kalmar-nagy mode-coupled regenerative machine tool vibrations

8/14/2019 Kalmar-Nagy Mode-Coupled Regenerative Machine Tool Vibrations

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Mode-Coupled Regenerative Machine Tool Vibrations

Tams Kalmr-Nagy1, Francis C. Moon21United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06108

2Sibley School of Mechanical and Aerospace Engineering

Cornell University, Ithaca, NY 14853, USA

Abstract

In this paper a new 3 degree-of-freedom lumped-parameter model for machine tool vibrations is developed andanalyzed. One mode is shown to be stable and decoupled from the other two, and thus the stability of the systemcan be determined by analyzing these two modes. It is shown that this mode-coupled nonconservative cuttingtool model including the regenerative effect (time delay) can produce an instability criteria that admits low-levelor zero chip thickness chatter.

1 Introduction

One of the unsolved problems of metal cutting is the existence of low-level, random-looking (maybe chaotic)vibrations (or pre-chatter dynamics, see Johnson and Moon [17]). Some possible sources of this vibration arethe elasto-plastic separation of the chip from the workpiece and the stick-slip friction of the chip over the tool.Recent papers of Davies and Burns [9], Wiercigroch and Krivtsov [43], Wiercigroch and Budak [41] and Moonand Kalmr-Nagy [27] have addressed some of these issues. Numerous researchers investigated single degree-of-freedom regenerative tool models (Tobias [39], Hanna and Tobias [13], Shi and Tobias [34], Fofana [11], Johnson[18], Nayfeh et al. [28], Kalmr-Nagy et al. [20], Stpn [36], Kalmr-Nagy [21], Stone and Campbell [38], Stpnet al. [37]). Even though the classical model (Tobias [39]) with nonlinear cutting force is quite successful inpredicting the onset of chatter (Kalmr-Nagy et al., [19]), it cannot possibly account for all phenomena displayedin real cutting experiments. Single degree-of-freedom deterministic time-delay models have been insufficient sofar to explain low-amplitude dynamics below the stability boundary. Also, real tools have multiple degrees offreedom. In addition to horizontal and vertical displacements, tools can twist and bend. Higher degree-of-freedommodels have also been studied in turning, as well as in boring, milling and drilling (Pratt [32], Batzer et al.[2], Balachandran [1], van de Wouw et al. [44]). In this paper we will examine the coupling between multiple

degree-of-freedom tool dynamics and the regenerative effect in order to see if this chatter instability criteria willpermit low-level instabilities.

Coupled-mode models in aeroelasticity or vehicle dynamics may exhibit so-called flutter or dynamics insta-bilities (see e.g. Chu and Moon [8]) when there exists a non-conservative force in the problem. One example is thefollower force torsion-beam problem as in Hsu [15]. In the present work we assume that the chip removal forcesrotate with the tool thereby introducing an unsymmetric stiffness matrix which can lead to flutter and chatter.Tobias called this mode-coupled chatter. Often this model of chatter is analyzed without the regenerative e ffect.In this paper we will show that the combination of mode-coupling nonconservative model and a time delay canproduce an instability criteria that admits low-level or zero chip thickness chatter. There is no claim in this paperto having solved the random- or chaotic low level dynamics since only linear stability analysis is presented in thispaper. But the results shown below provide an incentive to extend this model into the nonlinear regime. A dy-namic model with the combination of 2-degree-of-freedom flutter model with time delay may also be applicable toaeroelastic problems in rotating machinery where the fluid forces in the current cycle depend on eddies generatedin the previous cycle. However the focus of this paper is on the physics of cutting dynamics.

The structure of the paper is as follows. In Section 2 an overview of the turning operation is given, togetherwith the description of chatter and the regenerative effect. The equations of motion are developed in Section 3.The model parameters are estimated in Section 4. Analysis of the model is performed in Section 5 and conclusionsare drawn in Section 6.

2 Metal Cutting

The most common feature of machining operations (such as turning, milling, and drilling) is the removal of athin layer of material (the chip) from the workpiece using a wedge-shaped tool. They also involve relative motion

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Machined surfaceWorkpiece

vtoolW

Figure 1: Turning

between the workpiece and the tool. In turning the material is removed from a rotating workpiece, as shown inFigure 1.

