Experimental and Analytical Investigation of the Subcritical Instability in Turning

Proceedings of DETC’99 17th ASME Biennial Conference on Mechanical Vibration and Noise September 12-15, 1999, Las Vegas, Nevada, USA

DETC99/VIB-8060

EXPERIMENTAL AND ANALYTICAL INVESTIGATION OF THE SUBCRITICAL INSTABILITY IN METAL CUTTING

´ Kalmár-Nagy Tamas Department of Theoretical and Applied Mechanics Cornell University Ithaca, New York 14853 Email: [email protected]

Jon R. Pratt Matthew A. Davies, Michael D. Kennedy Manufacturing Engineering Laboratory National Institute of Standards and Technology Gaithersburg, Maryland 20899 Email: [email protected]

ABSTRACT A single-degree-of-freedom dynamic cutting fixture is used to map out a part of the lobed stability boundary in a simple high-speed machining experiment. The experiment reveals the hysteretic nature of the instability. A 1 DOF mechanical model is derived using parameters identified from the experiment. We then show the existence of a subcritical Hopf bifurcation in this delay-differential equation model which corresponds to the observed experimental instability. The calculation is based on center manifold reduction. Then time domain simulation is used to solve the full nonlinear equation of motion that allows for the tool to leave the workpiece giving excellent agreement with the experiment.

Spacek (1954), Tobias (1965)). The corresponding mathematical model is a delay-differential equation. Recently, there has been increased interest in the subject. The PhD theses of Johnson (1996) and Fofana (1993), and the paper of Nayfeh, Chin and Pratt (1997) presented the analysis of the Hopf bifurcation in different models using different methods, like the method of multiple scales, harmonic balance, Floquet Theory (see also Nayfeh and Balachandran (1995)) and of course, numerical simulations. Experimental results of Shi and Tobias (1984) clearly showed the existence of ’finite amplitude instability’, that is unstable periodic motion of the tool around its asymptotically stable position related to the stationary cutting. Linear stability analysis can not account for this regime of conditional stability. A single-degree-of-freedom active cutting fixture is employed to reveal and analyse the hysteretic nature of the lobed stability boundary in a simple high-speed machining experiment. Specifically, the seventh stability lobe of a defined cutting process is mapped using experimental, analytical and computational techniques. Then, taking width of cut as a control parameter, the transition from stable cutting to chatter is observed experimentally. The cutting stability is found to possess a substantial hysteresis. This behavior is predicted by applying nonlinear regenerative chatter theory to an empirical characterization of the cutting force dependence on chip thickness. Simulations that incorporate both the nonlinear cutting force and the multiple-regenerative effect due to the tool leaving the cut are shown to be in excellent agreement with the experiments.

INTRODUCTION High amplitude tool vibrations (chatter) detoriate surface finish. These vibrations are partly due to delay effects. Because of some external disturbances (deviations in the workpiece, jamming up of chips) the tool starts a damped oscillation relative to the workpiece thus making its surface uneven. After one revolution of the workpiece the chip thickness will vary at the tool. The cutting force thus depends not only on the current position of the tool and the workpiece but also on a delayed value of the displacement. The length of this delay is the time-period τ of one revolution of the workpiece. This is the so-called regenerative effect (see, for example Moon (1994), Stépán (1989), Tlusty and

Address

all correspondence to this author.

1

Copyright  1999 by ASME

Figure 2. 1 DOF MECHANICAL MODEL.

written as Figure 1.

