Mode-Coupled Regenerative Machine Tool Vibrations

Tamás Kalmár-Nagy1, Francis C. Moon2

1United Technologies Research Center, 411 Silver Lane, East Hartford, CT 061082Sibley 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 e¤ect (time delay) can produce an instability criteria that admits low-levelor zero chip thickness chatter.

1 IntroductionOne 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 Kalmár-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], Kalmár-Nagy et al. [20], Stépán [36], Kalmár-Nagy [21], Stone and Campbell [38], Stépánet al. [37]). Even though the classical model (Tobias [39]) with nonlinear cutting force is quite successful inpredicting the onset of chatter (Kalmár-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 insu¢cient 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 multipledegree-of-freedom tool dynamics and the regenerative e¤ect 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 ’‡utter’ 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 sti¤ness matrix which can lead to ‡utter and chatter.Tobias called this mode-coupled chatter. Often this model of chatter is analyzed without the regenerative e¤ect.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 ‡utter model with time delay may also be applicable toaeroelastic problems in rotating machinery where the ‡uid 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 e¤ect. 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 CuttingThe 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

1

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 =­2π

f (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 e¤ect. 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 τ of one revolution of the workpiece

τ =2π­

(2)

The phenomenon of the large amplitude vibration of the tool is known as chatter. A good description of chatter isgiven 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 a¤ects on surface …nish, 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

² The tool always appears to vibrate while cutting. The amplitude of the vibration distinguishes chatter fromsmall-amplitude vibrations.

² The tool vibration typically has a strong periodic component which approximately coincides with a naturalfrequency of the tool.

² 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], Gradišek et al. [12]).

2

Figure 2: Oblique chip formation model

2.1 Oblique CuttingAlthough many practical machining processes can adequately be modeled as single degree-of-freedom and orthog-onal, more accurate models demand a chip formation model in which the cutting velocity is not normal to thecutting edge.Figure 2 shows the usual oblique chip formation model, where the inclination angle i (measuredbetween the cutting edge and the normal to the cutting velocity in the plane of the machined surface) is not zero,as in orthogonal cutting. The cutting velocity is denoted by vC, the chip ‡ow angle is ηc, the thickness of theundeformed chip is f , the deformed chip thickness is f2 and the chip width is w. The three dimensional cuttingforce acting on the tool insert is decomposed into three mutually orthogonal forces: FC, FT , FR. The cuttingforce FC is the force in the cutting direction, the thrust force FT is the force normal to the cutting directionand machined surface, while the radial force FR is normal to both FC and FT . While orthogonal cuting can bemodeled as a 2-dimensional process, oblique cutting is a true three-dimensional plastic ‡ow problem (Oxley [30]).

3 3 DOF Model of Metal CuttingFigure 3 shows a tool with a cutting chip (insert) both in undeformed and deformed state of the tool.The three

Figure 3: 3 DOF metal cutting model

3

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

Figure 4: 3 DOF lumped-parameter model

The equations of motion are the following

mÄz + cz _z + kzz = Fz (3)

mÄx + cx _x + kxx = Fx (4)

I Äφ+ cφ _φ + kφφ = 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…ned. An inertialframe (I,J,K) …xed to the tool and a moving frame (i, j,k) …xed to the cutting velocity.The force acting on theinsert can then be written as

F = ¡FT I+ FRJ ¡ FCK (6)

orF = Fxi+ Fy j+ Fzk (7)

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

Fx = ¡FT (8)Fy = FC sinβ + FR cosβ (9)Fz = FR sinβ ¡ FC cosβ (10)

The bending also results in a pitch ψ (shown in Figure 6). This is not a separate degree of freedom, but nonethelessit will in‡uence the inclination angle.

