35.7 Case Study

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A case study of regenerative chatter for a face-milling operation is now presented. Further details are

presented in Landers (1997). The machine tool is a three-axis vertical milling machine (Figure 35.25).

Each axis has a linear encoder with a resolution of 10 mm mounted on it. The axis motors (186 W)

drive pulleys that rotate leadscrews and provide motion to the linear axes. The spindle (2240 W) drives

the face mill (Carboloy R/L220.13-02.00-12, 50 mm diameter). The tool holds four carbide inserts

(Carboloy SEAN 42AFTN-M14 HX, 458 lead angle). The part is 6061 aluminum. The spindle is run

open-loop. A Kistler 9293 piezoelectric three-component dynamometer was utilized for force process

modeling and chatter detection. The x and y channels have a natural frequency of 4.5 kHz, rigidity

of 0.7 kN/mm, and range of 2 20 to 20 kN. The z channel has a natural frequency of 5 kHz, rigidity of

7 kN/mm, and range of 2 100 to 200 kN. A Bently Nevada 3000 Series Type 190 proximity transducer

was utilized to measure the static stiffnesses of the structural components. The sensor gain is 8 V/mm,

the response is flat to 10 kHz, and the range is 1.02 mm. A Kistler Quartz Model #802A accelerometer

(resonant frequency 36.7 kHz) was utilized to measure the dynamic characteristics of the structural

components.

The cutting and thrust pressures, respectively, are

PC ¼ 0:29f 20:25d20:13 V

1000

􀀏 􀀐20:72

ð35:68Þ

PT ¼ 0:16f 20:40d20:41 V

1000

􀀏 􀀐20:58

ð35:69Þ

The transfer function matrices, respectively, between the tool structure and machining forces, and the

part structure and machining forces are modeled as

xtðsÞ

ytðsÞ

" #

¼

ð45002=14Þ

s2 þ 2ð0:07Þð4500Þs þ 45002 0

0 ð41002=14Þ

s2 þ 2ð0:11Þð4100Þs þ 41002

2

66664

3

77775

Fx ðsÞ

Fy ðsÞ

" #

ð35:70Þ

xpðsÞ

ypðsÞ

" #

¼ 2

ð26002=9:5Þ

s2 þ 2ð0:09Þð2600Þs þ 26002 0

0 ð21002=9:5Þ

s2 þ 2ð0:22Þð2100Þs þ 21002

2

66664

3

77775

Fx ðsÞ

Fy ðsÞ

" #

ð35:71Þ

This section presented several techniques to suppress regenerative chatter. The three major

techniques to suppress chatter are spindle-speed selection, feed selection, and SSV. In spindlespeed

selection, the spindle speed is adjusted to be a multiple of the chatter frequency to place the

spindle speed in a pocket of the stability lobe diagram. In feed selection, the feed is increased to

suppress chatter. In SSV, the spindle speed is varied in a sinusoidal manner to decrease the phase

difference between the current and previous tooth passes.

35-24 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Using the machining force and structural models,

a stability lobe diagram was constructed using time

domain simulations. Experimental data were

collected by adjusting the depth-of-cut in increments

of 0.1 mm until chatter occurred. The timedomain

simulations and the experimental data are

plotted in Figure 35.26. The cutting conditions

were ft ¼ 0.10 mm/tooth, Nt ¼ 4 teeth,

uen ¼ 2 908, and uex ¼ 908. The chatter detection

methodology for this system was described in

Example 5.

Spindle-speed adjustment is typically a more

productive chatter suppression option. However,

the machine tool in this case study is not equipped

with automatic spindle speed control and, thus,

the depth-of-cut is adjusted to suppress chatter.

When chatter is detected, the chatter suppressor rewrites the part program to accommodate one

additional tool pass (Figure 35.27). Therefore, the new operation depth-of-cut is

dn ¼

dp

1 þ Nc ð35:72Þ

FIGURE 35.25 Three-axis vertical machine tool schematic.

0.0

0.5

1.0

1.5

2.0

1000 1500 2000 2500

Spindle speed (rpm)

Depth-of-cut (mm)

FIGURE 35.26 Stability lobe diagram for a facemilling

operation — time-domain simulations (empty

boxes) and experimental points (filled circles).

Regenerative Chatter in Machine Tools 35-25

© 2005 by Taylor & Francis Group, LLC

where dp is the previous operation depth-of-cut and Nc is the number of times the chatter suppression

routine has been invoked. The new value may be well below the stability limit; however, making all passes

an equal depth-of-cut provides a good balance between productivity and the search for a stable depthof-

cut. Results of this controller are presented in Landers and Ulsoy (1998, 2001).

WORKPIECE

TOOTH 1a

2a

1b

2b

3a

4a

3b

4b

chatter

detected

original tool path (a) new tool path (b)

5b

6b

7b

8b

FIGURE 35.27 Original tool path (a) is rewritten when chatter occurs. New tool path (b) contains an additional

tool pass.

Nomenclature

Symbol Quantity

d depth-of-cut (mm)

dlim limiting stable depth-of-cut (mm)

f feed (mm)

ft feed per tooth (mm)

F machining force (kN)

FC cutting force (kN)

FT thrust force (kN)

Fx force acting on cutting tool in x

direction (kN)

Fy force acting on cutting tool in y

direction (kN)

Fz force acting on cutting tool in z

direction (kN)

k structural stiffness (kN/mm)

Ns spindle speed (rpm)

Nt number of teeth

P machining pressure (kN/mm2)

PC cutting pressure (kN/mm2)

PT thrust pressure (kN/mm2)

t time (sec)

Symbol Quantity

T spindle rotational period (sec)

Tt tooth rotational period (sec)

xp part structural displacement in x

direction (mm)

xt cutting tool structural displacement in

x direction (mm)

yp part structural displacement in y

direction (mm)

yt cutting tool structural displacement in

y direction (mm)

z cutting tool structural displacement in

z direction (mm)

u tooth angle (rad)

uen tooth entry angle (rad)

uex tooth exit angle (rad)

vc chatter frequency (rad/sec)

vn structural natural frequency

(rad/sec)

cr lead angle (rad)

z structural damping ratio

35-26 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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