EXPERIMENTAL APPARATUS.

p

1 x¨ + 2ζωn x˙ + ω2n x = ; ∆Fx m

(1)

where ωn = s=m is the natural frequency of the undamped free oscillating system, and ζ = c=(2mωn ) is the so-called relative damping factor. These parameters m s c were identified from the machine-tool response function as follows

EXPERIMENTAL SETUP A dynamic test fixture (1 DOF flexible tool holder with actuators and sensors for dynamic cutting measurements) has been developed to explore fundamental issues in cutting process dynamics. Experiments were performed on a diamond turning lathe equipped with an air-bearing spindle capable of speeds between 0 and 1500 rpm. Figure 1 shows the experimental apparatus. The dynamic test fixture consists of a right hand cutting tool with a triangular tungsten carbide insert (rake angle=5 , clearance angle=11), held in a standard tool post mounted to a rigid steel plate. This plate is supported at both ends by thin steel plates thus creating a table structure having high stiffness in the directions perpendicular to the feed direction. This structure is rigidly mounted to the cross slide. Between the two thin plates, two voice coil actuators are placed. The pole pieces are grounded to the cross-slide, while the coils move with the table. Forces can be exerted on the flexible structure by varying the current in the coils. In this investigation they were only used to damp out unwanted vibrations during cleaning up of the workpiece. Workpieces are flanged aluminium cylinders with nominal diameter of 100 mm. These are rigidly mounted to the spindle and assumed rigid compared to the tool. The relative displacement between the tool and the workpiece is measured by the table deflection using an optical displacement sensor. An accelerometer is also used to characterize tool motions.

m = 10 kg s = 3:35 MN/m c = 156 kg/s

ωn = 578:79 1/s ζ = 0:0135

=)

(2)

The calculation of the cutting force variation ∆Fx requires an expression of the cutting force as a function of the chip thickness f . Evidence of nonlinear dependence of cutting force on feed can be found as early as 1906 (Taylor). Here we measured the thrust force component of the static cutting force as a function of feed at a fixed cutting speed (by replacing the dynamic cutting fixture with a three-component force dynamometer). Figure 3 shows the results of static cutting force measurements. The thrust force in Newtons is plotted as a function of the feed per revolution (chip thickness) in micrometers for a rotational speed of Ω = 836 rpm and a nominal width of cut w0 = 0:25 mm. We use a simple empirical way to calculate the cutting force by using a power-law curve fitted to data obtained from the cutting tests (Taylor (1906)) as opposed to the cubic polynomials of Hanna and Tobias (1974) and Shi and Tobias (1984). Fx ( f ) = 553 f 0:41 N ]

MECHANICAL MODEL Figure 2 shows a 1 DOF mechanical model of the tool. f denotes chip thickness. The corresponding Free Body Diagram (ignoring horizontal forces) is also shown in Figure 2. The equation of motion can be

(3)

Since this law is strongly material dependent, in the following calculations we will use the general power law (Figure 4) Fx ( f ) = 2

0 Kw f a

f 0 f >0

(4)


as the difference of the present tool edge position x(t ) and the delayed one x(t ; τ) in the form ∆f

= f ; f0 = x(t ); x(t ; τ) = x ; xτ

(6)

where the delay τ = 2π=Ω is the time period of one revolution with Ω being the constant angular velocity of the rotating workpiece (or τ = 60=Ω if Ω is given in rpm’s). Let us also define ∆x = ;∆ f

= xτ ; x

(7)

The slope of the power-law curve at the nominal chip thickness f0 is called the and usually denoted by cutting forceα;coefficient ∂Fx 1 k1 (k1 = ∂ f = αKw f0 ), see Figure 4. k1 will be used as f = f0

a bifurcation parameter (it is linearly proportional to the width of cut, which is easy to vary experimentally). Expressing ∆Fx ( f ) in terms of k1 yields the following equation of motion Figure 3. NONLINEAR CUTTING FORCE DEPENDENCE ON FEED.

x¨ + 2ζωn x˙ +

ω2n x

=

(

k1 f 0 mα k1 f 0 mα

1;

∆ f ; f0 ∆ f > ; f0

f α f0

(8)

Let us introduce the nondimensional time t˜ and displacement xe t˜ = ωnt

xe =

x X

(9)

where X is a length scale chosen below. Then the nondimensional equation of motion is

8 < x˜00 + 2ζx˜0 + x˜ = : Figure 4.