The cylindrical workpiece rotates with constant angular velocity [rad/s] and the tool is moving along theaxis of the workpiece at a constant rate. The feed f is the longitudinal displacement of the tool per revolution ofthe workpiece, and thus it is also the nominal chip thickness. The translational speed of the tool is then given by

vtool =

2f (1)

The interaction between the workpiece and the tool gives rise to vibrations. One of the most important sourceof vibrations in a cutting process is the regenerative effect. The present cut and the one made one revolutionearlier might overlap, causing chip thickness (and thus cutting force) variations. The associated time delay is thetime-period of one revolution of the workpiece

=2

(2)

The phenomenon of the large amplitude vibration of the tool is known as chatter. A good description of chatter is

given by S. A. Tobias [39], one of the pioneers of modern machine tool vibrations research: The machining of metalis often accompanied by a violent relative vibration between work and tool which is called the chatter. Chatteris undesirable because of its adverse affects on surface finish, machining accuracy, and tool life. Furthermore,chatter is also responsible for reducing output because, if no remedy can be found, metal removal rates have tobe lowered until vibration-free performance is obtained.

Johnson [18] summarizes several qualitative features of tool vibration

The tool always appears to vibrate while cutting. The amplitude of the vibration distinguishes chatter fromsmall-amplitude vibrations.

The tool vibration typically has a strong periodic component which approximately coincides with a naturalfrequency of the tool.

The amplitude of the oscillation is typically modulated and often in a random way. The amplitude modu-lation is present in both the chattering and non-chattering cases.

Tool vibrations can be categorized as self-excited vibrations (Litak et al. [24], Milisavljevich et al. [25]) orvibrations due to external sources of excitation (such as resonances of the machine structure) and can be periodic,quasiperiodic, chaotic or stochastic (or combinations thereof). A great deal of experimental work has been carriedout in machining to characterize and quantify the dynamics of metal cutting. Recently a number of researchershave provided experimental evidence that tool vibrations in turning may be chaotic (Moon and Abarbanel [26],Bukkapatnam et al. [5], Johnson [18], Berger et al. [3]). Other groups however now disavow the chaos theory forcutting and claim that the vibrations are random noise (Wiercigroch and Cheng [42], Gradiek et al. [12]).

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degrees of freedom are horizontal position (x), vertical position (z), and twist (). In the lumped parameter model(Figure 4) all the mass m of the beam is placed at its end (this effective mass is equivalent to modal mass for adistributed beam).

x

z

f

chip

m k ,cf

kx

cx

kz cz

FT

FC

Figure 4: 3 DOF lumped-parameter model

The equations of motion are the following

mz + cz z + kzz = Fz (3)

mx + cxx + kxx = Fx (4)

I + c + k = My (5)

Figure 5 shows the forces acting on the tooltip.As the tool bends about the x axis, the direction of the cutting velocity (and main cutting force) changes, as

shown in Figure 6.In order to derive the equations of motion, two coordinate systems are de fined. An inertialframe (I, J, K) fixed to the tool and a moving frame (i,j, k) fixed to the cutting velocity.The force acting on theinsert can then be written as

F = FTI + FRJ FCK (6)or

F = Fxi + Fyj + Fzk (7)

where i, j, k are unit vectors in the x, y, z directions, respectively.

Fx = FT (8)Fy = FC sin + FR cos (9)

Fz = FR sin FCcos (10)The bending also results in a pitch (shown in Figure 6). This is not a separate degree of freedom, but nonethelessit will influence the inclination angle.

The following assumptions are used in deriving the equations of motion

The forces that act on the insert are steady-state forces

The width of cut w (y-position) is constant

All displacements are small

Yaw is negligible

Steady-state forces refer to time averaged quantities. The effect of rate-dependent cutting forces were studiedby Saravanja-Fabris and DSouza [33], Chiriacescu [7], Moon and Kalmr-Nagy [27]. Next we find the position ofthe tooltip in the fixed system of the platform. To do so we have to find the rotation matrix R that describes therelationship between the moving frame (i,j, k) and the fixed frame (I, J, K).

i j k

= R

I J K

(11)

Using the Tait-Bryant angles {, } we express R as a product of two consecutive planar rotations (Pitch-Rollsystem)

R = R2R1 (12)

