ¼ fyc ðtÞ  yc ðt  tÞg  fyw ðtÞ  yw ðt  tÞg

(3)

ARTICLE IN PRESS J. Srinivas, K. Rama Kotaiah / International Journal of Machine Tools & Manufacture 49 (2009) 1042–1047

Cy

Ky

In terms of natural frequencies (ocx, ocy, owx, owy), damping ratios (xcx, xcy, xwx, xwy) and stiffness coefficients (kx, ky, Kx, Ky) of cutting tool and work-piece in x and y directions, they can be arranged as

x

Kx

y b

Ff ψ

Cx

Fr kx

2

1 6 60 6 6 60 4

h (t) cy

cx ky

0 Fig. 1. Model under consideration.

where f is the nominal feed per revolution, hs ¼ f cos c the static chip thickness, xc(t), yc(t), xw(t) and yw(t) are the cutter and workpiece displacements in the current revolution and xc(tt), yc(tt), xw(tt), yw(tt) are those in the previous revolution along x and y directions, respectively. Also t is the time delay required for one spindle revolution. The cutting forces on the face of the insert along x and y directions are F x ¼ F f cos c þ F r sin c

(4)

F y ¼ F f sin c þ F r cos c

(5)

here Fr and Ff are components of cutting forces in a direction perpendicular to the cutting plane, which can be written as nonlinear force–feed expressions Kf bfhðtÞga cos c

Kr bfhðtÞgb Fr ¼ cos c

Kf a a1 b½hs þ ahs ðDx cos c  Dy sin cÞ cos c

Kr b b1 b½hs þ bhs ðDx cos c  Dy sin cÞ Fr ¼ cos c

(7)

(8)

(9)

Using the Eqs. (4), (5), (8) and (9), the component forces Fx and Fy can be written in terms of static feed hs as: fFg ¼ b½AfDdg

0

1

0

0

1

0 2

0

9 38 0 > > x€ c > > > > > 7> > < y€ c > = 07 7 7 > 7 € x 0 5> w> > > > > > > : y€ > ; 1 w

2xcx ocx

0

0

2xcy ocy

0

0

0

2xwx owx

0

0

ocx 0

6 60 6 þ6 60 6 4 0

0

o

2xwy owy w 9 38 9 8 2 0 o F =k  > x x x > c > cx > > > > > > > > > > > 7> > > > > > > 2 > 0 7 7< yc = < ocy F y =ky = 7 ¼ . > 0 7 o2wx F x =K x > > xw > > > > 7> > > > > > > > 5> > > > > : o2 F y =K y > ; o2wy : yw ; > wy 0

0 2 cy

0

0

o2wx

0

0

9 38 x_ c > > > > > > > 7> < y_ c > = 7> 7 7 _ > 7> > xw > > 5> > > > : y_ > ;

(16)

Time-domain solution is obtained by solving these set of coupled delay-differential equations using revised Runge–Kutta’s scheme with initial conditions taken as zero.

(6)

where Kf, a and Kr, b are the empirically determined cutting coefficients in the feed and radial directions, respectively. Substituting h(t) and linearizing the forces about hs the following equations are obtained: Ff ¼

0

6 60 6 þ6 60 4 0 2 2

Table 1 Cutting and material data. Cutting tool

Work-piece

Natural frequency oncx ¼ oncy ¼ 1100 Hz Stiffness kx ¼ ky ¼ 1.2  107 N/m2 Damping ratio: xcx ¼ xcy ¼ 0.015 Cutting coefficients: Kf ¼ 800 MPa Kr ¼ 128 MPa Feed exponents: a ¼ b ¼ 0.75

Modulus of elasticity E ¼ 2.11011 N/m2 Density r ¼ 7800 kg/m3 Diameter d ¼ 39 mm Length L ¼ 75 mm

Side cutting edge angle c ¼ 101 Normal rake angle ¼ 51 Inclination angle ¼ 51 Nose radius ¼ 0.4 mm Working feed ¼ 0.01–0.6 mm

Average damping ratio: xwx ¼ xwy ¼ 0.025

(10)

where {Dd} ¼ [Dx Dy]T is dynamic displacement vector and [A] the directional coefficient matrix given by 9 " #8 a1  cos c sin c < K f ahs = 1 ½cos c  sin c (11) ½A ¼ cos c : K r bhb1 ; cos c sin c s 2.1. Equations of motion of the system If mc and mw are the masses of cutting tool and work-piece, respectively, the dynamic equations of motion of the cutting tool and work-piece can be written along x and y directions using freebody diagrams. That is (12) mc x€ c þ cx x_ c þ kx xc ¼ F x mc y€ c þ cy y_ c þ ky yc ¼ F y

(13)

mw x€ w þ C x x_ w þ K x xw ¼ F x

(14)

mw y€ w þ C x y_ w þ K x yw ¼ F y

(15)

0.18

f = 0.05mm f = 0.25mm f = 0.45mm

0.16

depth of cut (mm)

Ff ¼

1043

0.14 0.12 0.1 0.08 0.06 0.04 2200

2400

2600

2800 3000 speed (rpm)

Fig. 2. Stability lobe diagram.

