Tony L. Schmitz1
e-mail: tschmitz@ufl.edu
G. Scott Duncan
Department of Mechanical and Aerospace
Engineering,
University of Florida,
Gainesville, FL 32611
Three-Component Receptance
Coupling Substructure Analysis
for Tool Point Dynamics
Prediction
In this paper we present the second generation receptance coupling substructure analysis
(RCSA) method, which is used to predict the tool point response for high-speed machining applications. This method divides the spindle-holder-tool assembly into three substructures: the spindle-holder base; the extended holder; and the tool. The tool and
extended holder receptances are modeled, while the spindle-holder base subassembly
receptances are measured using a “standard” test holder and finite difference calculations. To predict the tool point dynamics, RCSA is used to couple the three substructures.
Experimental validation is provided. 关DOI: 10.1115/1.2039102兴
Keywords: high-speed machining, milling, stability, beam, finite element
1
Introduction
One area of manufacturing research that has made significant
technological advancements in recent years is high-speed machining. Machine improvements include new spindle designs for
higher rotational speed, torque, and power; increased slide speeds
and accelerations; direct drive linear motor technology; and new
machine designs for lower moving mass. The combination of new
machine technology and tool material/coating developments often
makes high-speed machining a viable alternative to other manufacturing processes. A key application example is the aerospace
industry, where dramatic increases in material removal rates made
possible using high-speed machining techniques have allowed designers to replace assembly-intensive sheet metal build-ups with
monolithic aluminum components resulting in substantial cost
savings 关1兴.
A primary obstacle to the successful implementation of highspeed machining and full use of the available technology is chatter, or unstable machining. Many research efforts geared toward
the understanding and avoidance of chatter have been carried out
共e.g., see early studies in 关2–11兴兲. This work has led to the development of stability lobe diagrams that identify stable and unstable
cutting zones as a function of the chip width and spindle speed.
However, the methods used to produce these diagrams, whether
analytic or time-domain, require knowledge of the tool point dynamics. The required dynamic model is typically obtained using
impact testing, where an instrumented hammer is used to excite
the tool at its free end 共i.e., the tool point兲 and the resulting vibration is measured using an appropriate transducer, typically a low
mass accelerometer. However, due to the large number of spindle,
holder, and tool combinations, the required testing time can be
significant. Therefore, a model which is able to predict the tool
point response based on minimum input data is the preferred alternative.
The purpose of this paper is to build on the previous work of
Schmitz et al. 关12–15兴, which describes the tool point frequency
response function, or receptance, prediction using the receptance
coupling substructure analysis 共RCSA兲 method. In these previous
1
Author to whom correspondence should be addressed.
Contributed by the Manufacturing Engineering Division for publication in the
ASME JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received
April 6, 2004; final revision received February 4, 2005. Review conducted by: Y. C.
Shin.
studies, a two component model of the machine-spindle-holdertool assembly was defined. The machine-spindle-holder
displacement-to-force receptance was recorded using impact testing, while the tool was modeled analytically. The tool and
machine-spindle-holder substructure receptances were then
coupled through translational and rotational springs and dampers;
see the model in Fig. 1, where kx and k are the translational and
rotational springs, cx and c are the translational and rotational
viscous dampers, component A represents the tool, and component
B the machine-spindle-holder. While the purpose of the springs
and dampers between the tool and holder was to capture the effects of a potentially nonrigid, damped connection, it is likely that
these connections also served to compensate for the fact that the
displacement-to-moment, rotation-to-force, and rotation-tomoment receptances at the free end of the holder were assumed
zero 共i.e., perfectly rigid兲. Although it was shown in Ref. 关15兴 that
this two component model provides a valid approximation for a
flexible tool clamped in a stiff spindle-holder, it does not offer the
most generalized solution.
In order to enable RCSA predictions for a wider variety of
machine-spindle-holder-tool combinations, an improved threecomponent model is presented here. In this model, the machinespindle-holder substructure is separated into two parts: 共1兲 the
machine, spindle, holder taper, and portion of the holder nearest
the spindle with standard geometry from one holder to another
共hereafter referred to as the spindle-holder base subassembly兲; and
共2兲 the remaining portion of the holder from the base to the free
end 共hereafter referred to as the extended holder subassembly兲. A
technique for determining the rotation-to-force/moment and
displacement-to-moment receptances for the free end of the
spindle-holder base subassembly using only displacement-to-force
measurements is also described. The experimental procedure involves direct and cross displacement-to-force measurements of a
simple geometry “standard” holder clamped in the spindle to be
modeled. The portion of the standard holder beyond the section
with consistent geometry from holder-to-holder is then removed
in simulation using an inverse receptance coupling approach 共i.e.,
decomposition兲 to identify the four spindle-holder base subassembly receptances. These receptances are then coupled to models of
the actual holder and tool. In the following sections, the method is
described and experimental validation is presented.
