Kalmar-nagy Mode-coupled Regenerative Machine Tool Vibrations

pp. 129-149, in Guenter Radons and Raimund Neugebauer eds.: Nonlinear Dynamics of Production Systems. Wiley-VCH, Berlin, 2004

Mode-Coupled Regenerative Machine Tool Vibrations 1

Tamás Kalmár-Nagy1 , Francis C. Moon2 United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06108 2 Sibley 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 and analyzed. One mode is shown to be stable and decoupled from the other two, and thus the stability of the system can be determined by analyzing these two modes. It is shown that this mode-coupled nonconservative cutting tool model including the regenerative effect (time delay) can produce an instability criteria that admits low-level or 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 are the 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 Moon and Kalmár-Nagy [27] have addressed some of these issues. Numerous researchers investigated single degree-offreedom 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], Stépán et al. [37]). Even though the classical model (Tobias [39]) with nonlinear cutting force is quite successful in predicting the onset of chatter (Kalmár-Nagy et al., [19]), it cannot possibly account for all phenomena displayed in real cutting experiments. Single degree-of-freedom deterministic time-delay models have been insufficient so far to explain low-amplitude dynamics below the stability boundary. Also, real tools have multiple degrees of freedom. In addition to horizontal and vertical displacements, tools can twist and bend. Higher degree-of-freedom models 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 will permit low-level instabilities. Coupled-mode models in aeroelasticity or vehicle dynamics may exhibit so-called ’flutter’ or dynamics instabilities (see e.g. Chu and Moon [8]) when there exists a non-conservative force in the problem. One example is the follower force torsion-beam problem as in Hsu [15]. In the present work we assume that the chip removal forces rotate 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 effect. In this paper we will show that the combination of mode-coupling nonconservative model and a time delay can produce an instability criteria that admits low-level or zero chip thickness chatter. There is no claim in this paper to having solved the random- or chaotic low level dynamics since only linear stability analysis is presented in this paper. But the results shown below provide an incentive to extend this model into the nonlinear regime. A dynamic model with the combination of 2-degree-of-freedom flutter model with time delay may also be applicable to aeroelastic problems in rotating machinery where the fluid forces in the current cycle depend on eddies generated in 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, together with 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 conclusions are drawn in Section 6.

2

Metal Cutting

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

1

Workpiece Machined surface

W

v tool

Figure 1: Turning between the workpiece and the tool. In turning the material is removed from a rotating workpiece, as shown in Figure 1. The cylindrical workpiece rotates with constant angular velocity Ω [rad/s] and the tool is moving along the axis of the workpiece at a constant rate. The feed f is the longitudinal displacement of the tool per revolution of the workpiece, and thus it is also the nominal chip thickness. The translational speed of the tool is then given by vtool =

Ω f 2π

(1)

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

2π Ω

(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 metal is often accompanied by a violent relative vibration between work and tool which is called the chatter. Chatter is 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 to be 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 from small-amplitude vibrations. • The tool vibration typically has a strong periodic component which approximately coincides with a natural frequency of the tool. • The amplitude of the oscillation is typically modulated and often in a random way. The amplitude modulation 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]) or vibrations 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 carried out in machining to characterize and quantify the dynamics of metal cutting. Recently a number of researchers have 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 for cutting and claim that the vibrations are random noise (Wiercigroch and Cheng [42], Gradišek et al. [12]).

2

cutting tool insert f2 i

h

C

FR FC

FT

f W

vC

machined surface workpiece

Figure 2: Oblique chip formation model

2.1

Oblique Cutting

Although many practical machining processes can adequately be modeled as single degree-of-freedom and orthogonal, more accurate models demand a chip formation model in which the cutting velocity is not normal to the cutting edge.Figure 2 shows the usual oblique chip formation model, where the inclination angle i (measured between 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 flow angle is ηc , the thickness of the undeformed chip is f , the deformed chip thickness is f2 and the chip width is w. The three dimensional cutting force acting on the tool insert is decomposed into three mutually orthogonal forces: FC , FT , FR . The cutting force FC is the force in the cutting direction, the thrust force FT is the force normal to the cutting direction and machined surface, while the radial force FR is normal to both FC and FT . While orthogonal cuting can be modeled as a 2-dimensional process, oblique cutting is a true three-dimensional plastic flow problem (Oxley [30]).

