Runout effects in milling : Surface finish, surface location error, and stability

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International Journal of Machine Tools & Manufacture ] (]]]]) ]]]–]]]
www.elsevier.com/locate/ijmactool
Runout effects in milling: Surface ?nish, surface location error,
and stability
Tony L. Schmitza,Ã, Jeremiah Coueyb, Eric Marshb, Nathan Mauntlera, Duke Hughesa
aDepartment of Mechanical and Aerospace Engineering, University of Florida, P.O. Box 116300, 237 MAE-B, Gainesville, FL, 32611, USA
bDepartment of Mechanical Engineering, Penn State University, University Park, PA, USA
Received 20 June 2005; received in revised form 15 April 2006; accepted 20 June 2006
Abstract
This paper investigates the effect of milling cutter teeth runout on surface topography, surface location error, and stability in end
milling. Runout remains an important issue in machining because commercially-available cutter bodies often exhibit signi?cant variation
in the teeth/insert radial locations; therefore, the chip load on the individual cutting teeth varies periodically. This varying chip load
in?uences the machining process and can lead to premature failure of the cutting edges. The effect of runout on cutting force and surface
?nish for proportional and non-proportional tooth spacing is isolated here by completing experiments on a precision milling machine
with 0.1 mm positioning repeatability and 0.02 mm spindle error motion. Experimental tests are completed with different amounts of
radial runout and the results are compared with a comprehensive time-domain simulation. After veri?cation, the simulation is used to
explore the relationships between runout, surface ?nish, stability, and surface location error. A new instability that occurs when
harmonics of the runout frequency coincide with the dominant system natural frequency is identi?ed.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Machining; Eccentricity; Dynamics; Simulation; Chatter; Bifurcation
1. Introduction
include runout effects in a chatter suppression scheme for
end milling that relies on continuously variable spindle
Radial runout, or eccentricity, of the cutter teeth is a
speed [9]. The identi?cation of cutting force coef?cients in
common problem in multiple cutting edge, interrupted
the presence of runout is described by Wang and Zheng
machining operations. Well-known effects include prema-
[10], Ko et al. [11], and Yun and Cho [12]. Efforts focused
ture cutting edge failure due to periodic variations in the
on in-process monitoring and rejection of runout contribu-
chip load and force, as well as increased machined surface
tions to the cutting force are described by Heckman and
roughness. Several previous runout studies are available in
Liang [13], Yan et al. [14], Stevens and Liang [15], and
the literature. Kline and DeVor introduce the problems
Liang and Wang [16]. Additionally, Baek et al. [17]
associated with radial runout in end milling operations and
describe an optimum selection of feed rate considering
show the importance of the relationship between the
runout in face milling operations.
runout and chip load on surface ?nish [1]. Lazoglu [2]
In this paper, we build on these previous efforts by
and Feng and Menq [3,4] include runout in ball end milling
combining time-domain simulation with a machining setup
simulations. Zheng et al. [5] and Baek et al. [6] describe face
that allows continuous variation of runout while minimiz-
milling models which consider runout. Atabey et al. detail a
ing other surface roughness contributors, such as spindle
boring model that includes runout. [7]. Mezentsev et al.
and slide error motions. Additionally, we consider endmills
outline a model-based method for fault detection in
with both proportional and non-proportional teeth spa-
tapping which includes runout [8]. Altintas and Chan
cing.1 The paper is organized as follows: Section 2 provides
ÃCorresponding author. Tel.: +1 352 392 8909; fax: +1 352 392 1071.
1Non-proportional teeth spacing can be used to eliminate chatter by
E-mail address: tschmitz@u?.edu (T.L. Schmitz).
interrupting the regeneration of surface waviness (caused by tool
0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmachtools.2006.06.014

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T.L. Schmitz et al. / International Journal of Machine Tools & Manufacture ] (]]]]) ]]]–]]]
a description of the time-domain simulation; Section 3
is the incremental feed during the time step dt, and xtool
describes the setup used to determine the speci?c cutting
and ytool are the tool center coordinates determined in the
energy coef?cients for the force model and validate the
previous time step (set equal to zero for the ?rst simulation
model; Section 4 presents surface measurements and
time step). For the tooth-dependent radius values, the ROj
predictions for cutting conditions that differ from the
entries are referenced relative to the tooth with the largest
force model development; Section 5 includes a discussion
radius (assumed equal to the nominal cutter radius). The
of the in?uence of runout on surface ?nish, stability and
ROj value for this tooth is zero, while the rest are negative
surface location error; and Section 6 summarizes the paper.
(or zero) according to their difference from the largest
radius value.