Journal of Manufacturing Science and Engineering
Copyright © 2005 by ASME
NOVEMBER 2005, Vol. 127 / 781
Fig. 1 Previous two-component RCSA model. An external
force, Fa„t…, is applied to the free end of the tool „A… to determine the assembly Xa / Fa receptance. The tool is coupled to the
machine-spindle-holder „B… through springs and dampers
2
Background and Notation
Substructure analysis, or component mode synthesis, methods
have been used for several decades to predict the dynamic response of complicated assemblies using measurements and/or
models of the individual components, or substructures. These
components can be represented by spatial mass, stiffness, and
damping data, modal data, or receptances 共e.g., 关16–30兴兲. The
latter representation is preferred in situations where the assembly
receptances are the desired analysis output, as is the case in this
research. For an assembly consisting of two rigidly connected
substructures, as shown in Fig. 2, the assembly receptance,
G jk共兲, can be expressed as shown in Eq. 共1兲, where is the
frequency, X j and ⍜ j are the assembly displacement and rotation
at coordinate j, and Fk and M k are the force and moment applied
to the assembly at coordinate k. If coordinate j is coincident with
coordinate k, the receptance is referred to as a direct receptance;
otherwise, it is a cross receptance. For the purposes of this paper,
the nomenclature G jk共兲 is used to describe the receptances that
are produced when two substructures 共or subassemblies兲 are
coupled to produce the final assembly. The nomenclature GS jk共兲
will replace G jk共兲 in all relevant equations when two substructures 共or subassemblies兲 are coupled that do not form the final
assembly.
G jk共兲 =
冋 册
Xj
Fk
Xj
Mk
⍜j
Fk
⍜j
Mk
=
冋
H jk L jk
N jk P jk
册
R jk共兲 =
冋 册冋
j
fk
xj
mk
j
mk
=
h jk l jk
n jk p jk
册
共2兲
Based on the coordinates defined in Fig. 2, the equations to
determine the assembly direct receptances, Gaa共兲 and Gdd共兲,
and the assembly cross receptances, Gad共兲 and Gda共兲, can be
written as a function of the substructure receptances as shown in
Eqs. 共3兲–共6兲, where rigid connections have been applied 关32兴.
Fig. 2 Two-component assembly. The component responses
are coupled through a rigid connection to give the assembly
receptance„s…
782 / Vol. 127, NOVEMBER 2005
Gaa共兲 =
Gdd共兲 =
共1兲
The substructure receptances, R jk共兲, are defined in Eq. 共2兲,
where x j and j are the substructure displacement and rotation at
coordinate j, and f k and mk are the force and moment applied to
the substructure at coordinate k 关15,31兴.
xj
fk
Fig. 3 Example standard holder for spindle-holder base subassembly receptance identification „dimensions provided in
Table 1…. Hammer impacts are completed at locations 3, 3b, and
3c to identify the required direct and cross receptances
冋 册
冋 册
冋 册
冋 册
Xa
Fa
⍜a
Fa
Xd
Fd
⍜d
Fd
Xa
Ma
⍜a
Ma
Xd
Md
⍜d
Md
= Raa共兲 − Rab共兲关Rbb共兲 + Rcc共兲兴−1Rba共兲
共3兲
= Rdd共兲 − Rdc共兲关Rbb共兲 + Rcc共兲兴−1Rcd共兲
共4兲
Gad共兲 =
Xa
Fd
⍜a
Fd
Xa
Md
⍜a
Md
= Rab共兲关Rbb共兲 + Rcc共兲兴−1Rcd共兲 共5兲
Gda共兲 =
Xd
Fa
⍜d
Fa
Xd
Ma
⍜d
Ma
= Rdc共兲关Rbb共兲 + Rcc共兲兴−1Rba共兲 共6兲
As noted, in order to populate the substructure receptance matrices, we apply measurement and modeling. Common modeling
options include closed-form expressions for uniform EulerBernoulli beams 关33兴 and finite element solutions 共which can incorporate the more accurate Timoshenko beam model 关34兴兲. We
consider both approaches in this study. As a convenience to the
reader, the relevant analytical formulas and finite element Timoshenko stiffness and mass matrices are included in the Appendix.
