35.3 Chatter in Face-Milling Operations

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A schematic of a face-milling operation is shown in Figure 35.8. In milling operations, multiple teeth may

be in contact with the part simultaneously, the feed naturally varies as a function of the tooth angle even

when structural vibrations are not present, and each tooth enters and leaves contact with the part every

spindle revolution. The depth-of-cut is the chip thickness in the z direction and is assumed to be

constant, since the machine tool and part structures are typically much stiffer in the z direction than in

the x and y directions.

The instantaneous feed of the ith tooth, illustrated in Figure 35.9, is

fiðtÞ ¼ ft cos½uiðtÞ􀀉 þ DxðtÞ cos½uiðtÞ􀀉 þ DyðtÞ sin½uiðtÞ􀀉 ð35:30Þ

where

DxðtÞ ¼ {xtðtÞ 2 xtðt 2 TtÞ} 2 {xpðtÞ 2 xpðt 2 TtÞ} ð35:31Þ

DyðtÞ ¼ {ytðtÞ 2 ytðt 2 TtÞ} 2 {ypðtÞ 2 ypðt 2 TtÞ} ð35:32Þ

The term ft cos½uiðtÞ􀀉 in Equation 35.30 represents the feed due to the distance the part advances relative

to the cutting tool each tooth rotation and is known as the static feed. The terms DxðtÞ cos½uiðtÞ􀀉

Ns

y

x

z

part

direction of

part motion

structural

damping (x)

structural

stiffness (x)

ft

structural

stiffness (y)

structural

damping (y)

ith tooth

fi

qex

qen

qi

FIGURE 35.8 Face milling operation schematic: current pass (solid line), previous pass (dotted line), and depth-ofcut

in z direction.

fi = ft cos (qi)

No vibration Vibrations in phase Vibrations out of phase

ith tooth fi

ith tooth

fi

ith tooth

qi

qi

qi

FIGURE 35.9 Modulation in feed due to structural vibrations in a face-milling operation: current pass (solid line)

and previous pass (dotted line).

Regenerative Chatter in Machine Tools 35-9

© 2005 by Taylor & Francis Group, LLC

and DyðtÞ sin½uiðtÞ􀀉 in Equation 35.30 represent the feed due to tool and part vibrations in the x and y

directions, respectively, at the tooth angle uiðtÞ; and are known as the dynamic feed.

The machining forces in the x and y directions, respectively, are

Fx ðtÞ ¼ dft

XNt

i¼1

n

2PT cosðcrÞ cos2½uiðtÞ􀀉 þ PC cos½uiðtÞ􀀉 sin½uiðtÞ􀀉

o

s½uiðtÞ􀀉

þ dDxðtÞ

XNt

i¼1

n

2PT cosðcrÞ cos2½uiðtÞ􀀉 þ PC cos½uiðtÞ􀀉 sin½uiðtÞ􀀉

o

s½uiðtÞ􀀉

þ dDyðtÞ

XNt

i¼1

n

2PT cosðcrÞ sin½uiðtÞ􀀉 cos½uiðtÞ􀀉 þ PC sin2½uiðtÞ􀀉

o

s½uiðtÞ􀀉 ð35:33Þ

Fy ðtÞ ¼ dft

XNt

i¼1

n

2PT cosðcrÞ cos½uiðtÞ􀀉 sin½uiðtÞ􀀉 2 PC cos2½uiðtÞ􀀉

o

s½uiðtÞ􀀉

þ dDxðtÞ

XNt

i¼1

n

2PT cosðcrÞ cos½uiðtÞ􀀉 sin½uiðtÞ􀀉 2 PC cos2½uiðtÞ􀀉

o

s½uiðtÞ􀀉

þ dDyðtÞ

XNt

i¼1

n

2PT cosðcrÞ sin2½uiðtÞ􀀉 2 PC cos½uiðtÞ􀀉 sin½uiðtÞ􀀉

o

s½uiðtÞ􀀉 ð35:34Þ

where

s½uiðtÞ􀀉 ¼

1 if uen # uiðtÞ # uex

0 if uen . uiðtÞ . uex

(

ð35:35Þ

The function s½uiðtÞ􀀉 determines if the ith tooth is in contact with the part at the tooth angle, uiðtÞ:

The first terms in Equation 35.33 and Equation 35.34 are the machining forces acting on the tool

in the x and y directions, respectively, due to the static feed. The second terms in Equation 35.33

and Equation 35.34 are the machining forces acting on the tool in the x and y directions, respectively,

due to the dynamic feed resulting from structural vibrations in the x direction. The third terms in

Equation 35.33 and Equation 35.34 are the machining forces acting on the tool in the x and y

directions, respectively, due to the dynamic feed resulting from structural vibrations in the y

direction.