∆Fx ( f ) = Fx ; Fx ( f0 ) =

∆ f ; f0 ∆ f > ; f0

1; 1;

α

X f 0 ∆x˜

∆x˜ ∆x˜ <

f0 X f0 X

; k1 X (α ; 1) 2 k1 X 2 (α ; 1)(α ; 2) 3 k1 ∆x˜ + ∆x˜ + ∆x˜ + O ∆x˜4 2 2 2 2 mωn 2 f0 mωn 6 f0 mωn (11) Let us choose X > 0 in such a way that the absolute value of the coefficients of the second and the third order terms are equal, i.e

where w is the width of cut, α < 1 is the exponent and the parameter K depends on further technological parameters considered to be constant in the present analysis. When f 0 there is no contact between the tool and the workpiece. Then cutting force variation can then be expressed as

;Fx;( f0 ) Kw f a ; f0α

(10) where 0 denotes time derivative w.r.t the nondimensional time. Expanding the RHS about ∆x˜ = 0 yields

CUTTING FORCE CHIP THICKNESS RELATION.

k1 f 0 αmω2n X k1 f 0 αmω2n X

2 k1X (α ;21) = k1X (α ;2 1)(2α ; 2) 2 f0 mωn 6 f0 mωn

(5)

(12)

This gives X = 23;f0α . Introducing the nondimensional bifurcak1 tion parameter p = mω 2 and dropping the tilde results in the full

where f0 is the nominal chip thickness (feed) in steady state cutting. The chip thickness variation ∆ f can easily be expressed

n

3


THEORETICAL AND EXPERIMENTAL STABILITY CHART The characteristic function of equation (17) can be obtained by substituting the trial solution x(t ) = c exp(λt ) into its linear part:

nondimensional equation of motion

(

p(2;α) 3α p(2;α) 3α

x00 + 2ζx0 + x =

;

1; 1;

α 3 2;α ∆x

∆x ∆x <

2;α 3 2;α 3

(13)

D(λ p) = det(λI ; L( p); R( p)e;λτ ) = λ2 + 2ζλ +(1 + p); pe;λτ (18) The stability chart consists of curves on which the characteristic equation has one pair of pure imaginary roots, while the infinitely many others have negative real part. To find these curves

However, ;here the definition of ∆x must be modified according to ∆x = min xτ ; x 2;3 α , because when the tool is not in contact with the workpiece, the chip thickness will not be affected. The local bifurcation study will assume that the tool does not leave the material at any time and thus will be based on the power series expansion of the RHS of (13) provided that x τ ; x < 2;3 α

D(iω p) = 1 + p ; ω2 ; p cosωτ + i (2ζω + p sinωτ) = 0 (19)

3p (1 ; α) ( xτ ; x)2 +(xτ ; x)3 2 (2 ; α) (14) The second order equation of motion (14) is transformed into a 2 dimensional system by introducing x00 + 2ζx0 + x = p (xτ ; x)+

x(t ) =

x1 (t ) x2 (t )

= xx˙ ((tt ))

has to be solved. Eliminating the trigonometric terms from this equation yields p=

(15)

τ=

0

1

;(1 + p) ;2ζ

R( p) =

0

1 ; ω2 2 arctan + jπ ω 2ζω

Ω=

j = 1 2 : : :

(21)

2π τ

=

ωπ ;ω2 arctan 12ζω

+ jπ

j = 1 2 : : :

(22)

where j corresponds to the jth ’lobe’ (parameterized by ω) from the right in the stability diagram 5. It has long been realized that productivity can be well improved by exploiting the lobed nature of the stability chart, especially at higher speeds (Tlusty and Spacek (1954)). In this high-speed regime, the lobes are less densely packed, thus leaving room for changing control parameters. However Figure 6 clearly shows that linear stability does not describe the behavior of tool oscillations correctly. Here we show the rms amplitudes of tool vibrations when the rotation rate is fixed at 836 rpm, and w is swept forward and backwards through wc , a plot referred to as experimental bifurcation diagram. Linear stability analysis predicts asymptotically stable behavior for all widths of cut below wc =225 µm. For the forward sweep, this is the case, however for the reverse sweep chatter persists well below the critical value. This hysteretic behavior occurs in many practical machining operations, and has been observed by other researchers (Hanna and Tobias (1974),