3
Spindle-Holder Base Subassembly Identification
The experimental procedure used to determine the receptances
at the free end of the spindle-holder base subassembly, GS jk共兲, is
described in this section. It is composed of three primary steps.
First, the standard holder displacement-to-force direct and cross
receptances are determined by impact testing. The standard holder
geometry, which was selected to approximate a broad range of
potential holders, is provided in Fig. 3. Second, these results are
used to determine the three other direct receptances at the free end
of the standard holder. Third, the section of the standard holder
which is not common to other holders 共see Fig. 4兲 is removed
using inverse receptance coupling to determine all four spindleholder base subassembly receptances. Each step of the procedure
is described in the following sections. Example results are included.
3.1 Standard Test Holder Receptances Once the standard
holder is mounted in a spindle 共see Fig. 3兲, the four subassembly
Transactions of the ASME
Table 1 Standard holder substructure parameters
Coordinate j
Coordinate k
do 共mm兲
L 共mm兲
共kg/ m3兲
E 共N / m2兲
GS34共兲 =
receptances are determined by measuring the direct, H33, and
cross, H33b, and H33c, displacement-to-force receptances on the
standard holder, applying a second-order backward finite difference method to find L33 共and, equivalently, N33兲 关35兴, and then
synthesizing P33. For the cross displacement-to-force measurements, the distance S should be selected to increase the difference
in relative amplitudes between H33, H33b, and H33c without leading to a poor signal-to-noise ratio for the H33c measurement 共i.e.,
many of the lower frequency spindle-holder modes resemble a
fixed-free fundamental mode shape and have very small amplitudes near the spindle face for the bandwidth of interest兲. Practically, we have observed that the finite difference results improve
as S is increased; however, care must be taken to ensure that the
location of the H33c measurement provides sufficient signal-tonoise. The receptance L33 is determined from the measured
displacement-to-force receptances using Eq. 共7兲. By reciprocity,
N33 can be set equal to L33. The remaining receptance, P33, is
synthesized from H33, L33, and N33, as shown in Eq. 共8兲 关27兴.
L33 =
3H33 − 4H33b + H33c
2S
L332
1
⍜3 F3 X3 ⍜3
P33 =
=
=
L33N33 =
M 3 X3 M 3 F3 H33
H33
共7兲
3.2 Extended Holder Subassembly Model. The extended
holder subassembly for the steel standard holder consisted of
solid, cylindrical substructures I and II as shown in Fig. 4. Equations 共9兲–共12兲 provide the direct and cross extended holder subassembly receptance matrices, where rigid coupling has been applied. These equations were determined from Eqs. 共3兲–共6兲 by
appropriate substitutions.
GS33共兲 =
GS44共兲 =
冋 册
X3
M3
⍜3
M3
冋 册
X4
F4
⍜4
F4
X4
M4
⍜4
M4
= R33共兲 − R33a共兲关R3a3a共兲
+ R3b3b共兲兴−1R3a3共兲
共9兲
= R44共兲 − R43b共兲关R3b3b共兲
+ R3a3a共兲兴−1R3b4共兲
Journal of Manufacturing Science and Engineering
冋 册
冋 册
共10兲
X3
M4
⍜3
M4
X3
F4
⍜3
F4
X4
M3
⍜4
M3
X4
F3
⍜4
F3
3
3a
63.3
62.8
3b
4
52.7
16.3
= R33a共兲关R3a3a共兲 + R3b3b共兲兴−1R3b4共兲
共11兲
= R43b共兲关R3a3a共兲 + R3b3b共兲兴−1R3a3共兲
共12兲
3.3 Spindle-Holder Base Subassembly Receptance. The
spindle-holder base subassembly receptance matrix, G33共兲, can
be expressed as shown in Eq. 共13兲 by rewriting Eq. 共3兲. The lefthand side of this equation is known once the steps described in
Sec. 3.1 are completed. Also, the extended holder subassembly
receptances, GS33, GS44, GS34, and GS43, are determined using the
equations provided in Sec. 3.2. Therefore, Eq. 共13兲 can be rewritten to solve for the spindle-holder base subassembly receptances,
GS55共兲. See Eq. 共14兲.