3

3 DOF Model of Metal Cutting

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

tool insert cutting edge

a z

rz

x

vC

rx

y

Figure 3: 3 DOF metal cutting model

3

f

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 effective mass is equivalent to modal mass for a distributed beam).

FC FT

chip cx

m kf ,cf

z

kx

f

x

cz

kz

Figure 4: 3 DOF lumped-parameter model The equations of motion are the following m¨ z + cz z˙ + kz z = Fz

(3)

m¨ x + cx x˙ + kx x = Fx

(4)

I φ¨ + cφ φ˙ + kφ φ = 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 defined. An inertial frame (I, J, K) fixed to the tool and a moving frame (i, j, k) fixed to the cutting velocity.The force acting on the insert can then be written as F = −FT I + FR J − FC K (6) or

(7)

F = Fx i + Fy j + Fz k where i, j, k are unit vectors in the x, y, z directions, respectively.

(8)

Fx = −FT

Fy = FC sin β + FR cos β

(9)

Fz = FR sin β − FC cos β

(10)

The bending also results in a pitch ψ (shown in Figure 6). This is not a separate degree of freedom, but nonetheless it 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 studied by Saravanja-Fabris and D’Souza [33], Chiriacescu [7], Moon and Kalmár-Nagy [27]. Next we find the position of the tooltip in the fixed system of the platform. To do so we have to find the rotation matrix R that describes the relationship 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 {ψ, φ} we express R as a product of two consecutive planar rotations (Pitch-Roll system) R = R2 R1 (12)

4

k

z

i

j

x

FC

y

vC K

J

I

FR

FT

Figure 5: Forces on the tooltip

z b y

FC y

FR Figure 6: Direction of cutting velocity

5

The cross section is first rotated about I by the pitch angle ψ.  1 0 R1 =  0 cψ 0 −sψ where the abbreviations c = cos, s = sin were used. The through the roll angle φ (with respect to the toolholder)  cφ R2 =  0 −sφ

R can then be calculated by (12)



cφ R= 0 −sφ

The corresponding rotation matrix is  0 sψ  cψ

(13)

 sφ 0  cφ

(14)

second rotation is about the J2 (the rotated J) axis 0 1 0

−sφsψ cψ −cφsψ

 cψsφ sψ  cψcφ

The position of the tooltip can be expressed in the fixed frame as     rx rx cφ + rz cψsφ ∗  rz sψ r = R 0  =  rz rz cψcφ − rx sφ

(15)

(16)

The roll producing moment can then be calculated as

My = (r∗ × F) · j = FT (rx sφ − rz cφcψ) + FC cβ (rx cφ + rz cψsφ) − FR sβ (rx cφ + rz cψsφ)

(17)

In the following we assume small displacements and small angles and neglect nonlinear terms. The angle β is taken to be proportional to the vertical displacement, i.e. β = −nz (n > 0) and so is the pitch, i.e. ψ = kz (k > 0). m¨ x + cx x˙ + kx x = −FT (18) ¯ m¨ z + cz z˙ + kz z = − FC + nz FR (19) I φ¨ + cφ φ˙ + kφ φ = My = φ rx F¯T + rz F¯C − rz FT + rx FC + nzrx F¯R

(20)

where F¯C , F¯R , F¯T 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 α (see Figure 3). We again emphasize that the chip width w is considered constant in the present analysis. Our hypothesis here is that FC and FT depend linearly on both the rake angle and chip thickness (see Section 4.2) in the following manner FC = −lC α + mC t1 + FC0

(21)

FT = −lT α + mT t1 + FT 0

(22)

where mC and mT are cutting force coefficients, while lC and lT are angular cutting force coefficients (they show how strong the force dependence is on rake angle). The variable t1 is the chip thickness variation (the deviation from the nominal chip thickness). The constant forces FC0 and FT 0 arise from cutting at a nominal chip thickness. The radial cutting force can be expressed as (Oxley [30]) FR = sin i