2. Time-domain simulation description
Cxj ¼ rj sin fj þ df þ xtool,
Cy
The time-domain simulation applied in this study is
j ¼ rj cos fj þ ytool.
ð1Þ
based on the ‘Regenerative Force, Dynamic De?ection
In order to determine the instantaneous radial chip
Model’ described by Smith and Tlusty [18], which includes
thickness at each time step of the simulation, the {Cxj, Cyj}
the contribution of the tool/workpiece vibrations to the
coordinates of the current tooth (i.e., point C in Fig. 1) are
instantaneous chip thickness, and provides predictions for
compared to the surface coordinates during the prior tooth
both force and de?ection in the x- and y-directions in the
passage at the same angular orientation. However, because
plane of the cut (vibrations along the tool axis, or
we are not applying the circular tool path assumption, it is
z-direction, are not considered here). We also model the
not required that a data point exist at this angle from the
actual trochoidal motion of the cutter teeth, rather than
prior pass. Therefore, a search is completed to determine
assuming a circular tool path. Other instances of trochoidal
the two points from the previous tooth passage which
tool path simulations from the literature are provided in
bound this angle, referred to as points A and B in Fig. 1.
Refs. [19–21], for example. Our approach is similar in
Linear interpolation is then completed to determine point
nature to that described by Campomanes and Altintas [21];
D, which lies on the line between point C and the cutter
however, the authors did not consider runout in the
origin [23]. The coordinates of point D, {Dxj, Dyj}, are
referenced study. One effect which is not modeled is elastic
provided in Eq. (2):
spring back of the work surface after chip removal. This
tan f
has been shown to be important in micro-milling, for
Dx
j Á Axj C1 À tan fj Á Ayj þ tan fj Á Cyj À Cxj
j ¼
,
example, but is not included here [22].
tan fj Á C1 À 1
The present milling simulation is carried out by ?rst
Dyj ¼ Ayj À AxjC1 þ DxjC1,
ð2Þ
de?ning the cutting parameters, including spindle speed, O,
where C
feed/tooth, f
1 ¼ ðAyj À Byj Þ=ðAxj À Bxj Þ. The nonlinearity that
t, radial and axial depths of cut, a and b,
is exhibited when the vibration amplitude is large enough
respectively, and the tool geometry, including the number
that a tooth leaves the cut is included by setting the chip
of teeth, Nt, teeth pitch (both proportional and non-
thickness, h
proportional spacing are allowed), and radial runout, RO
j, equal to zero if
q????????????????????????????????????????????????????????????
of each tooth (straight teeth were used in the experiments
ðCx
and runout was assumed constant over the low axial depths
j À xtoolÞ2 þ ðCyj À ytoolÞ2
q????????????????????????????????????????????????????????????
applied here, although this is not a necessary condition for
o ðDxj À xtoolÞ2 þ ðDyj À ytoolÞ2.
ð3Þ
the simulation). The system dynamics are then de?ned.
These are included as modal mass, m, damping, c, and
stiffness, k, values for any number of modes (i.e., degrees-
of-freedom) in the x- (feed) and y-directions and are
y
obtained from dynamic, typically impact, tests.
Tooth
The force and de?ection are then determined by
numerical integration over small steps in time,dt ¼
C
60=ðSR Á OÞ, where dt is given in s, SR is the number of
A
steps per cutter revolution, and O is expressed in rev/min,
D
or rpm. In each time step, the cutter is rotated by an angle,
B
df ¼ 360/SR (in degrees). The current nominal {Cx
y
New surface
j, Cyj}
tool
j
coordinates of each tooth on the cutter are then calculated
according to Eq. (1), where rj is the tooth-dependent cutter
Old
radius (including runout, ROj), fj is the tooth angle, j is the
tooth number (which varies from 1 to Nt), df ¼ Ntf t=SR,
x
x
tool
(footnote continued)
vibrations) in situations where it is inconvenient to adjust the spindle
Fig. 1. Determination of instantaneous chip thickness by linear inter-
speed [32].
polation for trochoidal tool path.

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3
Two other conditions must also be queried for the chip
provided in Eq. (5).
thickness calculation. First, it must be determined if the
Dx0 ¼ ðy
current tooth is bounded by the speci?ed radial immersion.
j
test À ytoolÞ tan fj þ xtest,
Second, it must be veri?ed that the chip thickness has not
Dy0 ¼ y
j
test.
ð5Þ
been reduced during cut entry for down milling or cut exit
Under these conditions, the chip thickness is then
for up milling. The chip thickness reduction that occurs
calculated using Eq. (6) for up or down milling with less
at the cut exit for up milling, for example, is exhibited in
than or equal to 50% radial immersion or Eq. (7) for
Fig. 2.
greater than 50% radial immersion.
