35.2 Chatter in Turning Operations

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A schematic of a turning operation is shown in

Figure 35.2. The part structure is assumed to be

perfectly rigid, while the cutting-tool structure

is capable of vibrations in the longitudinal (i.e.,

the z) direction only. The machining force in the

longitudinal direction is

FðtÞ ¼ Pdf ðtÞ ð35:1Þ

The depth-of-cut is assumed to be constant;

however, the feed, and hence the machining

force, is time-varying due to structural vibrations.

It is assumed here that the machining force does

not explicitly depend upon the cutting speed.

The feed is the chip thickness in the longitudinal

direction. The nominal feed is the distance the tool advances relative to the part each spindle revolution

and is constant once the tool fully engages the part. However, the cutting tool vibrates, leaving an

undulated surface on the part and, thus, modulates the feed. The instantaneous feed is

f ðtÞ ¼ fnom þ DzðtÞ ¼ fnom þ zðtÞ 2 zðt 2 TÞ ð35:2Þ

The term fnom is the nominal feed, also known as the static feed. The term DzðtÞ is the feed due to the

cutting-tool vibrations and is known as the dynamic feed. The parameter T is the spindle-rotation

period. The structural vibration, zðtÞ; known as the inner modulation, is the cutting-tool vibration at the

current time. The delayed structural vibration zðt 2 TÞ; known as the outer modulation, is the cuttingtool

vibration as of when the part was at the current angular during the previous spindle rotation. The

modulation in feed due to structural vibrations is illustrated in Figure 35.3.

Inserting Equation 35.2 into Equation 35.1:

FðtÞ ¼ Pdfnom þ PdDzðtÞ ¼ Fnom þ DFðtÞ ð35:3Þ

The force Fnom ¼ Pdfnom is due to the nominal chip thickness and does not vary since the depth-of-cut

and nominal feed are constant. The force DFðtÞ ¼ PdDzðtÞ is due to changes in the nominal feed caused

by structural vibrations.

The structural vibrations are related to the machining force by

zðsÞ ¼ 2gðsÞFðsÞ ð35:4Þ

where gðsÞ is the transfer function relating the structural vibrations to the machining forces.

Since FðtÞ 2 Fðt 2 TÞ ¼ DFðtÞ 2 DFðt 2 TÞ; the structural vibrations are related to the machining

Ns

x

z

d

y

part

tool

direction of

tool motion

structural damping

structural stiffness

fnom

FIGURE 35.2 Turning operation schematic: current

pass (solid line) and previous pass (dotted line).

f = fnom

f

f

No vibration Vibrations in phase Vibrations out of phase

FIGURE 35.3 Modulation in feed due to structural vibrations in a turning operation: current pass (solid line) and

previous pass (dotted line).

Regenerative Chatter in Machine Tools 35-3

© 2005 by Taylor & Francis Group, LLC

forces by

DzðsÞ ¼ 2ð1 2 e2sT ÞgðsÞDFðsÞ ð35:5Þ

Substituting for Dz in Equation 35.5 and rearranging:

DFðsÞ 1 þ Pd

􀀑

1 2 e2sT 􀀜

gðsÞ

n o

¼ 0 ð35:6Þ

Equation 35.6 is now solved, based on the method presented by Budak and Altintas (1998a, 1998b), to

determine the stability lobe diagram. Assuming the steady-state solution is a harmonic function at a single

chatter frequency vc; Equation 35.6 becomes

DFðjvcÞe jvc t 1 þ Pd

􀀑

1 2 e2jvc T 􀀜

gð jvcÞ

n o

¼ 0 ð35:7Þ

where j2 ¼ 21: For nontrivial solutions of Equation 35.7, the following eigenvalue problem is derived:

det 1 þ Pd

􀀑

1 2 e2jvc T 􀀜

gðjvcÞ

n o

¼ 0 ð35:8Þ

Since the structural dynamics are one-dimensional, Equation 35.8 reduces to

1 þ Pd

􀀑

1 2 e2jvc T 􀀜

gð jvcÞ ¼ 0 ð35:9Þ

The parameter L is defined as

L ¼ Pd

􀀑

1 2 e2jvc T 􀀜

¼ LR þ jLI ð35:10Þ

Using the Euler identity 1 2 e2jvc T ¼ 1 2 cosðvcTÞ þ j sinðvcTÞ; the limiting stable depth-of-cut is

dlim ¼

1

P

􀀏 􀀐

LR þ jLI

1 2 cosðvcTÞ þ j sinðvcTÞ ð35:11Þ

Equation 35.11 is rewritten as

dlim ¼

1

2P

􀀏 􀀐

LR½1 2 cosðvcTÞ􀀉 þ LI sinðvcTÞ

1 2 cosðvcTÞ þ j

LR sinðvcTÞ þ LI½1 2 cosðvcTÞ􀀉

1 2 cosðvcTÞ

( )