FC cos i (i − sin α) − FT sin2 i sin α + cos2 i

(23)

where Stabler’s Flow Rule (Stabler [35]) ηC = i was used. The effective rake angle depends on the initial rake angle 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 delayed ones). The displacement of the tooltip is due to translational and rotational motion as shown in Figure 7.Here the dashed line corresponds to the position vector of the tooltip in the undeformed configuration, while the solid

6

rx

f

z

rz

k

x

i

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 is then given by t1 = t10 + x − xτ + rz sin (φ − φτ ) ≈ 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 be written as ¯C F

(27)

FC = mC (x − xτ ) + (lC + rz mC ) φ − rz mC φτ + mC t10 + FC0 − α0 lC FT = mT (x − xτ ) + (lT + rz mT ) φ − rz mT φτ + mT t10 + FT 0 − α0 lT

(28)

FR = k F¯T + (sin α0 − 1) F¯C + t10 (mC (1 − sin α0 ) − mT ) z

(29)

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

3.2

The Equations of Motion

Substituting (27-28) into equations (18-20) and eliminating the constant (by translation of the variables) results m¨ z + cz z˙ + kz z = −nF¯R z − mC (x − xτ ) − (lC + rz mC ) φ + rz mC φτ

(30)

m¨ x + cx x˙ + kx x = −mT (x − xτ ) − (lT + rz mT ) φ + rz mT φτ ˙ ¨ I φ + cφ φ + kφ φ = −rz mT (x − xτ ) − rz (lT + rz mT − mC t10 − FC0 + α0 lC ) φ + rz2 mT φτ

(31) (32)

where now (x, z, φ) represent deviations from the steady values of the original displacements. As we can see, the x 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 1 mT mT (lT + rz mT ) φ = xτ + rz φτ m m m rz rz mT rz mT mT x + ωφ2 + (lT + rz mT − mC t10 − FC0 + α0 lC ) φ = xτ + rz2 φτ φ¨ + 2ζφ ωφ φ˙ + I I I I x ¨ + 2ζx ωx x˙ + ωx2 +

mT m

x+

(33) (34)

where kx , ωφ = m By introducing the nondimensional time and displacement ωx =

tˆ = t/T ˆ0 + ωx2 + x ˆ00 + 2ζx ωx T x φ00 + 2ζφ ωφ T φ0 +

mT m

T 2x ˆ+

kφ I

(35)

(36)

x ˆ = x/X 2

2

mT 2 1 T mT T (lT + rz mT ) φ= T xτ + rz φτ m X m m X

rmT 2 rz T Xx (lT + rz mT − FC0 − mC t10 + α0 lC ) T 2 φ = ˆ + ωφ2 + I I rz mT 2 mT 2 T Xxτ + rz2 T φτ I I

7

(37)

(38)

With the choice of the following scales T =

1 ωx

X=

I m

(39)

the equations assume the form (ˆ τ = ωx τ ) ˆ0 + k11 x ˆ + k12 φ = r11 x ˆτˆ + r12 φτˆ x ˆ00 + 2ζx x

(40)

ˆ + k22 φ = r21 x ˆτˆ + r22 φτˆ φ00 + 2ζˆφ φ0 + k21 x

(41)

where mT ωx2 m rz mT √ = ωx2 Im

lT + rz mT √ ωx2 Im ωφ ζˆφ = ζφ ωx k12 =

k11 = 1 + k21

ωφ ωx mT = 2 ωx m

2

k22 = r11

r22 =

rz (lT + rz mT − mC t10 − FC0 + α0 lC ) ωx2 I rz mT √ r12 = r21 = ωx2 Im

+

rz2 mT √ ωx2 Im

(42) (43) (44) (45) (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 the model.