To determine if the current tooth is bounded by the
q??????????????????????????????????????????????????????????
selected radial depth of cut (i.e., engaged in the cut),
hj ¼
ðCxj À Dx0Þ2 þ ðCy
Þ2.
(6)
the value y
j
j À Dy0j
test, which gives the y direction coordinate of the
desired surface as shown in Fig. 2, is used. For up milling
q??????????????????????????????????????????????????????????
with less than or equal to 50% radial immersion, cutting
hj ¼
ðDx0 À Dx
À Dy
.
(7)
j
j Þ2 þ ðDy0j
j Þ2
occurs if Cyj is greater than ytest. This situation is depicted
in Fig. 2. If the up milling radial immersion is greater than
In any case that the computed chip thickness is greater
50%, then Dy
than zero, the tangential and radial force components,
j must be greater than ytest if cutting is to
occur (note that y
Ftan,j and Frad,j, respectively, for tooth j are calculated
test is negative in this case). For down
milling, Cy
according to Eq. (8):
j must be less than ytest if the radial immersion is
less than or equal to 50% and cutting is to take place (ytest
F tan;j ¼ Ktcbhj þ Kteb,
is again negative). If the radial immersion is greater than
F
50%, it is required that Dy
rad;j ¼ K rcbhj þ K reb,
ð8Þ
j be less than ytest if cutting is to
occur. In each case, provided the chip thickness is not
where Ktc and Krc are the force model cutting coef?cients
reduced at the cut exit (up milling) or entry (down milling),
and Kte and Kre are the edge coef?cients [24]. We presume
as shown in Fig. 2, and the tooth has not vibrated out of
that these coef?cients indirectly account for complicated
the cut (Eq. (3)), hj is calculated according to Eq. (4):
tool–chip formation effects such as work hardening and
q??????????????????????????????????????????????????????????
?ow stress temperature sensitivity, but have not modeled
h
these effects directly. The forces are summed over all teeth
j ¼
ðCxj À DxjÞ2 þ ðCyj À DyjÞ2.
(4)
engaged in the cut at the given instant in time. The total
PN
P
t
Nt
To check if the chip thickness reduction condition is met,
forces, F tan ¼
F
F
j¼1
tan;j and F rad ¼
j¼1
rad;j , are then
the tooth coordinates are again compared to y
projected into the x- and y-directions using Eq. (9). These
test. The
thickness reduction occurs if the following circumstances
force values are then used to determine the instantaneous
are satis?ed: (1) up milling, less than or equal to 50% radial
displacement values xtool and ytool for the next time step
immersion—Dy
using numerical integration and the measured modal
j is less than ytest; (2) up milling, greater
than 50% radial immersion—Cy
parameters. If multiple vibration modes are included, the
j is less than ytest; (3) down
milling, less than or equal to 50% radial immersion—Dy
displacement contributions from each mode are summed to
j is
greater than y
determine the total displacement. Provided the modal
test; and 4) down milling, greater than 50%
radial immersion—Cy
parameters were determined from a direct frequency
j is greater than ytest. In these cases,
Eq. (4) can no longer be used to compute the instantaneous
response function measurement (or model), the same forces
chip thickness. Rather, point D0 identi?ed in Fig. 2 must be
are used for each vibration mode.
considered. The coordinates of this point, {Dx0j, Dy0j}, are
F x ¼ À F tan cos fj À Frad sin fj,
F y ¼ F tan sin fj À Frad cos fj,
ð9Þ
y
In the case of a helical cutting edge, the tool can be
Current
segmented along its axis into several disks, each of which is
tooth path
treated as having a zero helix angle [24,25]. The forces for
each disk are then summed to determine the total radial
and tangential cutting force components for that particular
Previous
D’
simulation time step. Eq. (9) is then applied to project the
tooth path
A
C
forces in the x- and y-directions and the numerical
integration is completed. The difference, Df (in deg),
y
D
tool

between the tooth angle, f
j
j, for tooth j on disk k and the
angle for the same tooth j on disk k+1 (located farther
ytest
away from the tool tip by a distance b/SA) is provided in
B
Eq. (10):
x
x
tool
2b tan b 180
Df ¼
Á
,
(10)
Fig. 2. Reduced instantaneous chip thickness at cut exit in up milling.
SA Á d
p

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where b is the helix angle, SA is the number of axial disks,
and d is the nominal cutter diameter. The reader may also
note that the runout variable may be represented as a
matrix where the entries represent a tooth runout value for
each axial disk.