00 p0

f(x(t ) x(t ; τ) p) =

(20)

(16)

where the dependence on the bifurcation parameter p is also emphasized: L( p) =

2

1 ; ω2 + 4ζ2ω2 2 (ω2 ; 1)

which also implies ω > 1. Then

and we obtain the delay-differential equation x˙ (t ) = L( p)x(t )+ R( p)x(t ; τ)+ f(x(t ) x(t ; τ) p)

;

(17)

3p (1 ; α) 2 (2 ; α)

(x1 (t ; τ); x1 (t ))2 +(x1 (t ; τ); x1 (t ))3 4


Figure 5.

STABILITY CHART. Figure 7.

Figure 6.

EXPERIMENTAL STABILITY CHART.

creased in 25 µm increments using the manual slide in a ramp and hold fashion. The experiment is stopped and recorded when selfsustaining oscillations are observed in the accelerometer voltage monitored on a digital oscilloscope. Figure 7 shows the experimental stability chart as a function of speed rpm and width of cut w mm. Material removal can be increased between 840 rpm and 925 rpm, since w grows from approx. 200 µm to 800 µm over this range. To create the experimental bifurcation diagram (Figure 6) the width of cut w is swept as for the lobe determination, except that the ramp and hold increments are 10 µm and the outputs of the displacement and acceleration signals are continously recorded. At each increment, the tool cuts for '5 seconds, these intervals being logged on a separate channel of the scope via a manual switch. The forward sweep is executed first, then the part is remachined to a smooth surface, and the backward sweep is conducted.

EXPERIMENTAL BIFURCATION DIAGRAM.

Shi and Tobias (1984)). In the case shown here, the region of the subcritical instability makes up nearly 40% of the predicted chatter free operating regime. Within this region, disturbances, as might be caused by deviations in the workpiece, jamming up of chips, etc. can push the system into unstable vibrations. A lobe of the stability boundary is traced in a series of experiments. All cutting measurements are made at the same feedrate s = 508 µm/s. The feed per revolution f 0 thus varies with the rotational rate, but we do not account for this in this paper. Beginning with a small stable cut at a fixed speed, the width of cut w is in-

OPERATOR DIFFERENTIAL EQUATION FORMULATION The necessary condition for the existence of periodic orbits is that by varying the bifurcation parameter (p) the critical characteristic roots cross the imaginary axis with non-zero velocity, ( p) 6= 0. that is Re dλdp The change of the real parts of these critical characteristic roots can be determined via implicit differentiation of the characteristic function (18) with respect to the bifurcation parameter 5


For a heuristic argument of how these operators and bilinear form arise, see Kalmár-Nagy, Stépán and Moon (1999). A first order approximation to this center manifold can be given by the center subspace of the associated linear problem, which is spanned by the real and imaginary parts of the infinite dimensional complex eigenfunction s(ϑ) 2 H corresponding to the critical characteristic root iω. This eigenvector satisfies

p to get γ: = Re

dλ( p) dp

; 2 ; 2 = pτ ω ;2 1 + 2ζ 2ω + 1 2 p (2ζ + pτ) + 4ζτ (ω + 1)+ 4ω

(23)

Since this number is always positive and all the other infinitely many characteristic roots but the critical ones are located in the left half of the complex plane, the conditions of an infinite dimensional version of the Hopf Bifurcation Theorem given in Hassard, Kazarinoff and Wan (1981) are satisfied. γ will later be used in the estimation of the vibration amplitude. In order to study the critical infinite dimensional problem on a 2-dimensional center manifold we need the operator differential equation representation of (17). The delay-differential equation (17) can then be expressed as an abstract evolution equation (Campbell, Bélair, Ohira, Milton (1995), Hale (1977), Kuang (1993), Kalmár-Nagy, Stépán and Moon (1999)) on the Banach space H of continuously differentiable functions u: ;τ 0] ! R2

x˙ t

= A xt + F (xt )