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x

y

z

FTFR

FC IJK

ij

k

vC

Figure 5: Forces on the tooltip

FC

FR

z

y

b y

Figure 6: Direction of cutting velocity

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The cross section is first rotated about I by the pitch angle . The corresponding rotation matrix is

R1 =

1 0 00 c s

0 s c

(13)

where the abbreviations c = cos, s = sin were used. The second rotation is about the J2 (the rotated J) axisthrough the roll angle (with respect to the toolholder)

R2 =

c 0 s0 1 0s 0 c

(14)

R can then be calculated by (12)

R =

c ss cs0 c ss cs cc

(15)

The position of the tooltip can be expressed in the fixed frame as

r = R

rx0

rz

=

rxc + rzcsrzs

rzcc rxs

(16)

The roll producing moment can then be calculated as

My = (r

F) j = FT (rxs rzcc) + FCc(rxc + rzcs) FRs(rxc + rzcs) (17)In the following we assume small displacements and small angles and neglect nonlinear terms. The angle istaken to be proportional to the vertical displacement, i.e. = nz (n > 0) and so is the pitch, i.e. = kz(k > 0).

mx + cxx + kxx = FT (18)mz + cz z + kzz =

FC + nzFR

(19)

I + c + k = My =

rxFT + rzFC rzFT + rxFC + nzrxFR (20)

where FC, FR, FT denotes the constant term in FC, FR and FT, respectively.

3.1 Cutting Forces

Generally we assume that the cutting forces FC, FT, FR depend only on the inclination angle i and chip thickness

f (see Figure 2), and the rake angle (see Figure 3). We again emphasize that the chip width w is consideredconstant in the present analysis. Our hypothesis here is that FC and FT depend linearly on both the rake angleand chip thickness (see Section 4.2) in the following manner

FC = lC + mCt1 + FC0 (21)

FT = lT + mTt1 + FT0 (22)where mC and mT are cutting force coefficients, while lC and lT are angular cutting force coefficients (they showhow strong the force dependence is on rake angle). The variable t1 is the chip thickness variation (the deviationfrom the nominal chip thickness). The constant forces FC0 and FT0 arise from cutting at a nominal chip thickness.The radial cutting force can be expressed as (Oxley [30])

FR = sin iFCcos i (i sin ) FT

sin2 i sin + cos2 i(23)

where Stablers Flow Rule (Stabler [35]) C = i was used. The effective rake angle depends on the initial rakeangle and the roll

= 0 (24)while the inclination angle will depend on the initial inclination angle (i0) as well as the pitch

i = i0 (25)

The chip thickness depends on the nominal feed and the position of the tooltip (both the present and the delayedones). The displacement of the tooltip is due to translational and rotational motion as shown in Figure 7.Herethe dashed line corresponds to the position vector of the tooltip in the undeformed configuration, while the solid

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rx

rz

f

i

k

z

x

Figure 7: Motion of the tooltip

line depicts how this vector rotates () and translates (due to the displacements x and z). The chip thickness isthen given by

t1 = t10 + x x + rz sin( ) t10 + x x + rz ( ) (26)where x and denote the delayed values x (t ) and (t ), respectively. Then the cutting forces can bewritten as

FC = mC (x x) + (lC + rzmC) rzmC +FC

mCt10 + FC0 0lC (27)FT = mT (x x) + (lT + rzmT) rzmT + mTt10 + FT0 0lT (28)

If the initial inclination angle is assumed to be zero, the expression for FR will simplify

FR = k

FT + (sin 0 1) FC + t10 (mC (1 sin 0)mT)

z (29)

3.2 The Equations of Motion

Substituting (27-28) into equations (18-20) and eliminating the constant (by translation of the variables) results

mz + czz + kzz = nFRz mC (x x) (lC + rzmC) + rzmC (30)mx + cxx + kxx = mT (x x) (lT + rzmT) + rzmT (31)

I + c + k = rzmT (x x) rz (lT + rzmT mCt10 FC0 + 0lC) + r2zmT (32)where now (x, z, ) represent deviations from the steady values of the original displacements. As we can see, thex and equations are uncoupled from the z equation, so the stability of the system is determined by (31, 32).Equations (31, 32) can also be written as

x + 2xxx +

2x +mTm

x +

1

m(lT + rzmT) =

mTm

x + rzmTm

(33)