3. Experimental setup for force model validation
Experiments for this work were carried out on a two-axis
Moore 450 computer numerically-controlled machining
center with a programmable resolution of 0.1 mm and a
Professional Instruments 4R Twin Mount air bearing
milling spindle with 20 nm-level error motions [26]. See
Fig. 3. This machine and air bearing spindle were
Fig. 4. End mill radial runout measurement using capacitance probe
speci?cally selected to avoid the typical convolution
setup.
between spindle error motions and tool runout [27]. In
the experiments reported here, because the spindle error
motions are one –three orders of magnitude smaller than
the prescribed tool runout values, we can assume the
runout is constant with spindle speed and dependent only
on the tool runout, although this is not the case in general
for high-speed rolling element bearing spindles. The cutting
forces were recorded using a Kistler mini-dynamometer
(Type 9256A2) with a 2 mN resolution. The three force
components were recorded using a DSPT Siglab unit at a
sampling frequency of 5.12 kHz. Fig. 3 also shows the angle
plate used to support the dynamometer and 6061-T6
aluminum workpieces. The cutting tool was an SGS
12.7 mm diameter, two straight ?ute end mill (proportional
teeth spacing) heat shrunk into an aluminum chuck that
was precision ground in assembly and subsequently
dynamically balanced. Alumicut oil coolant was sprayed
onto the cutting tool during cutting.
The custom designed end mill chuck allowed the ?ute-to-
?ute radial runout to be varied from 0 to 400 mm by
loosening the chuck ?ange bolts and manually adjusting
the assembly to the desired runout value. The runout was
measured using an air bearing capacitance probe with a
diamond stylus (i.e., the air bearing supported a follower
with a diamond stylus that contacted the cutting tool
during manual rotation—the capacitance probe sensed the
Fig. 5. Dynamometer-mounted workpiece (dynamometer not shown).
displacement of the follower). Fig. 4 shows the 25 mm/V
Peripheral up milling was completed by machining the top of the front
Lion Precision air bearing capacitance probe setup (the
raised edge. The workpiece was repositioned to complete four tests with a
rigid probe holder is not shown). As noted in the ?gure, the
single sample.
tool was rotated by hand opposite the cutting direction.
The runout was then determined by differencing the peak
displacement values recorded for the two teeth.
Two workpiece geometries and mounting con?gurations
were used in the cutting tests. First, 33 Â 48 Â 16 mm
samples were mounted directly on the dynamometer. The
reversible, relieved geometry shown in Fig. 5 was used to
allow pure peripheral up milling with nominal axial depths
of 0.5 and 1.0 mm and a nominal radial depth of 1.27 mm
(10% radial immersion). Second, 38.1 Â 50.8 Â 6.35 mm
samples were mounted to a T-shaped base as shown in
Fig. 3. Two-axis milling machine with air bearing spindle. The
dynamometer and T-base mount with workpiece are also shown.
Fig. 6. Again, the relieved geometry enabled pure

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Fig. 6. T-base workpiece mount. The assembly dynamic stiffness was
signi?cantly less than the tool. Four tests were completed on each sample.
Fig. 7. Dynamometer mount measured (solid line) and simulated (dotted
line) x-direction force data: (top) RO ¼ 2 mm; (middle) RO ¼ 4.9 mm;
(bottom) RO ¼ 33 mm. Note changes in force axis scales.
Table 1
Experimental cutting coef?cients.
Ktc (N/mm2)
Krc (N/mm2)
Kte (N/mm)
Kre (N/mm)
values: {2, 4.9, and 33} mm. Comparisons between mea-
1000
520
2
3
sured and simulated x-direction forces for the dynamo-
meter mount are shown in Fig. 7. The cutting conditions
for the three panels were: (top) ft ¼ 0.01 mm/tooth,
peripheral up milling. This base-workpiece assembly was
RO ¼ 2.0 mm,
a ¼ 1.245 mm,
b ¼ 0.635 mm;
(middle)
signi?cantly more ?exible than the cutting tool (natural
ft ¼ 0.02 mm/tooth, RO ¼ 4.9 mm, a ¼ 1.168 mm, b ¼
frequency, fn ¼ 1075 Hz, k ¼ 1.1 Â 106 N/m, damping
1.194 mm; and (bottom) ft ¼ 0.02 mm/tooth, RO ¼ 33 mm,
ratio, z ¼ 0.009 determined by impact testing); modal
a ¼ 1.524 mm, b ¼ 0.572 mm (the reader may note that for
parameters were determined by impact testing and least
this RO value, only a single tooth is engaged in the cut).
squares polynomial curve ?tting and used to de?ne the
Comparable agreement was seen for y direction data, but
dynamics in simulation.
has not been included here for brevity.