The dynamic portion of the face milling force process model may be written compactly as

DFx ðtÞ

DFy ðtÞ

" #

¼ dAðtÞ

DxðtÞ

DyðtÞ

" #

¼ d

A11ðtÞ A12ðtÞ

A21ðtÞ A22ðtÞ

" #

DxðtÞ

DyðtÞ

" #

ð35:36Þ

where

A11ðtÞ ¼

XNt

i¼1

n

2PT cosðcrÞ cos2½uiðtÞ􀀉 þ PC cos½uiðtÞ􀀉 sin½uiðtÞ􀀉

o

s½uiðtÞ􀀉 ð35:37Þ

A12ðtÞ ¼

XNt

i¼1

n

2PT cosðcrÞ sin½uiðtÞ􀀉cos½uiðtÞ􀀉 þ PC sin2½uiðtÞ􀀉

o

s½uiðtÞ􀀉 ð35:38Þ

A21ðtÞ ¼

XNt

i¼1

n

2PT cosðcrÞ cos½uiðtÞ􀀉 sin½uiðtÞ􀀉 2 PC cos2½uiðtÞ􀀉

o

s½uiðtÞ􀀉 ð35:39Þ

A22ðtÞ ¼

XNt

i¼1

n

2PT cosðcrÞ sin2½uiðtÞ􀀉 2 PC cos½uiðtÞ􀀉 sin½uiðtÞ􀀉

o

s½uiðtÞ􀀉 ð35:40Þ

35-10 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

These coefficients modulate the instantaneous feed as the tooth angular displacement changes.

The summation from i ¼ 1 to Nt represents the contribution to this modulation for each of the Nt teeth.

Note the matrix AðtÞ is time-varying and periodic with the tooth-passing period, Tt: For chatter analysis,

the matrix AðtÞ is typically expanded in a Fourier series using the zeroth term (Minis and Yanushevsky,

1993; Budak and Altintas, 1998a). The zeroth term of the Fourier expansion of the force process matrix

AðtÞ is

A0 ¼

Nt

2p

A0

11 A0

12

A0

21 A0

22

" #

ð35:41Þ

where

A0

11 ¼

1

2

2PT cosðcrÞ u þ

1

2

sinð2uÞ

􀀘 􀀙

þ PC sin2ðuÞ

􀀒 􀀓u¼uex

u¼uen ð35:42Þ

A0

12 ¼

1

2

2PT cosðcrÞ sin2ðuÞ þ PC u 2

1

2

sinð2uÞ

􀀒 􀀘 􀀙􀀓u¼uex

u¼uen ð35:43Þ

A0

21 ¼

1

2

2PT cosðcrÞ sin2ðuÞ 2 PC u þ

1

2

sinð2uÞ

􀀒 􀀘 􀀙􀀓u¼uex

u¼uen ð35:44Þ

A0

22 ¼

1

2

2PT cosðcrÞ u 2

1

2

sinð2uÞ

􀀘 􀀙

2 PC sin2ðuÞ

􀀒 􀀓u¼uex

u¼uen ð35:45Þ

The dynamic force process is now approximated by the linear, time-invariant relationship:

DFx ðtÞ

DFy ðtÞ

" #

¼ dA0

DxðtÞ

DyðtÞ

" #

ð35:46Þ

The tool and part vibrations, respectively, are related to the machining forces by

xtðsÞ

ytðsÞ

" #

¼ GtðsÞ

Fx ðsÞ

Fy ðsÞ

" #

¼

Gt11 ðsÞ Gt12 ðsÞ

Gt21 ðsÞ Gt22 ðsÞ

" #

Fx ðsÞ

Fy ðsÞ

" #

ð35:47Þ

xpðsÞ

ypðsÞ

" #

¼ 2GpðsÞ

Fx ðsÞ

Fy ðsÞ

" #

¼ 2

Gp11 ðsÞ Gp12 ðsÞ

Gp21 ðsÞ Gp22 ðsÞ

" #

Fx ðsÞ

Fy ðsÞ

" #

ð35:48Þ

where GtðsÞ and GpðsÞ are the transfer functions relating the tool structural and part structural vibrations,

respectively, to the machining forces. The negative sign in Equation 35.48 is due to the fact that the forces

acting on the part are equal in magnitude and opposite in direction to the machining forces given in

Equation 35.33 and Equation 35.34. Since

Fx ðtÞ

Fy ðtÞ

" #

2

Fx ðt 2 TtÞ

Fy ðt 2 TtÞ

" #

¼

DFx ðtÞ

DFy ðtÞ

" #

2

DFx ðt 2 TtÞ

DFy ðt 2 TtÞ

" #

the structural vibrations can be related to the machining forces by

Dx

Dy

" #

¼ ð1 2 e2sTt Þ½GtðsÞ þ GpðsÞ􀀉

DFx ðsÞ

DFy ðsÞ

" #

ð35:49Þ

The machine tool and part vibrations are assumed to occur at a chatter frequency, vc; when a marginally

stable depth-of-cut is taken. Assuming the steady-state solution is a harmonic function at a chatter