The following assumptions are used in deriving the equations of motion

² The forces that act on the insert are steady-state forces

² The width of cut w (y-position) is constant

² All displacements are small² Yaw is negligible

Steady-state forces refer to time averaged quantities. The e¤ect of rate-dependent cutting forces were studiedby Saravanja-Fabris and D’Souza [33], Chiriacescu [7], Moon and Kalmár-Nagy [27]. Next we …nd the position ofthe tooltip in the …xed system of the platform. To do so we have to …nd the rotation matrix R that describes therelationship between the moving frame (i, j,k) and the …xed frame (I,J,K).

(i j k

)= R

(I J K

)(11)

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

R = R2R1 (12)

4

Figure 5: Forces on the tooltip

Figure 6: Direction of cutting velocity

5

The cross section is …rst rotated about I by the pitch angle ψ. The corresponding rotation matrix is

R1 =

1 0 00 cψ sψ0 ¡sψ cψ

(13)

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

R2 =

cφ 0 sφ0 1 0¡sφ 0 cφ

(14)

R can then be calculated by (12)

R =

cφ ¡sφsψ cψsφ0 cψ sψ¡sφ ¡cφsψ cψcφ

(15)

The position of the tooltip can be expressed in the …xed frame as

r¤ = R

rx

0rz

=

rxcφ + rzcψsφrzsψ

rzcψcφ ¡ rxsφ

(16)

The roll producing moment can then be calculated as

My = (r¤ £ F) ¢ j= FT (rxsφ¡ rzcφcψ) + FC cβ (rxcφ + rzcψsφ)¡ FRsβ (rxcφ+ rzcψsφ) (17)

In the following we assume small displacements and small angles and neglect nonlinear terms. The angle β istaken to be proportional to the vertical displacement, i.e. β = ¡nz (n > 0) and so is the pitch, i.e. ψ = kz(k > 0).

mÄx + cx _x + kxx = ¡FT (18)

mÄz + cz _z + kzz = ¡(FC + nz ¹FR

)(19)

I Äφ+ cφ _φ + kφφ = My = φ(rx ¹FT + rz ¹FC

)¡ rzFT + rxFC + nzrx ¹FR (20)

where ¹FC, ¹FR, ¹FT denotes the constant term in FC, FR and FT , respectively.

3.1 Cutting ForcesGenerally we assume that the cutting forces FC, FT , FR depend only on the inclination angle i and chip thicknessf (see Figure 2), and the rake angle α (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 α+mCt1 + FC0 (21)

FT = ¡lT α+mT t1 + FT0 (22)

where mC and mT are cutting force coe¢cients, while lC and lT are angular cutting force coe¢cients (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 iFC cos i (i¡ sinα)¡ FT

sin2 i sinα + cos2 i(23)

where Stabler’s Flow Rule (Stabler [35]) ηC = i was used. The e¤ective rake angle depends on the initial rakeangle and the roll

α = α0 ¡ φ (24)

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

i = i0 ¡ψ (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 con…guration, while the solid

6

Figure 7: Motion of the tooltip

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

t1 = t10 + x¡ xτ + rz sin (φ ¡ φτ ) ¼ t10 + x¡ xτ + rz (φ ¡ φτ ) (26)where xτ and φτ denote the delayed values x (t¡ τ ) and φ (t¡ τ), respectively. Then the cutting forces can bewritten as

FC = mC (x¡ xτ) + (lC + rzmC )φ ¡ rzmCφτ +

¹FC︷ ︸︸ ︷mC t10 + FC0 ¡ α0lC (27)

FT = mT (x¡ xτ ) + (lT + rzmT )φ ¡ rzmTφτ +mT t10 + FT0 ¡ α0lT (28)If the initial inclination angle is assumed to be zero, the expression for FR will simplify

FR = k( ¹FT + (sinα0 ¡ 1) ¹FC + t10 (mC (1¡ sinα0)¡mT )

)z (29)

3.2 The Equations of MotionSubstituting (27-28) into equations (18-20) and eliminating the constant (by translation of the variables) results

mÄz + cz _z + kzz = ¡n ¹FRz ¡ mC (x¡ xτ )¡ (lC + rzmC) φ + rzmCφτ (30)

mÄx + cx _x + kxx = ¡mT (x¡ xτ )¡ (lT + rzmT ) φ+ rzmT φτ (31)