To enable comparison between experimental and simu-
In order to make a direct comparison with forces
lated results, the four coef?cients for the force model
measured using the T-base (?exible)-workpiece assembly
shown in Eq. (8) were determined experimentally using the
shown in Fig. 6, the simulated data was ?ltered by the
mechanistic identi?cation procedure described by Altintas
dynamometer force-to-cutting force frequency response
[24]. In this approach, slotting cuts with constant axial
[28]. This response was obtained by exciting the free end of
depth were made in the rigid workpieces over a range of
the T-base-workpiece assembly with an impact hammer
feed/tooth values. The average force/tooth is recorded and
and recording both the hammer input and the dynam-
the four cutting coef?cients are identi?ed by a linear
ometer force output. The resulting dynamometer force-to-
regression of feed/tooth vs. average force. In our tests, the
cutting force frequency response was least squares ?t with a
axial depth was 1.0 mm, the feed/tooth values were
2nd-order (in both the numerator and denominator) ?lter,
{0.02, 0.04, 0.06, 0.08} mm/tooth, and the spindle speed
F dynamometer=F cuttingðoÞ, and used to correct the simulated
was 5290 rpm. The average force per tooth was determined
force according to Eq. (11), where Fy,d (o) is the frequency-
by sectioning the digital cutting force data into single tooth
domain ?ltered force. This result was then inverse Fourier
passages using a once-per-revolution capacitance probe
transformed to obtain the new simulated force that was
signal (also recorded during cutting at 5.12 kHz). The
compared to the measured y direction force. A 2nd-order
cutting coef?cients are provided in Table 1. These cutting
?t to the required frequency response was applied because
force values are somewhat higher than would be expected
it exhibited a single mode at the T-base-workpiece
for 6061-T6 aluminum. This may be due to the straight
fundamental clamped-free natural frequency within the
?ute geometry or the cutting edge preparation for the tools
bandwidth of interest (5 kHz).
used in this study (i.e., increased cutting forces for small
F dynamometer
feed/tooth values due to the non-zero cutting edge radius).
F y;dðoÞ ¼
Á F
F
y,
(11)
The reader may note that predictions for much higher or
cutting
lower feed/tooth values could require additional testing to
Comparisons between measured and simulated y direc-
identify any dependence of the cutting coef?cients on the
tion forces for the T-base mount with the Eq. (11) ?ltering
nominal feed per tooth. Cutting tests for both mounting
applied are shown in Fig. 8. The cutting conditions for the
con?gurations were completed at three different runout
three panels were: (top) ft ¼ 0.02 mm/tooth, RO ¼ 2.0 mm,

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metric pro?ler (i.e., a scanning white light interferometer)
using a 2.5 Â objective with a 1.9 Â 2.5 mm ?eld of view.
Once the surface ?nish topograph was collected for each
cutting condition, a line trace was then extracted from the
center of the surface pro?le for direct comparison to
simulation.
4.1. Proportional teeth spacing results
The surface ?nish of the ?exible workpieces was
compared to the predicted results for up milling using the
two
?ute,
straight
tooth
cutters
with:
a ¼ 0.5 mm;
b ¼ 0.5 mm; ft ¼ {0.51 and 0.24} mm/tooth; and O ¼ {150
and 300} rpm. The runout values varied between 0.3 and
Fig. 8. T-base mount measured (solid line) and simulated (dotted line) y
37 mm. A progression of representative results is shown in
direction force data with dynamics correction: (top) RO ¼ 2 mm; (middle)
RO ¼ 4.9 mm; and (bottom) RO ¼ 33 mm. Note changes in force axis
the four panels provided in Fig. 9. In all cases, the
scales.
experimental results are represented by the heavy solid line,
while the simulated tool path is given by the lighter solid
and dotted lines, where the dotted line represents the tooth
with a smaller radius (i.e., lower runout value). In panel (a),
a ¼ 1.295 mm, b ¼ 0.584 mm; (middle) ft ¼ 0.02 mm/tooth,
the combination of small runout and lower feed per tooth
RO ¼ 4.9 mm, a ¼ 1.124 mm, b ¼ 1.054 mm; and (bottom)
gives the expected cusped surface with a spatial period
ft ¼ 0.02 mm/tooth, RO ¼ 33 mm, a ¼ 1.461 mm, b ¼
nominally equal to the feed/tooth. As the runout is
1.041 mm (again, for this runout value only a single tooth
increased to 5 mm while the feed/tooth is held constant,
is cutting). The agreement between the measured and
the surface is de?ned only by the larger radius tooth as
simulated results is reasonable, but less accurate than the x-
shown in panel (b). In panel (c), the runout is slightly
direction data shown in Fig. 7. The reason for the
smaller, 4.2 mm, but the feed/tooth has been doubled. Now
discrepancy is due to the dif?culty of measuring the
the surface is left by the combination of both teeth;
dynamics without in?uencing the system under test. Also,
however, the spatial period is not constant, but varies from
the ?xture design led to slight variation in the workpiece
one tooth to the next. Panel (d) shows the result for a 33 mm
mounting between the multiple cutting tests. Impact tests
runout; again, only one tooth leaves the ?nal surface.