4.1

Structural Parameters

The toolholder is assumed to be a rectangular steel beam. The length of the toolholder is relatively short for normal cutting, while it can be longer for boring operations (see Kato et al. [22]). So we assume l to be between 0.05 m and 0.3 m. The width and height are usually of order of a centimeter. The stiffnesses for such a cantilevered beam can be in the following ranges N m N kz ' 105 ÷ 107 m

kx ' 104 ÷ 107

kφ ' 1000 ÷ 10000

(47) (48) N rad

(49)

Since a lumped parameter approximation is used, the mass at the end of the massless beam is assumed to be the modal mass. The vibration frequencies are then rad s rad ωz ' 100 ÷ 10000 s rad ωφ ' 1000 ÷ 10000 s ωx ' 100 ÷ 5000

The ratio

4.2

ωφ ωx

(50) (51) (52)

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

Cutting Force Parameters

Experimental cutting force data during machining of 0.2% carbon steel is shown in Figure 8 (Oxley [30]).The graph shows the forces FC and FT for different 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. After Oxley (1989)

FC [kN] 4 3.5 3 2.5 2 1.5 1 0.5

a [rad]

FT [kN] 2 1.75 1.5 1.25 1 0.75 0.5 0.25

a [rad]

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 against rake angle (Figure 9).The constants lC and lT were found as the slope of the lines corresponding to t1 = 0.25 mm N N , lT = 3150 rad rad A linear relationship is assumed between forces at zero rake angle and chip thickness, i.e. lC = 1580

(53)

FC = FC0 + mc t1

(54)

FT = FT 0 + mT t1

(55)

where these coefficients were determined to be mC = 6 ∗ 106

N m

FC0 = 458 N

mT = 1.65 ∗ 106

4.3

N m

(56)

FT 0 = 784 N

(57)

Model Parameters

Since rz (lT + rz mT − mC t10 − FC0 + α0 lC ) ¿ ωx2 I this term will be neglected, i.e. 2 ωφ k22 = ωx Also, the term r22 is very small, so it is neglected

ωφ ωx

2

(58)

(59)

(60)

r22 = 0

5

Analysis of the Model

With the approximations (59, 60) the model (40, 41) can be written as the matrix equation (61)

x ¨ + Cx˙ + Kx = Rxτ where x ˆ φ

x= and the matrices are given by C=

2ζx 0

0 2ζφ

, R=

K= p q

1+p pq

a + pq k22

q 0

,

(62) (63)

Here we introduced the parameters p=

mT , ωx2 m

q=

where constants a and k22 are

rz = rz X

m I

(64)

2

lT ωφ √ , k22 = (65) ωx ωx2 Im It is characteristic of systems with nonsymmetric stiffness matrix, that they can lose stability either by divergence (buckling) or by flutter. Chu and Moon [8] examined divergence and flutter instabilities in magnetically levitated models. Kiusalaas and Davis [23] studied stability of elastic systems under retarded follower forces. Recently several 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]). a=

5.1

Classical Limit

If q = 0 the equations reduce to x00 + 2ζx x0 + (1 + p) x + aφ = pxτ (66) φ00 + 2ζφ φ0 + cφ = 0 (67) The φ-equation is uncoupled from the x-equation and reduces to that of a damped oscillator. Its equilibrium φ = 0 is asymptotically stable and thus it does not affect the stability of the x-equation. In this case we recover the 1 DOF classical model (Tobias and Fishwick [40]).

10

5.2

Stability Analysis of the Undamped System without Delay

First we perform linear stability analysis of the system (68)

x ¨ + Kx = 0 where the matrix K is non-symmetric and of the form (k22 > 0) k12 k22

k11 k21

K=

(69)

Assuming the solutions in the form x = deiωt

(70)

K − ω2 I d = 0

(71)

= k11 − ω2

(72)

we obtain the characteristic polynomials which have nontrivial solution if the determinant of K − ω2 I is zero k11 − ω2 k21

k12 k22 − ω2

k22 − ω2 − k21 k12 = 0

The characteristic equation for the coupled system becomes ω4 − (k11 + k22 ) ω2 + k11 k22 − k21 k12 = 0

(73)