Regenerative Chatter in Machine Tools 35-11

© 2005 by Taylor & Francis Group, LLC

frequency, vc; and substituting for the structural vibrations, Equation 35.49 becomes

DFx

DFy

" #

ejvc t ¼

dNt

2p ð1 2 e jvc Tt ÞG0ðjvcÞ

DFx

DFy

" #

e jvc t ð35:50Þ

where the matrix G0 is

G0ðjvcÞ ¼

2p

Nt

A0½GtðjvcÞ þ GpðjvcÞ􀀉 ð35:51Þ

Equation 35.50 is now solved based on the method presented by Budak and Altintas (1998a, 1998b)

to determine the stability lobe diagram. The characteristic equation of Equation 35.50 is

det I2 2

dNt

2p ð1 2 ejvc Tt ÞG0ðjvcÞ

􀀒 􀀓

¼ 0 ð35:52Þ

where I2 is the 2 £ 2 identity matrix. The solution of Equation 35.52 yields the limiting stable depth-ofcut.

The inverse of the eigenvalue of G0 is defined as

LðjvcÞ ¼ LRðjvcÞ þ jLIðjvcÞ ¼ 2

dNt

2p ð1 2 e jvc TtÞ ð35:53Þ

Expanding the exponential term in Equation 35.53 and noting that the depth-of-cut must be a real

number, the limiting stable depth-of-cut may be written as

dlim ¼ 2

pLR

Nt ð1 þ k2Þ ð35:54Þ

where the parameter k is defined by the transcendental equation

k ¼

LI

LR ¼

sinðvcTtÞ

1 2 cosðvcTtÞ ð35:55Þ

Equation 35.55 is solved for the tooth-passing period of the lth stability lobe and the tooth-passing

period is related to the spindle speed to yield

Ns ¼

60vc

Nt½p 2 2w þ 2lp􀀉

; l ¼ 0; 1; 2; … ð35:56Þ

where, again, w ¼ tan21ðkÞ: A chatter frequency is selected and the limiting stable depth-of-cut is

calculated from Equation 35.54 corresponding to the spindle speed on the lth lobe as given by

Equation 35.56.

35.3.1 Example 2

The cutting and thrust pressures in a face-milling operation are given by PC ¼ 2:0 kN/mm2 and

PT ¼ 0:8 kN/mm2, respectively, and the lead angle is 458. The part is assumed to be perfectly rigid and

the tool structural dynamics for the x and y directions are given by Equation 35.57 and Equation

35.58, respectively. The nominal parameters are uen ¼ 2458; uex ¼ 458; Nt ¼ 4; kx ¼ 14 kN/mm,

ky ¼ 17 kN/mm, zx ¼ 0:15; zy ¼ 0:1; vx ¼ 3000 rad/sec, and vy ¼ 4000 rad/sec. Stability lobe

diagrams are generated for the nominal parameters and Nt ¼ 1; 2, and 8 teeth. Next, stability lobe

diagrams are generated for the nominal parameters and uex ¼ 308; 608, and 758. The first 15 lobes are

included for all stability lobe diagrams.

x€tðtÞ þ 2zxvxx_tðtÞ þv2

x xtðtÞ ¼

v2

x

kx

Fx ðtÞ ð35:57Þ

y€tðtÞ þ 2zyvyy_tðtÞ þv2y

ytðtÞ ¼

v2y

ky

Fy ðtÞ ð35:58Þ

The effect of the number of teeth is illustrated in Figure 35.10: as the number of teeth increases,

the lobes shift to the left and the asymptotic stability borderline decreases. In Figure 35.11, the effect of

the exit angle is illustrated: as the exit angle increases, the asymptotic stability borderline decreases.

35-12 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

0 10 20 30 40 50

10

15

20

25

30

Spindle speed (krpm)

Depth-of-cut (mm) Depth-of-cut (mm)

Depth-of-cut (mm) Depth-of-cut (mm)

Nt= 1

0 5 10 15 20 25

5

10

15

Spindle speed (krpm)

Nt= 2

0 2 4 6 8 10 12

3

4

5

6

7

8

Spindle speed (krpm)

Nt= 4

0 2 4 6

1

2

3

4

Spindle speed (krpm)

Nt = 8

FIGURE 35.10 Stability lobe diagrams for Example 2, with Nt ¼ 1, 2, 4, and 8.

0 5 10

2

4

6

8

10

Spindle speed (krpm)

Depth-of-cut (mm) Depth-of-cut (mm)

Depth-of-cut (mm) Depth-of-cut (mm)

qex = 30°

0 5 10

2

4

6

8

Spindle speed (krpm)

qex = 45°

0 5 10

2

4

6

8

Spindle speed (krpm)

qex = 60°

0 5 10

2

3

4

5

6

Spindle speed (krpm)

qex = 75°

FIGURE 35.11 Stability lobe diagrams for Example 2, with uex ¼ 308, 458, 608, and 758.

Regenerative Chatter in Machine Tools 35-13

© 2005 by Taylor & Francis Group, LLC

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