ð35:12Þ

Since the limiting depth-of-cut must be a real number:

LR sinðvcTÞ þ LI½1 2 cosðvcTÞ􀀉 ¼ 0 ð35:13Þ

The parameter k is defined as

k ¼

LI

LR ¼

sinðvcTÞ

1 2 cosðvcTÞ ð35:14Þ

The limiting stable depth-of-cut is solved explicitly as

dlim ¼

LR

2P ð1 þ k2Þ ð35:15Þ

Note that LR must be positive for dlim to be positive. From Equation 35.9, the parameter L is

L ¼ 2

1

gðjvcÞ ð35:16Þ

Equation 35.16 is used to determine LR and LI; and these values are used to solve for dlim:

Next, the spindle speed at which the limiting depth-of-cut occurs is determined. The trivial solution to

Equation 35.14 is

vcT ¼ 0 þ 2lp; l ¼ 0; 1; 2; … ð35:17Þ

35-4 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

The quantity vcT may be interpreted as the number of vibration cycles during a spindle rotation. The trivial

solution indicates that the successive vibrations are in phase (i.e., there is no regeneration). The nontrivial

solution to Equation 35.14 is

cosðvcTÞ ¼

k 2 2 1

k 2 þ 1 ð35:18Þ

and may be rewritten as

vcT ¼ 1 þ 2lp; l ¼ 0; 1; 2; … ð35:19Þ

where

1 ¼ cos21 k 2 2 1

k 2 þ 1

􀁻 !

ð35:20Þ

The parameter 1 is the fraction of the vibration cycles during a spindle rotation. The angle of L in the

complex plane is

w ¼ tan21 LI

LR

􀀏 􀀐

¼ tan21ðkÞ ð35:21Þ

Substituting k ¼ tanðwÞ into Equation 35.18 yields

cosðvcTÞ ¼ 2cosð2wÞ ð35:22Þ

A solution to Equation 35.22 is

vcT ¼ p 2 2w þ 2lp; l ¼ 0; 1; 2; … ð35:23Þ

Comparing Equation 35.19 and Equation 35.23, it is seen that the fraction of vibration cycles is

1 ¼ p 2 2w: Since 0 # 1 # 2p; one must ensure that 2p=2 # w # p=2 when computing w: For

example, if Equation 35.21 is solved using a four-quadrant inverse tangent function whose solution is

bounded between 2p and p, then 2p=2 # w # p=2 since LR is positive. For milling applications, it

will be seen that LR must be negative; therefore, the following conditions must be enforced to ensure

0 # 1 # 2p:

if LI , 0 then w ! w þ p

if LI . 0 then w ! w 2 p ð35:24Þ

The spindle speed is

Ns ¼

60

T ¼

60vc

1 þ 2lp

; l ¼ 0; 1; 2; … ð35:25Þ

To construct a stability lobe diagram, the following steps are implemented:

1. Select a chatter frequency ðvcÞ near a dominant structural frequency.

2. Calculate LR and LI using Equation 35.16.

3. Calculate dlim using Equation 35.15.

4. Select a stability lobe number ðlÞ and calculate Ns using Equation 35.25. The point ðNs; dlimÞ is the

point on the stability lobe diagram corresponding to the chatter frequency, vc; and the stability

lobe number, l:

5. Repeat Step 4 for the desired number of stability lobes. The result is a vector of spindle speeds,

N~s ¼ {Ns1 Ns2 · · · Nsn }: Each point {ðNs1

; dlimÞ ðNs2

; dlimÞ · · · ðNsn

; dlimÞ}: corresponds

to a different stability lobe, and all of the points correspond to the chatter frequency vc:

6. Select another chatter frequency and repeat Steps 2 to 5. In this manner, the stability lobe diagram

is constructed. The smaller the difference between successive chatter frequencies, the greater

the resolution of the stability lobe diagram. In general, the lobes will overlap. In this case, the

Regenerative Chatter in Machine Tools 35-5

© 2005 by Taylor & Francis Group, LLC

minimum limiting depth-of-cut is the smallest depth-of-cut. If the lobes do not overlap, then the

range of chatter frequencies must be increased.