+ 2 +rzmT

Ix +

2 +

rzI

(lT + rzmT mCt10 FC0 + 0lC)

=rzmT

Ix + r

2

z

mTI

(34)

where

x = kx

m , = k

I (35)By introducing the nondimensional time and displacement

t = t/T x = x/X (36)

x00 + 2xxTx0 +

2x +

mTm

T2x +

1

m(lT + rzmT)

T2

X =

mTm

T2x + rzmTm

T2

X (37)

00 + 2T 0 +

rmTI

T2Xx +

2 +rzI

(lT + rzmT FC0 mCt10 + 0lC)

T2 =

rzmTI

T2Xx + r2

z

mTI

T2 (38)

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With the choice of the following scales

T =1

xX =

I

m(39)

the equations assume the form ( = x)

x00 + 2xx0 + k11x + k12 = r11x + r12 (40)

00 + 20 + k21x + k22 = r21x + r22 (41)

where

k11 = 1 + mT2xm

k12 = lT + rzmT2x

Im(42)

k21 =rzmT

2x

Im =

x

(43)

k22 =

x

2

+rz

2xI(lT + rzmT mCt10 FC0 + 0lC) (44)

r11 =mT

2xmr12 = r21 =

rzmT

2x

Im(45)

r22 =r2zmT

2x

Im(46)

Note that the stiffnesses k12 and k21 are different. This is characteristic of nonconservative systems (Bolotin [4],Panovko and Gubanova [31]). In many mechanical systems this nonconservativeness is due to the presence of

following forces.

4 Estimation of Model Parameters

In the following we estimate different terms in (42-46) to establish their relative strengths in order to simplify themodel.

4.1 Structural Parameters

The toolholder is assumed to be a rectangular steel beam. The length of the toolholder is relatively short fornormal cutting, while it can be longer for b oring operations (see Kato et al. [22]). So we assume l to be between0.05 m and 0.3 m. The width and height are usually of order of a centimeter. The stiffnesses for such a cantileveredbeam can be in the following ranges

kx ' 104 107

N

m(47)

kz ' 105 107

N

m(48)

k ' 1000 10000N

rad(49)

Since a lumped parameter approximation is used, the mass at the end of the massless beam is assumed to be themodal mass. The vibration frequencies are then

x ' 100 5000rad

s(50)

z ' 100 10000rad

s(51)

' 1000 10000rad

s(52)

The ratio

xvaries between 2 and 10 (the shorter the tool is the higher the ratio).

4.2 Cutting Force Parameters

Experimental cutting force data during machining of 0.2% carbon steel is shown in Figure 8 (Oxley [30]).The graphshows the forces FC and FT for different rake angles ( = 5 and 5 for top and bottom Figures, respectively).The width of cut and chip thickness were 4 mm and 0.25 mm, respectively. Since our model assumes constant

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Figure 8: Forces in oblique cutting of 0.2% carbon steel. = 5

(top) and = 5

(bottom). f= 0

.125

mm. AfterOxley (1989)

a [rad]0.5

1

1.5

2

2.5

3

3.5

4

0.25

0.5

0.75

1

1.25

1.5

1.75

2

F [kN]C

F [kN]T

a [rad]

Figure 9: Forces vs. rake angle (derived from Oxley [30]) a, cutting force b, thrust force.

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5.2 Stability Analysis of the Undamped System without Delay

First we perform linear stability analysis of the system

x + Kx = 0 (68)

where the matrix K is non-symmetric and of the form (k22 > 0)

K =

k11 k12k21 k22

(69)

Assuming the solutions in the formx = deit (70)

we obtain the characteristic polynomials K 2I

d = 0 (71)

which have nontrivial solution if the determinant of K 2I is zero k11 2 k12

k21 k22 2 = k11 2 k22 2 k21k12 = 0 (72)

The characteristic equation for the coupled system becomes

4 (k11 + k22) 2 + k11k22 k21k12 = 0 (73)

Divergence (static deflection, buckling) occurs when = 0 (or det K = 0), that is when

k11k22

k21k12 = 0 (74)

If 6= 0, then the characteristic equation (73) can be solved for 2 as

2 =1

2

k11 + k22

(k11 + k22)

2 4 (k11k22 k21k12)

(75)

For stable solutions, both solutions should be positive. Since k22 > 0, this is the case if