I Äφ + cφ _φ+ kφφ = ¡rzmT (x ¡ xτ) ¡ rz (lT + rzmT ¡mCt10 ¡ FC0 + α0lC)φ + r2zmT φτ (32)where now (x, z, φ) represent deviations from the steady values of the original displacements. As we can see, thex and φ 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

Äx + 2ζxωx _x +(ω2

x +mT

m

)x +

1m

(lT + rzmT )φ =mT

mxτ + rz

mT

mφτ (33)

Äφ+ 2ζφωφ _φ+rzmT

Ix +

(ω2

φ +rz

I(lT + rzmT ¡mCt10 ¡ FC0 + α0lC)

)φ =

rzmT

Ixτ + r2z

mT

Iφτ (34)

where

ωx =√

kx

m, ωφ =

√kφ

I(35)

By introducing the nondimensional time and displacement

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

x00 + 2ζxωxT x0 +(ω2

x + mT

m

)T 2x + 1

m(lT + rzmT )

T 2

Xφ = mT

mT 2xτ + rz

mT

mT 2

Xφτ (37)

φ00+ 2ζφωφTφ0 + rmT

IT 2Xx +

[ω2

φ + rz

I(lT + rzmT ¡ FC0 ¡mCt10 + α0lC)

]T 2φ =

rzmT

IT 2Xxτ + r2z

mT

IT 2φτ (38)

7

With the choice of the following scales

T = 1ωx

X =√

Im

(39)

the equations assume the form (τ = ωxτ)

x00 + 2ζxx0 + k11x + k12φ = r11xτ + r12φτ (40)

φ00 + 2ζφφ0 + k21x + k22φ = r21xτ + r22φτ (41)

where

k11 = 1 +mT

ω2xm

k12 =lT + rzmT

ω2xp

Im(42)

k21 =rzmT

ω2xp

Imζφ = ζφ

ωφ

ωx(43)

k22 =(

ωφ

ωx

)2

+rz

ω2xI

(lT + rzmT ¡ mC t10 ¡ FC0 + α0lC ) (44)

r11 =mT

ω2xm

r12 = r21 =rzmT

ω2xp

Im(45)

r22 =r2zmT

ω2x

pIm

(46)

Note that the sti¤nesses k12 and k21 are di¤erent. This is characteristic of nonconservative systems (Bolotin [4],Panovko and Gubanova [31]). In many mechanical systems this nonconservativeness is due to the presence offollowing forces.

4 Estimation of Model ParametersIn the following we estimate di¤erent 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 boring 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 sti¤nesses for such a cantileveredbeam can be in the following ranges

kx ' 104 ¥ 107Nm

(47)

kz ' 105 ¥ 107 Nm

(48)

kφ ' 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

ωx ' 100¥ 5000 rads

(50)

ωz ' 100¥ 10000rads

(51)

ωφ ' 1000¥ 10000rads

(52)

The ratio ωφωx

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

4.2 Cutting Force ParametersExperimental 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 di¤erent rake angles (α = ¡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

8

Figure 8: Forces in oblique cutting of 0.2% carbon steel. α = ¡5± (top) and α = 5± (bottom). f = 0.125 mm. AfterOxley (1989)

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

9

cutting speed, forces were taken from these graphs at the value 200 m/s of the cutting speed and plotted againstrake angle (Figure 9).The constants lC and lT were found as the slope of the lines corresponding to t1 = 0.25 mm

lC = 1580 Nrad

, lT = 3150 Nrad

(53)

A linear relationship is assumed between forces at zero rake angle and chip thickness, i.e.