G33共兲 =
冋
共8兲
Due to the subtraction of the similarly scaled H33, H33b, and
H33c receptances, noise in the measurement data can detrimentally
affect the quality of L33 and N33 共produced by the finite-difference
method兲 and, therefore, P33. To reduce the noise effect, the measured receptance data were smoothed using a Savitzky-Golay filter, which performs a local polynomial regression to determine the
smoothed value for each data point 关36兴, prior to the application of
Eq. 共7兲. For this study, filters with polynomial orders of two or
three were applied over windows of 31 to 81 data points.
X3
F3
⍜3
F3
GS43共兲 =
II
7800
2 ⫻ 1011
0.0015
Fig. 4 Standard holder substructures for inverse receptance
coupling
I
GS55共兲 =
H33 L33
N33 P33
册
冋 册
x5
f5
5
f5
x5
m5
5
m5
= GS33共兲 − GS34共兲关GS44共兲
+ GS55共兲兴−1GS43共兲
共13兲
= GS34共兲关GS33共兲 − G33共兲兴−1
⫻GS43共兲 − GS44共兲
共14兲
Tests were completed to determine GS55共兲 for a
24,000 rpm/ 40 kW direct drive spindle 共HSK 63A interface兲 using a steel standard holder. The dimensions and material properties for the standard holder substructures are provided in Table 1,
where do is the diameter, L is the length, is the density, and is
the frequency-independent damping coefficient. The values
used in this study were determined experimentally from free-free
testing of representative cylindrical rods. During the measurement
of the direct and cross receptances for the mounted standard
holder, the distance S was selected as 25.40 mm. The resulting
spindle receptances, h55, l55, and p55, are shown in Fig. 5. These
results are based on the average of 15 complete measurement sets
共H33, H33b, and H33c—each the average of ten impacts兲. Our experience has shown that averaging is the most effective technique
for reducing the inherent noise amplification during the finite difference computations.
As shown in Eq. 共14兲, the Fig. 5 result was determined by
removing the extended holder subassembly for the standard holder
from the complete assembly in simulation. Because the measurement bandwidth for high-speed/high-power spindle testing is typically 5 kHz or less, we have found that it makes no practical
difference whether the Euler-Bernoulli or Timoshenko beam
model is used to describe the standard holder substructure共s兲. The
standard holder behaves basically as an inertial mass since its
NOVEMBER 2005, Vol. 127 / 783
Fig. 7 Measured „two nominally identical holders… and predicted H33 results for tapered thermal shrink fit holder „25.3 mm
bore…
Fig. 5 Spindle receptances G55„… determined from standard
holder direct and cross receptance measurements
clamped-free bending mode fundamental natural frequency, for
the geometry used in this study, is outside the bandwidth of interest.
3.4 Holder
Experimental
Verification.
Once
the
24,000 rpm/ 40 kW spindle-holder base subassembly receptances,
GS55共兲, were determined, it was possible to couple this result to
arbitrary holder geometries to predict the receptance at any coordinate on the machine-spindle-holder assembly. To validate the
procedure, a tapered thermal shrink fit holder 共25.3 mm bore兲 with
an HSK 63A spindle interface was divided into 12 substructures
beyond the spindle-holder base subassembly as shown in Fig. 6.
Each substructure was assumed to be a hollow or solid cylindrical
steel beam, as appropriate. Table 2 provides the holder geometry
and assumed material properties.
The first step in predicting the assembly response, as described
in Sec. 3.2, was to couple substructures I–XII to produce the
direct and cross extended holder subassembly receptances at coordinates 3 and 4. With the increase in substructures from 2 to 12,
the Sec. 3.2 procedure remained the same; however, substructure I
was first coupled to substructure II, then the resulting subassembly
was coupled to substructure III, and so on to produce the required
extended holder subassembly receptances.
The next step was to rigidly couple the spindle-holder base
subassembly 共determined in the previous section and shown in
Fig. 5兲 to the extended holder subassembly using Eq. 共13兲 to determine the receptances at the free end of the holder, G33共兲.