0 k11k22 k21k12

k11 + k222

2(76)

The two bounds correspond to divergence and flutter boundaries, respectively. With the stiffness matrix in (62)

k11 = 1 + p, k12 = a + pq (77)

k21 = pq (78)

In the plane of the bifurcation parameters q, p the divergence boundaries are given by

p =1

2q2

k22 aq

4k22q2 + (k22 aq)2

(79)

and the flutter boundary is characterized by

p =1

1 + 4q2(80)

k22 2aq 1 2

q

a (1 k22) + a2q q (k22 1)2

(81)

Figure 10 shows these boundaries on the (q, p) parameter plane for a = 1, k22 = 2.The different stability regionsare indicated by the root location plots.

5.3 Stability Analysis of the 2 DOF Model with Delay

In this section we include the delay terms in the analysis. In order to be able to study how these terms influencethe stability of the system, we introduce a new parameter, similar to the overlap factor (Tobias [39]).

First we analyze the system with no damping:

x + Kx = Rx (82)

When = 0 we recover the previously studied (68), while = 1 corresponds to equation (61) without damping.The characteristic equation is

det2I + K eR

= 0 (83)

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q

p

S

U

U

U

Figure 10: Stability boundaries of the undamped 2 DOF model without delay

4 +

k11 + k22 pe

2 + k11k22 k12k21+

e (q (k12 + k21)pk22) 2q2e2 = 0 (84)

Substituting = i, 0 yields a complex equation that can be separated into the two real ones (the secondequation was divided by sin( ) 6= 0)

4 (k11 + k12) 2 + k11k22 k12k21+ (85) cos( )

p2 + q (k12 + k21)pk22

2q2 cos(2 ) = 0

p2 + q (k12 + k21)pk22 + 2q2 cos( ) = 0 (86)We solve the second equation for cos( )

cos( ) =p2 + q (k12 + k21)pk22

2q2 (87)

Using this relation and the identity cos(2 ) = 2cos( )2 1 in the real part (85) results

4

(k11 + k22) 2

+ k11k22 k12k21 + 2

q2

= 0 (88)

Divergence occurs where = 0, that is where

k11k22 k12k21 + 2q2 = 0

Substituting the elements of the stiffness matrix as given in (62) yields

(q (a + q (p )) (p )) + k22 (1 + pp ) = 0 (89)

which can be solved for p as

1

2 q2

k22 (1 ) + q (2 q a)

(k22 ( 1) + q (a 2q))2 + 4 q2 (k22 + q (a q ))

(90)

The change of the divergence boundary is shown in Figure 11 (top, middle, bottom) for = 0.1, 0.5 and 1 while

the delay was set to 1. Flutter occurs for > 0, and the boundary can be found by numerically solving equations(86, 88) for p and q for a given . Figure 12 shows the flutter boundary for a small (0.01) together with theflutter boundary (80). Figure 13 shows how this boundary changes with increasing ( = 0.1, 0.5, 1). Figure 14shows the full stability chart, complete with both the divergence and flutter boundaries, for = 1. To validate thisstability chart the parameter space (p,q) was gridded and the delay-differential equation (61, 62) was integratedwith constant initial function (note that the amplitude does not matter for linear stability) at the gridpoints.The integration was carried out for 15 intervals of which the first 5 intervals were discarded. Stability wasdetermined by whether the amplitude of the solution grew or decayed. Dark dots correspond to stable numericalsolutions. This figure can also explain a practical trick used in machine shops: sometimes, to avoid chatter, thetool is placed slightly ABOVE the centerline. We note that increasing q moves the system into the stable regionof the chart.

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- 2 0 2q

0

2p ftfS

S

S

U

U

U

Figure 12: Flutter boundary of (80) with = 0.01. S and U denote Stable and Unstable regions

- 1 0 1q

0

2p

m m=0. mmmm

Figure 13: Flutter boundary as a function of ( = 0.1, 0.5, 1)

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Figure 15: Stability chart for the 3 DOF model. a, /x = 2 b, /x = 10

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Figure 16: Stability charts for the 3 DOF model with increasing q

[2] S. A. Batzer, A. M. Gouskov, and S. A. Voronov, Modeling the Vibratory Drilling Process. Proceedings ofthe 17th ASME Biennial Conference on Vibration and Noise 1-8 (1999).