FC = FC0 + mct1 (54)

FT = FT0 +mT t1 (55)where these coe¢cients were determined to be

mC = 6 ¤ 106 Nm

FC0 = 458 N (56)

mT = 1.65 ¤ 106 Nm

FT0 = 784 N (57)

4.3 Model Parameters

Sincerz

ω2xI

(lT + rzmT ¡mCt10 ¡ FC0 + α0lC )¿(

ωφ

ωx

)2

(58)

this term will be neglected, i.e.

k22 =(

ωφ

ωx

)2

(59)

Also, the term r22 is very small, so it is neglected

r22 = 0 (60)

5 Analysis of the ModelWith the approximations (59, 60) the model (40, 41) can be written as the matrix equation

Äx+ C_x +Kx = Rxτ (61)

where

x =(

xφ

)

and the matrices are given by

C =[

2ζx 00 2ζφ

], K =

[1 + p a+ pq

pq k22

], (62)

R =[

p qq 0

](63)

Here we introduced the parameters

p =mT

ω2xm

, q =rz

X= rz

√mI

(64)

where constants a and k22 are

a =lT

ω2xp

Im, k22 =

(ωφ

ωx

)2

(65)

It is characteristic of systems with nonsymmetric sti¤ness matrix, that they can lose stability either by divergence(buckling) or by ‡utter. Chu and Moon [8] examined divergence and ‡utter instabilities in magnetically levitatedmodels. Kiusalaas and Davis [23] studied stability of elastic systems under retarded follower forces. Recentlyseveral numerical methods were proposed to investigate stability of linear time-delay systems (see Chen et al. [6],Engelborghs and Roose [10], Insperger and Stépán [16], Olgac and Sipahi [29]).

5.1 Classical LimitIf q = 0 the equations reduce to

x00 + 2ζxx0+ (1 + p)x + aφ = pxτ (66)φ00 + 2ζφφ0 + cφ = 0 (67)

The φ-equation is uncoupled from the x-equation and reduces to that of a damped oscillator. Its equilibriumφ = 0 is asymptotically stable and thus it does not a¤ect the stability of the x-equation. In this case we recoverthe 1 DOF classical model (Tobias and Fishwick [40]).

10

5.2 Stability Analysis of the Undamped System without DelayFirst we perform linear stability analysis of the system

Äx+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 = deiωt (70)

we obtain the characteristic polynomials (K¡ ω2I

)d = 0 (71)

which have nontrivial solution if the determinant of K¡ ω2I is zero∣∣∣∣

k11 ¡ ω2 k12k21 k22 ¡ ω2

∣∣∣∣ =(k11 ¡ ω2) (

k22 ¡ ω2) ¡ k21k12 = 0 (72)

The characteristic equation for the coupled system becomes

ω4 ¡ (k11 + k22)ω2 + k11k22 ¡ k21k12 = 0 (73)

Divergence (static de‡ection, buckling) occurs when ω = 0 (or detK = 0), that is when

k11k22 ¡ k21k12 = 0 (74)

If ω 6= 0, then the characteristic equation (73) can be solved for ω2 as

ω2 =12

(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 ‡utter boundaries, respectively. With the sti¤ness 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 = 12q2

(k22 ¡ aq§

√4k22q2 + (k22 ¡ aq)2

)(79)

and the ‡utter 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 di¤erent stability regionsare indicated by the root location plots.

5.3 Stability Analysis of the 2 DOF Model with DelayIn this section we include the delay terms in the analysis. In order to be able to study how these terms in‡uencethe stability of the system, we introduce a new parameter, similar to the overlap factor (Tobias [39]).

First we analyze the system with no damping:

Äx +Kx = µRxτ (82)

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

det(¡λ2I+K¡ µe¡λτR

)= 0 (83)

11

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

λ4 +(k11 + k22 ¡ µpe¡λτ

)λ2 + k11k22 ¡ k12k21+

µe¡λτ (q (k12 + k21)¡ pk22) ¡ µ2q2e¡2λτ = 0 (84)

Substituting λ = iω, ω ¸ 0 yields a complex equation that can be separated into the two real ones (the secondequation was divided by µ sin (τω) 6= 0)

ω4 ¡ (k11 + k12)ω2 + k11k22 ¡ k12k21+ (85)