Figure 7 shows the predicted H33 result as well as measurements
for two nominally identical holders. The Euler-Bernoulli beam
model was applied to develop the extended holder receptances in
this case.
4
Tool Point Response Prediction
To predict the tool point dynamics, the modeling procedure was
again applied to the 24,000 rpm/ 40 kW spindle 共HSK 63A interface兲 using a tapered thermal shrink holder with a 19.1 mm carbide tool blank inserted as shown in Fig. 8. The assembly was
divided into the spindle-holder base subassembly and 13 cylindrical substructures of differing diameters; see Table 3. To model the
receptances, a composite modulus and mass were substituted for
substructures II–VIII to account for the material differences between the steel holder and the carbide tool blank. Also, the mass
expression for these substructures 共provided in the Appendix兲 was
replaced with the composite mass shown in Eq. 共15兲, where h
and t are the density of the holder and tool, respectively. Additionally, the product of the elastic modulus and second area moment of inertia, EI, was replaced by the product shown in Eq.
共16兲, where Eh is the holder modulus, Et is the tool material
modulus, and Ih and It are the second area moments of inertia for
the holder and tool, respectively. The substructure parameters are
shown in Table 3.
Fig. 6 Tapered thermal shrink fit holder „25.3 mm bore… substructure model
Table 2 Tapered thermal shrink fit holder „25.3 mm bore… substructure parameters
Substructure
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
di 共mm兲
do 共mm兲
L 共mm兲
共kg/ m3兲
E 共N / m2兲
25.3
44.2
5.5
25.3
45.1
5.5
25.3
46.1
5.5
25.3
47.0
5.5
25.3
47.9
5.5
25.3
48.9
5.5
25.3
49.8
5.5
26.0
50.7
5.5
26.0
51.7
5.5
26.0
52.6
5.5
26.0
52.6
15.7
52.6
30.3
784 / Vol. 127, NOVEMBER 2005
7800
2 ⫻ 1011
0.0015
Transactions of the ASME
Fig. 8 Tapered thermal shrink fit holder with 19.1-mm-diam tool blank substructure model
m=
共h共d2o − di2兲 + tdi2兲L
4
EI = EhIh + EtIt =
共15兲
Eh共d4o − di4兲 + Etdi4
64
共16兲
The next step was to rigidly couple substructures I through XIII
to produce the direct and cross extended holder-tool subassembly
receptances at coordinates 1 and 4. The final step in the procedure
was to predict the tool point dynamics by rigidly coupling the
extended holder-tool subassembly to the spindle-holder base subassembly. With the appropriate coordinate substitution in Eq. 共13兲,
the tool point receptance, G11共兲, was determined according to
Eq. 共17兲, where the receptances associated with coordinates 1 and
4 are the extended holder-tool subassembly direct and cross receptances. The predicted and measured assembly tool point
displacement-to-force receptances, H11, are displayed in Fig. 9. In
this figure, results for both Euler-Bernoulli and Timoshenko 共finite
element兲 beam models are provided. It is seen that the finite element model 共100 elements were used for each substructure兲 dominant natural frequency is closer to the measured result, as expected. However, the predicted natural frequency is still
approximately 50 Hz higher. This disagreement is explored in Sec.
5.3.
G11共兲 =
冋
H11 L11
N11 P11
册
= GS11共兲 − GS14共兲关GS44共兲
+ GS55共兲兴−1GS41共兲
5
共17兲
Case Studies
5.1 Geared Quill-Type Spindle. In this section, prediction
and measurement results are provided for two cutters coupled to a
geared, quill-type spindle with a CAT-50 spindle-holder interface
共Big-Plus tool holders were used which include both taper and
face contact兲. The spindle-holder base subassembly receptances
were determined using a steel cylindrical standard holder 共63.4
mm diameter and 89.0 mm length兲; the cross FRF measurements
were again recorded at distances of 25.4 mm and 50.8 mm from
the free end of the standard holder. The substructure receptances
for the solid body tools 共i.e., both cutting tools were composed of
solid steel modular bodies with carbide inserts attached兲 were then
computed and the tool point FRF predicted by rigidly coupling the
tool models to the spindle measurements.
Figure 10 displays the H11 results for an inserted endmill with 4
“flutes” 共20 total inserts兲. The tool body geometry is defined in
Table 4 共as before substructure I is nearest the free end of the
clamped cutter兲. Figure 11 shows the H11 measurement and prediction for a 28-insert facemill 共see Table 5兲. In both cases, EulerBernoulli beam models were employed to describe the standard
holder and cutter bodies.