Divergence (static deflection, buckling) occurs when ω = 0 (or det K = 0), that is when (74)

k11 k22 − k21 k12 = 0 2

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

1 2

(k11 + k22 )2 − 4 (k11 k22 − k21 k12 )

k11 + k22 ±

(75)

For stable solutions, both solutions should be positive. Since k22 > 0, this is the case if 0 ≤ k11 k22 − k21 k12 ≤

k11 + k22 2

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) (78)

k21 = pq In the plane of the bifurcation parameters q, p the divergence boundaries are given by p=

1 2q 2

k22 − aq ±

4k22 q 2 + (k22 − aq)2

(79)

and the flutter boundary is characterized by p= k22 − 2aq − 1 ± 2

1 1 + 4q 2

q a (1 − k22 ) + a2 q − q (k22 − 1)2

(80) (81)

Figure 10 shows these boundaries on the (q, p) parameter plane for a = 1, k22 = 2.The different stability regions are 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 influence the 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 = µRxτ

(82)

When µ = 0 we recover the previously studied (68), while µ = 1 corresponds to equation (61) without damping. The characteristic equation is (83) det −λ2 I + K − µe−λτ R = 0

11

p S

U

U U q Figure 10: Stability boundaries of the undamped 2 DOF model without delay λ4 + k11 + k22 − µpe−λτ λ2 + k11 k22 − k12 k21 + µe−λτ (q (k12 + k21 ) − pk22 ) − µ2 q 2 e−2λτ = 0

(84)

Substituting λ = iω, ω ≥ 0 yields a complex equation that can be separated into the two real ones (the second equation was divided by µ sin (τ ω) 6= 0) ω4 − (k11 + k12 ) ω2 + k11 k22 − k12 k21 + 2

(85)

2 2

µ cos (τ ω) pω + q (k12 + k21 ) − pk22 − µ q cos (2τ ω) = 0 pω2 + q (k12 + k21 ) − pk22 + 2µq 2 cos (τ ω) = 0

(86)

We solve the second equation for cos (τ ω)

cos (τ ω) =

pω2 + q (k12 + k21 ) − pk22 −2µq 2

(87)

Using this relation and the identity cos (2τ ω) = 2 cos (τ ω)2 − 1 in the real part (85) results ω4 − (k11 + k22 ) ω2 + k11 k22 − k12 k21 + µ2 q2 = 0

(88)

Divergence occurs where ω = 0, that is where k11 k22 − k12 k21 + µ2 q 2 = 0 Substituting the elements of the stiffness 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 1 2 q2

k22 (1 − µ) + q (2 q µ − a) ±

(k22 (µ − 1) + q (a − 2µq))2 + 4 q 2 (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 while the 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 flutter boundary for a small µ (0.01) together with the flutter boundary (80). Figure 13 shows how this boundary changes with increasing µ (µ = 0.1, 0.5, 1). Figure 14 shows the full stability chart, complete with both the divergence and flutter boundaries, for µ = 1. To validate this stability chart the parameter space (p, q) was gridded and the delay-differential equation (61, 62) was integrated with 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 first 5τ intervals were discarded. Stability was determined by whether the amplitude of the solution grew or decayed. Dark dots correspond to stable numerical solutions. This figure can also explain a practical trick used in machine shops: sometimes, to avoid chatter, the tool is placed slightly ABOVE the centerline. We note that increasing q moves the system into the stable region of the chart.

12

p 3 2.5

U

U

2 1.5 1

S

0.5

q p 3 2.5

U

U

2 1.5

S

1 0.5

U

q p 3 2.5

U

2

U

1.5

S U

1 0.5

U q

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

13

S

U

f

p

2

ft U

0 -2

S

U

S

0 q

2

p

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

2 m=1

m=0.5

m=0.1 m=0.1

0 -1

m = 0 .5

0 q

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

14

m=1

1

Figure 14: Stability chart for the undamped system (82), µ = 1, τ = 1 Now we examine the effect 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 effect in nonconservative systems. The damping coefficients ζx and ζφ are taken to be 0.01, while the ratio of frequencies ωφ /ωx was changed in Figure 15 (this is the same as keeping this ratio fixed and increasing ζφ ). As the figure shows, the size of the stability regions increases with added damping. And finally, we show how the lobes of the conventional stability chart deform with the 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 force coefficient which is positive). In our model these lobes become a new source of instability, where the classical model would predict stable behavior.