A s(ϑ) = iωs(ϑ) i.e.

s(ϑ) = ceiωϑ

ϑ 2 ;τ 0) ϑ=0

(ϑ) L ( p) u (0)+ R ( p) u (;τ) d dϑ u

A n(σ) = ;iωn(σ)

0 f (u (0) u (;τ) p)

ϑ 2 ;τ 0) ϑ=0

; dσd n (σ) = ;iωn(σ) L ( p) n (0)+ R ( p) n (τ) = ;iωn(0)

A

; dσd v (σ) v (σ) = L ( p) v (0)+ R ( p) v (τ)

as well as the bilinear form (

(v u) = v (0)u(0)+

)

Z

:H 0

;τ

σ 2 (0 τ] σ=0

n(σ) = deiωσ

(34)

(35)

The bilinear form (29) provides the ’orthonormality’ condition (27)

(s s ) = 1

(36)

Defining s1 = Re (s), s2 = Im (s), n1 = Re (n), n2 = Im (n), and c=

(28)

c11 + ic12 c21 + ic22

d=

d11 + id12 d21 + id22

(37)

calculations detailed in Kalmár-Nagy, Stépán and Moon (1999) give

H ! IR defined by

v (ξ + τ)R( p)u(ξ)dξ

σ 2 (0 τ] σ=0

that has the solution

(26)

where u 2 H . We also define the adjoint operator A on the space H of continuously differentiable functions v: 0 τ] ! R 2

(33)

which is again a boundary value problem for n(σ)

(25)

while the nonlinear operator F can be written as

F (u)(ϑ) =

(32)

Similarly the adjoint eigenfunction n(σ) satisfies

The linear operator A at the critical value of the bifurcation parameter assumes the form

A u (ϑ) =

(31)

which is a boundary value problem with the solution

(24)

ϕ 2 ;τ 0]

ϑ 2 ;τ 0) ϑ=0

(ϑ) = iωs(ϑ) L ( p) s (0)+ R ( p) s (;τ) = iωs(0) d dϑ s

Here xt (ϕ) 2 H is defined by the shift of time xt (ϕ) = x(t + ϕ)

(30)

c=

(29) 6

1 iω

(38)


d=δ

;

;

;

2 2 ω2; + 2ζ2 + ζτ (1 ;+ p) + i 2ζ + τ 1 + p ; ω 2 2 ω 4ζ τ + 2ζ ; τ 1 + p ; ω + i2ω (1 + ζτ)

δ=

4

2ω2 (1 + ζτ)2 +(2ζ + τ (1 + p ; ω2))

2

The time derivative of w can be expressed both by differentiating the right-hand-side of equation (46) via substituting equations (43, 44) and also by calculating equation (45). Equating coefficients of like powers we obtain a 6 dimensional linear boundary value problem for the unknown coefficients h 1 h2 h3 . For detailed calculations, the reader is referred to Kalmár-Nagy, Stépán and Moon (1999). In order to restrict a third order approximation of system (43, 44, 45) to the 2 dimensional center manifold, the second order approximation w(y 1 y2 ) of the center manifold has to be substituted into equations (43, 44). Then these equations will assume the form

(39)

(40)

The solution xt (ϑ) of equation (24) can be decomposed into two components y12 lying in the center subspace and into the infinite dimensional component w transverse to the center subspace xt (ϑ) = y1 (t )s1 (ϑ)+ y2 (t )s2 (ϑ)+ w(t )(ϑ)

(41)

y˙1 = ωy2 + a20 y21 + a11y1 y2 + a02y22 + +a30y31 + a21y21y2 + a12y1 y22 + a03y32

where y1 (t ) = (n1 xt )

y2 (t ) = (n2 xt )

(42)