showed up to a 20 Hz shift in the assembly natural
As expected, the spatial period doubles between panels (b)
frequency under different mounting conditions, primarily
and (d). In all cases, good agreement is seen between
due to clearance in the workpiece through holes (see
Fig. 6). For the simulated forces in Fig. 8, the modal
parameters for an average frequency response function
were used to predict all cases.
4. Surface prediction and measurement
Once the force model was veri?ed on both rigid and
?exible workpieces, comparisons were made between
machined and simulated surfaces using: (1) the pre-
viously-de?ned two ?ute, straight tooth cutter; and (2) a
modi?ed 12.7 mm diameter Woodruff cutter. Originally,
the Woodruff cutter had 12 proportionally-spaced teeth
(located at {0, 30, 60, y, 330}1). However, 10 of the
straight teeth were ground away to leave cutting edges
only at the 01 and 2101 orientations. The cutter was then
mounted in the chuck and dynamically balanced to
minimize the in?uence of forced vibrations (due to mass
imbalance) on the machined surfaces. Different cutting
conditions were used for the surface prediction tests than
Fig. 9. Two ?ute, straight tooth cutter: (a) ft ¼ 0.24 mm/tooth, O ¼
for the force model validation to demonstrate the general-
300 rpm, RO ¼ 0.3 mm; (b) ft ¼ 0.24 mm/tooth, O ¼ 300 rpm, RO ¼ 5 mm;
ity of the simulation model. The sample surface ?nish was
(c) ft ¼ 0.51 mm/tooth, O ¼ 150 rpm, RO ¼ 4.2 mm; and (d) ft ¼ 0.51 mm/
recorded using a Wyko NT 1000 non-contact interfero-
tooth, O ¼ 150 rpm, RO ¼ 33 mm.

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experiment and simulation (the reader may note the
Fig. 11 shows results for two 200 rpm tests. In the top
difference in vertical scales between the four panels).
panel, the radius of the tooth oriented at 2101 was 2.3 mm
less than the tooth at 01. The 01 tooth (solid line) primarily
creates the surface, while the 2101 tooth removes a portion
4.2. Non-proportional teeth spacing results
of the cusp off-center to the right. In the bottom panel,
the 2101 tooth had the larger radius (5.5 mm runout). The
The up milling cutting conditions for the ?exible
surface is now generated mainly by the 2101 tooth and the
workpiece cutting trials and simulations using the non-
01 tooth removes a portion of the cusp off-center to the left.
proportional Woodruff cutter were: a ¼ 1.0 mm; b ¼
The axis scales are again equal in the two panels.
0.65 mm; ft ¼ {0.36 and 0.19} mm/tooth; and O ¼ {200
and 400} rpm. The runout values varied between 2.4 and
5. Discussion
32 mm. Example results for the 400 rpm tests are provided
in Fig. 10. It is seen that the surface topography changes
Once the simulation force and surface ?nish predictions
very little between the three panels (note that the scales are
were veri?ed experimentally, the code was used to explore
equal). This is because, even for the smallest runout value
the global effects of runout on surface roughness, stability,
of 2.4 mm, the surface is generated predominantly by a
and surface location error.
single tooth. This result underscores the importance of
minimizing runout in situations that require low surface
5.1. Surface roughness analysis
roughness.
The ?rst task was to organize the surface roughness
prediction information into a useable format. Our intent
was to collect the ‘local view’ of the process available from
time-domain analysis into a ‘global view’ that could be
conveniently used by process planners. Because the surface
texture coef?cients described in ASME B46.1-2002 are
commonly used to express the surface ?nish requirements
on engineering drawings, we focused our attention on the
common metric referred to as roughness average, Ra [29].
The reader may note that we have assumed constant
cutting force coef?cients for all simulations. Practically, it
may be necessary to vary these coef?cients as a function of
feed/tooth and/or cutting speed to accurately capture the
complicated tool–chip interactions over a broad range of
cutting conditions.