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Conclusions

A new 3 DOF model derived may help explain at least two phenomena in metal cutting. The first is that offcentering the tool might help avoiding chatter. The second phenomenon is the observation that small amplitude tool vibrations can arise below the classical stability boundary. As shown, the added degrees of freedom result in unstable regions below the one predicted by the one DOF classical model. To summarize the important observations: • The 3-DOF model results in coupling between twist and lateral bending • The model can exhibit both divergence and flutter instabilities

• Damping seems to increase the size of stability regions

• The tool offset 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 exhibit hysteresis (Moon and Kalmár-Nagy [27]). It does not include temperature effects either (Davies and Burns [9]). Finally, only the analysis of a full nonlinear model could characterize the nature of vibrations and provide estimates of vibration amplitudes for the low chip thickness unstable regions.

References [1] B. Balachandran, Nonlinear Dynamics of Milling Processes, Philosophical Transactions of the Royal Society, 359, 793—819 (2001).

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Figure 15: Stability chart for the 3 DOF model. a, ωφ /ωx = 2 b, ωφ /ωx = 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 of the 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), 936—942 (1995). [4] V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability. New York: The MacMillan Company (1963). [5] S. T. S. Bukkapatnam, A. Lakhtakia, and S. R. T. Kumara, Analysis of Sensor Signals Shows Turning on a Lathe Exhibits Low-Dimensional Chaos. Physical Review E, 52(3), 2375—2387 (1995). [6] S.-G. Chen, A. G. Ulsoy, and Y. Koren, Computational Stability Analysis of Chatter in Turning. Journal of Manufacturing Science and Engineering, 119, 457—460 (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), 1619—1625 (1983). [9] M. A. Davies, and T. J. Burns, Thermomechanical Oscillations in Material Flow During High-Speed Machining. Philosophical Transactions of the Royal Society, 359, 821—846 (2001). [10] K. Engelborghs, and D. Roose, Numerical Computation of Stability and Detection of Hopf Bifurcations of Steady State Solutions of Delay Differential Equations. Advances in Computational Mathematics, 10, 271—289 (1999). [11] M. S. Fofana, Delay Dynamical Systems with Applications to Machine-Tool Chatter. Ph.D. thesis, University of Waterloo, Department of Civil Engineering (1993). [12] J. Gradišek, E. Govekar, and I. Grabec, I, Time Series Analysis in Metal Cutting: Chatter versus Chatter-Free Cutting. Mechanical Systems and Signal Processing, 12(6), 839—854 (1998). [13] N. H. Hanna, and S. A. Tobias, A Theory of Nonlinear Regenerative Chatter. Transactions of the American Society of Mechanical Engineers - Journal of Engineering for Industry, 96(1), 247—255 (1974). [14] G. Herrmann, and I.-C. Jong, On the Destabilizing Effect of Damping in Nonconservative Elastic Systems. Transactions of the ASME, 592—597 (1965). [15] C. S. Hsu, Application of the Tau-Decomposition Method to Dynamical Systems Subjected to Retarded Follower Forces. Transactions of the American Society of Mechanical Engineers - Journal of Applied Mechanics, 37(2), 259-266 (1970). [16] T. Insperger, and G. Stépán, Semi-discretization Method for Delayed Systems. International Journal for Numerical Methods in Engineering, 503-518 (2002).

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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), 715—726 (1997). [43] M. Wiercigroch, and A. M. Krivtsov, Frictional Chatter in Orthogonal Metal Cutting. Philosophical Transactions of the Royal Society, 359, 713—738 (2001). [44] N. van de Wouw, R. P. H. Faassen, J. A. J. Oosterling, and H. Nijmeijer, Modelling of High-Speed Milling for Prediction of Regenerative Chatter. This volume (2003).

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