35.2.1 Example 1

The feed force for a turning operation is given by Equation 35.1 and the structural dynamics are given

by Equation 35.26. An analytical expression for the limiting depth-of-cut and corresponding spindle

speed for a given chatter frequency and stability lobe number is developed. The stability lobe diagram is

plotted for P ¼ 0:6 kN/mm2, vn ¼ 600 Hz, z ¼ 0:2; and k ¼ 12 kN/mm. The stability lobe diagram is

compared to stability lobe diagrams for z ¼ 0:1; 0.3, and 0.4. The stability lobe diagram is then compared

with stability lobe diagrams for vn ¼ 500; 700, and 800 Hz. The first ten lobes are included for all stability

lobe diagrams:

z€ðtÞ þ 2zvnzðtÞ þv2

nzðtÞ ¼ 2

v2

n

k

FðtÞ ð35:26Þ

The parameter L is

L ¼

21

gðjvcÞ ¼ LR þ jLI ¼

k

v2

n ðv2

c 2 v2

nÞ 2 j

k

v2

n ð2zvcvnÞ ð35:27Þ

The limiting depth-of-cut is

dlim ¼

kðv2

c 2 v2

nÞ

2Kv2

n

1 þ

4z2v2

cv2

n

ðv2

c 2 v2

nÞ2

" #

ð35:28Þ

The spindle speed is

Ns ¼

60vc

p 2 2 tan21 22zvcvn

v2

c 2 v2

n

􀀏 􀀐

þ 2lp

; l ¼ 0; 1; 2; … ð35:29Þ

where l is the stability lobe number. Note that the chatter frequency must be greater than the

structural natural frequency for the limiting depth-of-cut to be positive. The first ten lobes of the

0 20 40 60

Spindle speed (krpm)

Depth-of-cut (mm)

80 100 120

5

10

15

20

25

30

lobe1 lobe 0

FIGURE 35.4 Unprocessed stability lobe diagram for Example 1.

35-6 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

stability lobe diagram are plotted in Figure 35.4 and Figure 35.5. In Figure 35.4, the entire solution

for each of the ten stability lobes is shown. The largest stability lobe is the zeroth lobe on the right.

The lobe number increases from right to left on the stability lobe diagram and successive lobes

become closer together. The stability lobe diagram is processed in Figure 35.5 such that the

minimum depth-of-cut is selected at each spindle speed showing the true stability borderline.

0 10 20 30 40 50

Spindle speed (krpm)

Depth-of-cut (mm)

60 70 80 90 100

5

10

15

20

25

FIGURE 35.5 Processed stability lobe diagram for Example 1.

0 10 20 30 40 50

5

10

15

20

Spindle speed (krpm)

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

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

z = 0.1

0 10 20 30 40 50

5

10

15

20

25

Spindle speed (krpm)

z = 0.2

0 10 20 30 40 50

15

20

25

30

35

Spindle speed (krpm)

z = 0.3

200 10 20 30 40 50 60

25

30

35

40

Spindle speed (krpm)

z = 0.4

FIGURE 35.6 Stability lobe diagrams for Example 1 with z ¼ 0.1, 0.2, 0.3, and 0.4.

Regenerative Chatter in Machine Tools 35-7

© 2005 by Taylor & Francis Group, LLC

In Figure 35.6, the effect of the structural damping ratio is illustrated: as the structural damping

ratio increases, the lobes shift slightly to the left and the asymptotic stability boundary shifts up

dramatically. The effect of the structural natural frequency is illustrated in Figure 35.7: as

the structural natural frequency increases, the lobes shift to the right but the magnitude remains

the same.

0 20 40

5

10

15

20

25

Spindle speed (krpm)

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

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

wn = 500 Hz

0 20 40

5

10

15

20

25

Spindle speed (krpm)

wn = 600 Hz

0 20 40 60

5

10

15

20

25

Spindle speed (krpm)

wn = 700 Hz

0 20 40 60

5

10

15

20

25

Spindle speed (krpm)

wn = 800 Hz

FIGURE 35.7 Stability lobe diagrams for Example 1 with vn ¼ 500, 600, 700, and 800 Hz.

This section presented an analytical method to generate stability lobe diagrams for turning

operations. The limiting depth-of-cut in a turning operation is given by

dlim ¼

LR

2P ð1 þ k2Þ

where LR is the real part of 21=gðjvcÞ; gðjvcÞ is the structural transfer function evaluated at the

chatter frequency, vc; k ¼ LI=LR; and LI is the imaginary part of 21=gðjvcÞ: The corresponding

spindle speed is

Ns ¼

60vc

1 þ 2lp

where l ¼ 0; 1; 2; … is the stability lobe number and

1 ¼ cos21 k2 2 1

k2 þ 1

􀁻 !

35-8 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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