Fig. 12 exhibits the relationship between Ra, ft, and
runout for the two ?ute, straight tooth cutter with
Fig. 10. Woodruff cutter with ft ¼ 0.19 mm/tooth and O ¼ 400 rpm: (top)
proportional tooth spacing. As expected, the tendency is
RO ¼ 2.4 mm; (middle) RO ¼ 5.5 mm; and (bottom) RO ¼ 32 mm.
toward higher roughness values with increases in ft, and
runout. However, closer examination shows counterintui-
tive local trends. Fig. 13 shows contours of constant Ra
values as a function of feed/tooth and radial runout (i.e.,
the planar projection of Fig. 12). It is seen that the
roughness average does not increase monotonically with ft,
and runout. Rather, for a constant feed per tooth (see the
vertical line in Fig. 13), Ra is seen to increase and then
decrease as the runout becomes larger. The information in
Fig. 13 allows a process planner to view the trade-off
between runout and surface roughness. For a priori
knowledge of the runout, the maximum feed/tooth value
that satis?es the surface quality requirements can be
selected.
5.2. Stability analysis
Next, simulated Ra data was collected into a map similar
Fig. 11. Woodruff cutter with ft ¼ 0.36 mm/tooth and O ¼ 200 rpm: (top)
to the well-known stability lobe diagram, which identi?es
RO ¼ 2.4 mm, radius for 01 tooth is largest; and (bottom) RO ¼ 5.5 mm,
radius for 2101 tooth is largest.
stable and unstable cutting zones as a function of spindle

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required surface ?nish without sacri?cing high material
removal rates.
5.3. Surface location error analysis
It is well known that time-domain simulation can be
applied to the computation of surface location error, or
part geometric error that arises from forced vibration of
the cutting tool. However, to the authors’ knowledge, the
effect of runout on surface location error has not been
explored in the literature.
The surface location error amplitude depends on the
position of the cutter as it leaves the surface (i.e., as it exits
Fig. 12. Simulated roughness average (Ra) as a function of the feed/tooth
the cut in down milling or enters the cut in up milling). This
(ft) and runout (RO). As expected the roughness increases with RO and ft.
position, in turn, depends on the frequency of forced
oscillations, which is de?ned by the tooth passing
frequency (or spindle speed). Surface location error is,
therefore, spindle speed dependent. The largest variation in
surface location occurs when the fundamental tooth
passing frequency (or one of its harmonics) is near the
natural frequency, fn (in Hz), which corresponds to the
most ?exible structural vibration mode. These ‘sensitive
speeds’, Os in rpm, are de?ned in Eq. (12). The reader may
note that these are the same speeds selected to take advan-
tage of the peaks in the stability lobes as seen in Fig. 14.
60 Á f
O
n
s ¼
;
j ¼ 1; 2; 3; . . . .
(12)
j Á Nt
In the presence of runout, additional frequency content
is observed at the spindle rotational frequency (or runout
frequency) and its harmonics. Therefore, it should be
expected that supplementary sensitive speeds will exist
where the runout frequency or its harmonics are near fn.
Fig. 13. Contour plot of simulated roughness average (Ra) as a function
This will also serve to excite the structural dynamics. The
of the feed per tooth (ft) and runout (RO). It is seen that increasing the RO
does not continuously increase the Ra (heavy dotted line).
full complement of Os values for a two ?ute cutter can then
be expressed as shown in Eq. (13), where the bottom
speed and chip width (axial depth of cut in peripheral end
equation identi?es the runout-dependent speeds.
milling operations). Fig. 14(a) shows the analytical stability
( 60 Á f =j Á N
boundary for a 50% radial immersion down milling cut
n
t
Os ¼
;
j ¼ 1; 2; 3; . . . .
(13)
[30]. Simulation parameters were: 25.4 mm diameter, four
60 Á f n=2j À 1
?ute end mill (proportional teeth spacing with 301 helix
angle) with no runout; a single vibration mode in the
To test these sensitive speeds, simulations were carried
x-
and
y-directions
(f
out for the following conditions: 10% radial immersion
n ¼ 500 Hz,
k ¼ 1 Â 107
N/m,
z ¼ 0.01);
K
down milling; 12.7 mm diameter, two ?ute end mill
tc ¼ 700
N/mm2,
Krc ¼ 210
N/mm2,
K
(proportional teeth spacing with 301 helix angle); a single
te ¼ Kre ¼ 0; and ft ¼ 0.1 mm/tooth. Fig. 14(b) shows
lines of constant Ra as a function of spindle speed and axial
vibration mode in the x- and y-directions (fn ¼ 500 Hz,
depth.2 Clearly, the contour plot captures the stability
k ¼ 1 Â 107 N/m, z ¼ 0.01); Ktc ¼ 700 N/mm2, Krc ¼ 210
behavior. However, rather than identifying the stability
N/mm2, Kte ¼ Kre ¼ 0; ft ¼ 0.1 mm/tooth; and a 3.0 mm
boundary as a single line, or step function, the transition
axial depth of cut (well below the critical stability limit of
from unstable to stable behavior is shown in a more usable
4.6 mm—see Fig. 17—so that stable cuts should be
fashion for process planners. Based on this diagram, once
observed at all spindle speeds). The spindle speed was
the desired surface roughness is known, the cutting
varied over the range from 5000 to 30 000 rpm in
conditions can be deterministically selected to achieve the
increments of 50 rpm and the runout was 20 mm. For this
spindle speed range, the sensitive spindle speeds from
2
Eq. (13) are: {30 000, 10 000, 6000, y} rpm for the runout
A similar plot can be found in Ref. [33], but the ratio of the simulated
roughness to theoretical (geometric) roughness was presented.