3200

3400

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2.2. Linear Stability

This equation has non-trivial solution if and only if its determinant is zero, yielding

The stability states are obtained in frequency-domain using oriented transfer function approach. Taking Laplace transforms on both sides of Eq. (10) st

FðsÞ ¼ b½Að1  e

(17)

ÞDðsÞ

Considering the transfer function matrix [G(s)] of the system, Eq. (17) can be written as FðsÞ ¼ b½Að1  est Þ½GðsÞFðsÞ

(18)

detf½I  l½A½GðjoÞg ¼ 0

(20)

where the complex eigenvalue l ¼ bð1" ejot Þ (21) # Gxx ðjoÞ 0 and the Here the transfer matrix [G(jo)] ¼ 0 Gyy ðjoÞ eigenvalue is obtained as

l¼

 cos c ðp cos2 c  q sin c cos cÞGxx þ ðp sin2 c þ q sin c cos cÞGyy (22)

Substituting s ¼ jo and simplifying (19)

a1

where p ¼ K f ahs

1

0.4

0.5

0.2 dyc /dt

dxc /dt

f½I  b½Að1  ejot Þ½GðjoÞgFðjoÞ ¼ 0

0 -0.5

b1

and q ¼ K r bhs

0 -0.2

-1

-0.4 -1

0

1

-5

2

0

x 10-4

xc

5 yc

800

10 x 10-5

340

Fy (N)

Fx (N)

330 750

700

320 310 300 290

650 0.01 0.02 time (seconds)

0.03

0

4

1

2

0.5 dyc /dt

dxc /dt

0

0

0.03

0 -0.5

-2

-1

-4 0

2

4 xc

6

0

8 x 10-4

3800

1700

3750

1680

3700 3650

1

2 yc

Fy (N)

Fx (N)

0.01 0.02 time (seconds)

3 x 10-4

1660 1640 1620

3600

1600 0

0.01 0.02 time (seconds)

0.03

0

0.01 0.02 time (seconds)

0.03

Fig. 3. Limit cycle behaviour at a depth of cut b ¼ 1.6 mm. (a) Feed f ¼ 0.05 mm and (b) feed f ¼ 0.45 mm.

(23)

ARTICLE IN PRESS J. Srinivas, K. Rama Kotaiah / International Journal of Machine Tools & Manufacture 49 (2009) 1042–1047

Writing l as   l ¼ lR þ jlI ¼ bð1  ejot Þ ¼ bð1  cos otÞ þ jb sin ot

1045

This gives the critical values of t as     2 1 l t¼ nþ p  a tan I where n ¼ 0; 1; 2 . . . o lR 2

(24)

then

(26a)

The corresponding critical values of b is given by

p lI sin ot ot ot ¼ tan þ np  ¼ ¼ 1= tan . lR 1  cos ot 2 2 2

b¼

(25)

lR

(26b)

1  cos ot

Using Eqs. (26), the stability lobe diagram can be obtained for different feed values.

10

3. Results and discussion

f = 0.05mm f = 0.25mm f = 0.45mm

9

The cutting and material parameters used in the numerical simulations are given in Table 1 [13]. The axial (x) and bending (y) stiffness (Kx, Ky) and corresponding natural frequencies (owx, owy) of the work are obtained by considering it as a cantilever. The tool and work transfer function matrices are evaluated from the following relations: 3 2

8

amplitude

7 6 5

o2cx

4

0

6 6 kx ðs2 þ 2xcx ocx s þ o2cx Þ 6 Gc ðsÞ ¼ 6 6 4 0

3 2

2

1 0.5

1

1.5

2

2.5

depth of cut (mm)

0

Ky

ðs2

o2wy þ 2xwy owy s þ o2wy Þ

7 7 7 7 7 5

Fig. 2 shows the stability boundaries in the plane (b, 60/t) at three different feed rates f. By considering the exponents: a ¼ b ¼ 1, the present lobe diagram matches with that shown in literature [13]. It is seen from Fig. 2 that minimum depths of cut increases with feed at all spindle speeds. The feed effect is also verified using the time-domain analysis. The system states are shown as limit cycles by keeping time as implicit coordinate. Fig. 3 shows the limit cycle behaviour of cutting tool in x and y

Fig. 4. Part of the sub-critical bifurcations.

Table 2 The cutting conditions considered in experiments. Cutting speed (m/min) Depth of cut (mm) Feed (mm/rev)

3

o2wx

6 6 K x ðs2 þ 2xwx owx s þ o2wx Þ 6 Gw ðsÞ ¼ 6 6 4 0

0 0

o2cy ky ðs2 þ 2xcy ocy s þ o2cy Þ

7 7 7 7, 7 5

65.6, 140.3, 218.7 0.25, 0.75, 1.5 0.05, 0.1, 0.2

Experimental dynamic cutting forces at f = 0.2mm/rev, d = 1.5mm, N = 1200rpm FFT (Fx)

Fx (N)

-30 -40 -50 -60

10 5 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

100

200

300

400

500

600

0

100

200

300

400

500

600

0

100

200

300

400

500

600

FFT (Fy)

Fy (N)

60 40 20

10 5 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 FFT (Fz)

Fz (N)

90 80 70 60

15 10 5 0

0

0.5

1

1.5

2

2.5 time (s)

3

3.5

4

4.5

5

frequency (Hz)

Fig. 5. Experimental stable cutting condition (f ¼ 0.2 mm/rev, b ¼ 1.5 mm, N ¼ 1200 rpm). (a) Time domain results and (b) force spectra.