µ cos (τω)(pω2 + q (k12 + k21)¡ pk22

)¡ µ2q2 cos (2τω) = 0

pω2 + q (k12 + k21) ¡ pk22 + 2µq2 cos (τω) = 0 (86)

We solve the second equation for cos (τω)

cos (τω) =pω2 + q (k12 + k21) ¡ pk22

¡2µq2(87)

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

ω4 ¡ (k11 + k22) ω2 + k11k22 ¡ k12k21 + µ2q2 = 0 (88)

Divergence occurs where ω = 0, that is where

k11k22 ¡ k12k21 + µ2q2 = 0

Substituting the elements of the sti¤ness matrix as given in (62) yields

¡ (q (a+ q (p¡ µ)) (p¡ µ)) + k22 (1 + p¡ p µ) = 0 (89)

which can be solved for p as

12 q2

(k22 (1¡ µ) + q (2 q µ¡ a)§

√(k22 (µ¡ 1) + q (a¡ 2µq))2 + 4 q2 (k22 + q µ (a¡ q µ))

)(90)

The change of the divergence boundary is shown in Figure 11 (top, middle, bottom) for µ = 0.1, 0.5 and 1 whilethe delay was set to 1. Flutter occurs for ω > 0, and the boundary can be found by numerically solving equations(86, 88) for p and q for a given µ. Figure 12 shows the ‡utter boundary for a small µ (0.01) together with the‡utter boundary (80). Figure 13 shows how this boundary changes with increasing µ (µ = 0.1, 0.5, 1). Figure 14shows the full stability chart, complete with both the divergence and ‡utter boundaries, for µ = 1. To validate thisstability chart the parameter space (p, q) was gridded and the delay-di¤erential 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 15τ intervals of which the …rst 5τ intervals were discarded. Stability wasdetermined by whether the amplitude of the solution grew or decayed. Dark dots correspond to stable numericalsolutions. This …gure 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.

12

Figure 11: The change of the divergence boundary for system (80), τ = 1. µ = 0.1, 0.5, 1 for top, middle and bottomFigures, respectively

13

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

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

14

Figure 14: Stability chart for the undamped system (82), µ = 1, τ = 1

Now we examine the e¤ect of damping on the size of stability regions. It is an important step, as it is known(Herrmann and Jong [14]) that damping can have a destabilizing e¤ect in nonconservative systems. The dampingcoe¢cients ζx and ζφ are taken to be 0.01, while the ratio of frequencies ωφ/ωx was changed in Figure 15 (thisis the same as keeping this ratio …xed and increasing ζφ). As the …gure shows, the size of the stability regionsincreases with added damping. And …nally, we show how the lobes of the conventional stability chart deform withthe added parameter q (0 · q · 1). Figure 16 shows that increasing q results in the ’birth’ of unstable regions.These upside-down lobes are actually lobes of the classical model for p < 0 (p is the nondimensional cutting forcecoe¢cient which is positive). In our model these lobes become a new source of instability, where the classicalmodel would predict stable behavior.

6 ConclusionsA new 3 DOF model derived may help explain at least two phenomena in metal cutting. The …rst is that o¤-centering the tool might help avoiding chatter. The second phenomenon is the observation that small amplitudetool vibrations can arise below the classical stability boundary. As shown, the added degrees of freedom resultin unstable regions below the one predicted by the one DOF classical model. To summarize the importantobservations:

² The 3-DOF model results in coupling between twist and lateral bending

² The model can exhibit both divergence and ‡utter instabilities

² Damping seems to increase the size of stability regions

² The tool o¤set produces new regions of instability (the upside-down lobes)

This model is based on the assumption of rate-independent cutting forces, i.e. forces that do not exhibithysteresis (Moon and Kalmár-Nagy [27]). It does not include temperature e¤ects either (Davies and Burns [9]).

Finally, only the analysis of a full nonlinear model could characterize the nature of vibrations and provideestimates of vibration amplitudes for the low chip thickness unstable regions.

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359, 793–819 (2001).

15

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

16

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

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