frequency content and {15 000, 7500, 5000, y} rpm for the

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9
Fig. 14. (a) Analytical stability lobe diagram for 50% radial immersion down milling cut. (b) Lines of constant Ra from time-domain simulation.
Fig. 15. (a) Surface location error (SLE) versus spindle speed. Variations in the machined surface position are seen when: (1) the tooth passing frequency
or its harmonics are near fn (dashed lines); and (2) when the runout frequency or its harmonics are near fn (dotted lines). (b) Ra versus spindle speed. Poor
surface ?nish is observed when the runout frequency ?rst and third harmonics are near fn; (c) surface pro?le at 15 000 rpm—the ?nal surface location is in
error by +55 mm relative to the commanded location (À6.35 mm), but the cut is stable (Ra ¼ 0.2 mm); (d) surface pro?le at 10 000 rpm—the cut is now
unstable (SLE ¼ À133 mm, Ra ¼ 30 mm).
tooth passing frequency content. The simulation results are
10 050 rpm are shown in Fig. 16. Here, the instability
shown in Fig. 15.
resembles the period-doubling instability (i.e., ?ip bifurca-
In Fig. 15(a) the traditional periodic variation in surface
tion) ?rst reported by Davies et al. [31], but it occurs at
location as the tooth passing frequency and its harmonics
different spindle speeds; the large y direction oscillations
pass through fn is observed (sensitive speeds identi?ed by
occur at fn ¼ 500 Hz. Note that the cut would be stable if
the dashed lines). Runout has no appreciable effect at these
no runout were present. It can also be observed that the cut
speeds. However, strong sensitivity of the error to spindle
is stable with small surface location error when the
speed is also observed when the third and ?fth runout
fundamental runout frequency is near fn ($30 000 rpm).
harmonics are near fn (dotted lines). Fig. 15(b) shows that
In this case, all runout and tooth passing frequency
the surface roughness is also high near these speeds. The
harmonics are located to the right of fn. However, for the
surface pro?les for 15 000 and 10 000 rpm simulations are
10 050 rpm case, the runout third harmonic near fn is
provided in panels Figs. 15(c) and (d), respectively. It is
accompanied by the fundamental tooth passing frequency
seen that the Ra and surface location error values are
on the left and its ?rst harmonic on the right. The
signi?cantly higher for the 10 000 rpm case.
combined forcing frequency content would appear to
To better understand these small regions of large error,
contribute to the local instability(s) observed in Figs. 15
results for a single time-domain simulation completed at
and 17.

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Fig. 16. (a) Force and displacement in the y direction. The displacement is sampled at the tooth passing frequency (circles) and the instability is observed.
(b) Phase space representation of y displacement versus velocity. The 1/tooth sampled points occur on two diverging trajectories.
that occur in the presence of runout when harmonics of the
runout frequency coincide with the system natural fre-
quency were demonstrated.
Acknowledgments
The authors gratefully acknowledge partial ?nancial
support for this research from the National Science
Foundation (DMI-0238019) and Of?ce of Naval Research
(2003 Young Investigator Program). The authors also wish
to recognize the contributions of M. Tummond in the
completion of Figs. 12 and 13.
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Fig. 17. Analytical stability limit and lines of constant Ra for 10% radial
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Document Outline

• Runout effects in milling: Surface finish, surface location error, and stability
  • Introduction
  • Time-domain simulation description
  • Experimental setup for force model validation
  • Surface prediction and measurement
    • Proportional teeth spacing results
    • Non-proportional teeth spacing results
  • Discussion
    • Surface roughness analysis
    • Stability analysis
    • Surface location error analysis
  • Conclusions
  • Acknowledgments
  • References

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