[3] B. S. Berger, I. Minis, Y. H. Chen, A. Chavali, and M. Rokni, Attractor Embedding in Metal Cutting.Journal of Sound and Vibration, 184(5), 936942 (1995).

[4] V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability. New York: The MacMillanCompany (1963).

[5] S. T. S. Bukkapatnam, A. Lakhtakia, and S. R. T. Kumara, Analysis of Sensor Signals Shows Turning on aLathe Exhibits Low-Dimensional Chaos. Physical Review E, 52(3), 23752387 (1995).

[6] S.-G. Chen, A. G. Ulsoy, and Y. Koren, Computational Stability Analysis of Chatter in Turning. Journal ofManufacturing Science and Engineering, 119, 457460 (1997).

[7] S. T. Chiriacescu, Stability in the Dynamics of Metal Cutting. Elsevier (1990).

[8] D. Chu, and F. C. Moon, Dynamic Instabilities in Magnetically Levitated Models. Journal of Applied Physics,54(3), 16191625 (1983).

[9] M. A. Davies, and T. J. Burns, Thermomechanical Oscillations in Material Flow During High-Speed Ma-chining. Philosophical Transactions of the Royal Society, 359, 821846 (2001).

[10] K. Engelborghs, and D. Roose, Numerical Computation of Stability and Detection of Hopf Bifurcations ofSteady State Solutions of Delay Differential Equations. Advances in Computational Mathematics, 10, 271289(1999).

[11] M. S. Fofana, Delay Dynamical Systems with Applications to Machine-Tool Chatter. Ph.D. thesis, Universityof Waterloo, Department of Civil Engineering (1993).

[12] J. Gradiek, E. Govekar, and I. Grabec, I, Time Series Analysis in Metal Cutting: Chatter versus Chatter-FreeCutting. Mechanical Systems and Signal Processing, 12(6), 839854 (1998).

[13] N. H. Hanna, and S. A . Tobias, A Theory of Nonlinear Regenerative Chatter. Transactions of the AmericanSociety of Mechanical Engineers - Journal of Engineering for Industry, 96(1), 247255 (1974).

[14] G. Herrmann, and I.-C. Jong, On the Destabilizing Effect of Damping in Nonconservative Elastic Systems.Transactions of the ASME, 592597 (1965).

[15] C. S. Hsu, Application of the Tau-Decomposition Method to Dynamical Systems Subjected to Retarded Fol-lower Forces. Transactions of the American Society of Mechanical Engineers - Journal of Applied Mechanics,37(2), 259-266 (1970).

[16] T. Insperger, and G. Stpn, Semi-discretization Method for Delayed Systems. International Journal forNumerical Methods in Engineering, 503-518 (2002).

17


18/19

[17] M. Johnson, and F. C. Moon, Nonlinear Techniques to Characterize Prechatter and Chatter Vibrations inthe Machining of Metals. International Journal of Bifurcation and Chaos, 11(2), 449467 (2001).

[18] M. A. Johnson, Nonlinear Differential Equations with Delay as Models for Vibrations in the Machining ofMetals. Ph.D. thesis, Cornell University (1996).

[19] T. Kalmr-Nagy, J. R. Pratt, M. A. Davies, and M. D. Kennedy, Experimental and Analytical Investigationof the Subcritical Instability in Turning. In: Proceedings of the 1999 ASME Design Engineering TechnicalConferences. DETC99/VIB-8060 (1999).

[20] T. Kalmr-Nagy, G. Stpn, and F. C. Moon, Subcritical Hopf Bifurcation in the Delay Equation Model forMachine Tool Vibrations. Nonlinear Dynamics 26, 121-142 (2001).

[21] T. Kalmr-Nagy, Delay-Differential Models of Cutting Tool Dynamics with Nonlinear and Mode-CouplingEffects. Ph.D. thesis, Cornell University (2002).

[22] S. Kato, E. Marui, and H. Kurita, Some Considerations on Prevention of Chatter Vibration in BoringOperations. Journal of Engineering for Industry, 717-730 (1969).

[23] J. Kiusalaas, and H. E. Davis, On the Stability of Elastic Systems Under Retarded Follower Forces. Inter-national Journal of Solids and Structures, 6(4), 399409 (1970).