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speed = 218.7 m/min, f = 0.05 mm/rev, b = 1.5 mm FFT (Fx)

Fx (N)

-25 -30 -35 -40

10 5 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

200

300

400

500

600

700

200

300

400

500

600

700

200

300

400 500 frequency (Hz)

600

700

FFT (Fy)

Fy (N)

40 30 20

4 2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 FFT (Fz)

Fz (N)

50 40 30

10 5 0

20 0

0.5

1

1.5

2

2.5 3 time (s)

3.5

4

4.5

5

Fig. 6. Experimental unstable cutting condition (f ¼ 0.05 mm/rev, b ¼ 1.5 mm, N ¼ 1200 rpm). (a) Time domain results and (b) force spectra.

directions at two different feed values. The corresponding cutting forces are also shown. It is seen that at the same depth of cut as feed is reduced, the process becomes unstable. A sketch showing the amplitude of the limit cycles as a function of depth of cut at 1200 rpm is depicted in Fig. 4. At all feed rates, amplitude is physically unreachable, confirming the presence of unstable limit cycles. It is observed that an unstable limit cycle (periodic) coexists with the stable equilibrium (stationary cutting). On the other hand in the supercritical case, a stable limit cycle coexists with the unstable equilibrium. In normal turning models only sub-critical Hopf bifurcation occurs. This sub-critical nature is clearly related to the nonlinear dependence of the cutting force on the chip thickness. For large amplitude vibrations, the tool may lose contact with the work-piece. This results in a fold back of the unstable branch to a periodic (or quasi-periodic or chaotic) attractor at certain depth of cut. Machine tool chatter corresponds to this large amplitude attractor. The effect of feed on stability in oblique turning is verified with an experimental analysis on a 7.5 kW Namseon engine lathe. To minimize the tool wear effects, coated carbide insert is employed for cutting an AISI 1045 work-piece without tailstock support. Three components of dynamic cutting forces are recorded with a tool dynamometer (Kistler 9121) and associated 5070 multichannel charge amplifier connected to PC employing Kistler Dynoware force measurement software. Measurements were taken within the first 5 s of cut. The sampling rate used is 10 kHz. The considered cutting conditions are shown in Table 2. A combination of these variables evolved into 27 experiments. Figs. 5 and 6 illustrate examples of stable and unstable operations, respectively at 1200 rpm. In stable operation, there are no highfrequency variations in the force plots after 260 Hz. The FFT plots show significant energy at 260 Hz in Fx signal, 260 Hz and 530 Hz in Fy signal and 260 Hz in Fz signal. In unstable operation, highfrequency variations are evident in force plots. Frequency-domain plots show significant energy in all three forces at 451 Hz. Results of these experiments demonstrate that the feed affects the presence of chatter instability and at higher feeds, the instability is dramatically suppressed.

4. Conclusions Some analytical and experimental results of bifurcation analysis in linear stability of oblique turning operation have been presented. The analysis showed that the criticality of the Hopf bifurcations along the stability lobes depends on the feed rate. For smaller feed rates, the bifurcation is sub-critical. Here an attractor (periodic orbit) coexists with the stable stationary cutting state that may lead to chatter within the stability boundaries. For larger feed rates, it is observed that bifurcation becomes supercritical at certain higher spindle speeds and no attractor coexists with the stable stationary cutting state. Supercritical Hopf bifurcation is more favorable than sub-critical case because, the system cannot experience chatter within the linear stability boundaries. Even, the feed minimizes the cycle time, the machining forces grow-up and it cannot be used to suppress chatter conditions. Due to the inclusion of feed in chatter relations, it is possible to calibrate the force models at any feed as reported in literature. Acknowledgments Authors acknowledge Mr. M. Sekar, KNU, Daegu, S. Korea for helping in the necessary experimental works. Authors also thankful to Sri Chandra Sekhara Mahavidyalaya Deemed University, Kanchipuram. References [1] H.E. Merrit, Theory of self-excited machine tool chatter, Journal of Engineering for Industry—Transactions of the ASME 87 (1965) 447–454. [2] J. Tlusty, M. Polacek, The stability of machine tools against self-excited vibrations in machining, International Research in Production Engineering ASME (1963) 465–474. [3] S.A. Tobias, W. Fishwick, The chatter of lathe tools under orthogonal cutting conditions, Transactions of ASME 80 (1958) 1079–1088. [4] T. Kaneko, H. Sato, Y. Tani, M. Ohori, Self-excited chatter and its marks in turning, Transactions of ASME 222 (1984) 106–228. [5] I.E. Minis, E.B. Magrab, I.O. Pandelidis, Improved methods for the prediction of chatter in turning Part-3: a generalized linear theory, Transactions of ASME 112 (1990) 12–20.

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