The so-called Poincarè-Liapunov constant ∆ can be calculated as shown in Hassard, Kazarinoff, Wan (1981) or Stépán (1989):

With these new coordinates the operator differential equation (24) can be transformed into a ’normal form’. y˙1 = (n1 x˙ t ) = ωy2 + d12F

(43)

y˙2 = (n2 x˙ t ) = ;ωy1 + d22 F

(44)

∆=

d dt

(45)

∆=

CENTER MANIFOLD REDUCTION AND HOPF BIFURCATION Although the tools for Center Manifold Reduction have been available for a long time (Hale (1977), Hassard, Kazarinoff, Wan (1981)) the closed form calculation regarding the existence and the nature of the corresponding Hopf bifurcation in delaydifferential equations is only feasible by using computer algebra (see also Campbell, Bélair, Ohira, Milton (1995)) The center manifold is tangent to the plane y 1 y2 at the origin, and it is locally invariant and attractive to the flow of system (24). Its equation can be assumed in the form of the truncated power series 1 (h1 (ϑ)y21 + 2h2(ϑ)y1 y2 + h3 (ϑ)y22 ) 2

(48)

The negative/positive sign of ∆ determines if the Hopf bifurcation is supercritical or subcritical. Despite a long and tedious calculations ∆ is quite simple:

(xt ; y1s1 ; y2s2 ) = A w + F (xt ); d12Fs1 ; d22Fs2

w(y1 y2 )(ϑ) =

1 (a20 + a02)(;a11 + b20 ; b02)+ 8ω +(b20 + b02)(a20 ; a02 + b11)]+

+ 18 (3a30 + a12 + b21 + 3b03)

where F= F (w + y1s1 + y2 s2 )(0)2 . Similarly w ˙ =

(47)

y˙2 = ;ωy1 + b20y21 + b11y1 y2 + b02 y22 + +b30y31 + b21y21y2 + b12y1 y22 + b03y32

;

qδ ω2 ; 1 4p2

;

; ;

2ζ 3 ω2 ; 1

+2σ

where σ =

(

;

;

3pτ ω2 ; 1 + 2pσ 1 + 2p ; ω2

+

(49)

+ 2 + 7p + 6p2 +(4 + 11p) ω2 ; 2ω2 ; 4ω6 (

2q ω2 ;1

)

p 4+9p;12ω4 +8ω6

) and q =

3p(1;α) 2(2;α) (the

coefficient of

the nonlinear term in (14)). ∆ can be shown to be positive for any 0 < α < 1, that is the Hopf bifurcation is subcritical (unstable periodic motion exists around the stable steady state cutting for cutting coefficients p which are somewhat smaller than the critical value pcr ). The estimation of the vibration amplitude has the simple form

(46) 7


and proved analytically with the help of the Center Manifold and Hopf Bifurcation Theory. This analysis is local in the sense that it does not account for nonlinear phenomena as the tool leaves the material (and also, because the series expansion is local). In this case the regenerative effect disappears, and the result of the local analysis is not valid anymore. In this case numerical simulation can be used and this revealed a subcritical Hopf bifurcation with a folded structure in close agreement with the experiments.

ACKNOWLEDGMENT T. Kalmár-Nagy would like to thank Francis C. Moon, Chris Evans and E. Clayton Teague for support through a summer research appointment. J. R. Pratt thanks the National Research Council, who have supported his research through a NRC/NIST Postdoctoral Research Associateship.

Figure 8.