[24] G. Litak, J. Warminski, and J. Lipski, Self-Excited Vibrations in Cutting Process. 4th Conference onDynamical Systems - Theory and Applications, 193-198 (1997).

[25] B. Milisavljevich, M. Zeljkovic, and R. Gatalo, The Nonlinear Model of a Chatter and Bifurcation. Pages391398 of: Proceedings of the 5th Engineering Systems Design and Analysis Conference. ASME (2000).

[26] F. C. Moon, and H. D. I. Abarbanel, Evidence for Chaotic Dynamics in Metal Cutting, and Classification ofChatter in Lathe Operations. Pages 1112,2829 of: Moon, F. C. (ed), Summary Report of a Workshop onNonlinear Dynamics and Material Processing and Manufacturing, organized by the Institute for Mechanicsand Materials, held at the University of California, San Diego (1995).

[27] F. C. Moon, and T. Kalmr-Nagy, Nonlinear Models for Complex Dynamics in Cutting Materials. Philo-sophical Transactions of the Royal Society, 359, 695711 (2001).

[28] A. Nayfeh, C. Chin, and J. Pratt, Applications of Perturbation Methods to Tool Chatter Dynamics. In:Moon, F. C. (ed), Dynamics and Chaos in Manufacturing Processes. John Wiley and Sons (1998).

[29] N. Olgac, and R. Sipahi, An Exact Method for the Stability Analysis of Time-Delayed LTI Systems. IEEETransactions on Automatic Control, 47(5), 793-797 (2002).

[30] P. L. B. Oxley, Mechanics of Machining: An Analytical Approach to Assessing Machinability. New York:John Wiley and Sons (1989).

[31] Y. G. Panovko, and I. I. Gubanova, Stability And Oscillations of Elastic Systems. New York: ConsultantsBureau (1965).

[32] J. Pratt, Vibration Control for Chatter Suppression with Application to Boring Bars. Ph.D. thesis, VirginiaPolytechnic Institute and State University (1996).

[33] N. Saravanja-Fabris, and A. F. DSouza, Nonlinear Stability Analysis of Chatter in Metal Cutting. Transac-tions of the American Society of Mechanical Engineers - Journal of Engineering for Industry, 96(2), 670675(1974).

[34] H. M. Shi, and S. A. Tobias, Theory of Finite Amplitude Machine Tool Instability. International Journal ofMachine Tool Design and Research, 24(1), 4569 (1984).

[35] G. V. Stabler, Advances in Machine Tool Design and Research. Pergamon Press Ltd., Oxford (1964). Chap.The Chip Flow Law and its Consequences, page 243.

[36] G. Stpn, Modelling Nonlinear Regenerative Effects in Metal Cutting. Philosophical Transactions of theRoyal Society, 359, 739757 (2001).

[37] G. Stpn, R. Szalai, and T. Insperger, Nonlinear Dynamics of High-Speed Milling Subjected to RegenerativeEffect. This volume (2003).

[38] E. Stone, and S. A. Campbell, Stability and Bifurcation Analysis of a Nonlinear DDE model for Drilling.Submitted to Journal of Nonlinear Science (2002).

[39] S. A. Tobias, Machine Tool Vibration. London: Blackie and Son Limited (1965).

[40] S. A. Tobias, and W. Fishwick, The Chatter of Lathe Tools Under Orthogonal Cutting Conditions. Trans-actions of the American Society of Mechanical Engineers, 80(5), 10791088 (1958).

[41] M. Wiercigroch, and E. Budak, Sources of Nonlinearities, Chatter Generation and Suppression in MetalCutting. Philosophical Transactions of the Royal Society, 359, 663693 (2001).

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[42] M. Wiercigroch, and A. H-D. Cheng, Chaotic and Stochastic Dynamics of Orthogonal Metal Cutting. Chaos,Solitons & Fractals, 8(4), 715726 (1997).

[43] M. Wiercigroch, and A. M. Krivtsov, Frictional Chatter in Orthogonal Metal Cutting. Philosophical Trans-actions of the Royal Society, 359, 713738 (2001).

[44] N. van de Wouw, R. P. H. Faassen, J. A. J. Oosterling, and H. Nijmeijer, Modelling of High-Speed Millingfor Prediction of Regenerative Chatter. This volume (2003).

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kalmar-nagy mode-coupled regenerative machine tool vibrations

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