REFERENCES Campbell, S. A.; Bélair, J.; Ohira, T.; and Milton, J. 1995, ’Complex dynamics and multistability in a damped oscillator with delayed negative feedback’, Journal of Dynamics and Differential Equations 7, 213-236. Doi, S.; Kato, S., 1956, ’Chatter vibration of lathe tools’, Transactions of the ASME 78, 1127-1134. Fofana, M., 1993, Nonlinear Dynamics of Cutting Process, PhD thesis, University of Waterloo, Waterloo. Guckenheimer, J.; Holmes, P., 1986, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, SpringerVerlag, New York. Hale, J. K., 1977, Theory of Functional Differential Equations, Springer, New York. Hanna, N. H.; Tobias S. A., 1974, ‘A theory of nonlinear regenerative chatter’, ASME J. Eng. Indust. 96, 247-255. Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., 1981 Theory and Applications of Hopf Bifurcations, London Mathematical Society Lecture Note Series 41, Cambridge. Johnson, M. A., 1996, Nonlinear differential equations with delay as models for vibrations in the machining of metals, PhD Thesis, Cornell University. Kalmár-Nagy, T.; Stépán, G.; Moon, F. C., 1999, ’Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations’, To appear in Nonlinear Dynamics. Kuang, Y., 1993, Delay Differential Equations, Academic Press, Boston. Moon, F. C., 1994, ’Chaotic dynamics and fractals in material removing processes’, in Nonlinearity and Chaos in Engineering Dynamics (Thompson, J. M. T., and Bishop, S. R., eds.), pp. 25-37, Wiley, Chichester. Nayfeh, A. H.; Balachandran, B., 1995, Applied Nonlinear Dynamics Wiley, New York. Nayfeh, A. H.; Chin, C.-M.; Pratt, J., 1997, ’Applications

BIFURCATION DIAGRAM.

r r r γ γpcr p r = ; ( p ; pcr ) = 1; ∆ ∆ p

(50)

cr

where γ was defined in equation (23). The approximation of the periodic solution of the delay-differential equation (17) can be obtained in the form x(t ) = xt (0)

y1 (t )s1 (0)+ y2(t )s2 (0) = r

cos (ωt ) sin (ωt )

(51)

NUMERICAL RESULTS The results of the above sections were confirmed numerically. A Mathematica procedure was written to integrate the full delay equation (17). To find the amplitude of the unstable limit cycles these equations with a suitably chosen initial function (i.e. one close to the stable manifold of the limit cycle) were integrated. The bifurcation diagram (presenting the amplitude of the unstable limit cycle as a function of the cutting force coefficient) is shown in Figure 8, together with the previously obtained experimental points and analytical approximation. The dashed line corresponds to the amplitude where the tool leaves the workpiece.

CONCLUSIONS The existence and nature of a Hopf bifurcation in the delaydifferential equation for self-excited tool vibration is presented 8


of perturbation methods to tool chatter dynamics’, in Nonlinear Dynamics of Material Processing and Manufacturing (Ed.: F. C. Moon) 193-213, Wiley, New York. Pratt, J. R.; Davies, M. A.; Kennedy, M. D.; Kalmár-Nagy, T., 1999, ’Predictive modeling of nonlinear regenerative chatter via measurement, analysis and simulation’, Submitted to IMECE’99. Shi, H. M.; Tobias, S. A., 1984, ‘Theory of finite amplitude machine tool instability’, Int. J. of Machine Tool Design and Research 24, 45-69. Stépán, G., 1989, Retarded Dynamical Systems, Longman, London. Stépán, G., 1997, ‘Delay-differential equation models for machine tool chatter’, in Nonlinear Dynamics of Material Processing and Manufacturing (Ed.: F. C. Moon) 165-192, Wiley, New York. Stépán, G.; T. Kalmár-Nagy, 1997, ’Nonlinear regenerative machine tool vibrations’, in Proceedings of the 1997 ASME Design Engineering Technical Conferences, 16th ASME Biennial Conference on Mechanical Vibration and Noise (Sacramento, 1997), DETC97/VIB-4021, 1-11. Taylor, F. W., 1906, ’On the art of cutting metals’, American Society of Mechanical Engineers, New York. Tlusty, J.; Spacek, L., 1954, Self-Excited Vibrations on Machine Tools (in Czech), Nakl CSAV, Prague. Tobias, S. A., 1965, Machine Tool Vibrations, Blackie, London.

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Experimental and Analytical Investigation of the Subcritical Instability in Turning

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