5.2 Geared Spindle Comparison. In this section, the
spindle-holder base subassembly receptances were measured on
two nominally identical, geared spindles 共CAT-50 holder-spindle
interface兲. The steel cylindrical standard holder was 63.4 mm in
diameter and 89.0 mm long. The cross FRF measurement locations were the same as specified previously. Figure 12 provides
standard holder direct FRF measurement results for both spindles.
Three curves are shown: the solid line 共line 1a兲 represents the
average of five measurement sets 共10 impacts each兲 completed
without removing the holder from the first spindle 共i.e., spindle 1兲;
the dotted line gives the average of three more spindle 1 measurements after removing and replacing the holder 共line 1b兲; and the
dashed line shows the average of five spindle 2 measurements
共line 2兲. These curves show that, although the spindles are similar,
the difference between the spindle dynamics is larger than the
measurement divergence.
Next, a 16-insert solid body facemill was inserted in spindle 1
and the tool point FRF recorded. Predictions were finally completed using both the spindle 1 and 2 receptances. This result is
provided in Fig. 13; the facemill geometry and material properties
are given in Table 6. It is seen that the prediction completed using
the spindle 1 receptances 共dashed line兲 more accurately identifies
the spindle 1 measured frequency content 共solid line兲. Therefore,
it would be necessary to measure both spindles to make accurate
predictions, rather than relying on manufacturing repeatability. It
has been our experience that the dynamic consistency between
spindles is manufacturer-dependent.
5.3 Shrink Fit Holder With Varying Tool Length. In this
study 30 carbide tool blanks were sequentially inserted in a tapered thermal shrink fit holder and the tool point response recorded. The insertion length was maintained at 22.9 mm while the
overhang length varied from 66.0 to 142.2 mm in increments of
2.5 mm 共the 139.7 overhang length test was not completed兲 for
the 19.1-mm-diam tool blanks. These measurements were completed on a 16,000 rpm direct drive spindle with an HSK 63A
spindle-holder interface. The substructure information is provided
in Table 7. The 30 measurement results are shown in the top panel
of Fig. 14, while the bottom panel shows the h55 spindle response
Table 3 Tapered thermal shrink fit holder and 19.1 mm diameter tool blank substructure parameters
Substructure
I
II
III
IV
di 共mm兲
do 共mm兲
L 共mm兲
共kg/ m3兲
E 共N / m2兲
19.1
111.9
19.1
33.4
5.8
19.1
34.4
5.8
19.1
35.4
5.8
V
VI
VII
VIII
IX
19.1
19.1
19.1
19.1
19.1
36.4
37.5
38.5
39.5
39.5
5.8
5.8
5.8
5.8
4.1
7800 共steel holder兲 14,500 共carbide tool blank兲
2 ⫻ 1011 共steel holder兲 5.85⫻ 1011 共carbide tool blank兲
0.0015
Journal of Manufacturing Science and Engineering
X
XI
XII
XIII
19.1
40.4
4.1
19.1
41.4
4.1
19.1
41.4
10.6
41.4
37.4
NOVEMBER 2005, Vol. 127 / 785
Fig. 9 Measured and predicted H11 results for tapered thermal
shrink fit holder with 19.1-mm-diam tool blank „111.9 mm overhang length…
Fig. 11 Measured and predicted H11 results for 28-insert
facemill
共i.e., after removing the extended portion of the standard holder in
simulation兲. It is seen in the top panel that, although the general
trend is increased amplitude and reduced frequency with increasing overhang length, the tool point magnitudes are attenuated near
800 and 1200 Hz. This is due to dynamic interaction between the
tool clamped-free mode and the spindle modes 关37兴. The fact that
the spindle natural frequencies agree with the locations of the
dynamic interactions 共see bottom panel of Fig. 14兲 suggests that
the spindle response has been properly identified.
Predictions of the tool point responses using the spindle receptances and Timoshenko beam elements 共100 for each substructure兲
to model the tool and holder showed similar disagreement in natural frequency to the results provided in Fig. 9. Reasonable perturbations to the model parameters were unable to close the approximately 50 Hz gap. Therefore, translational and rotational springs
and viscous dampers 共as shown in Fig. 1兲 were inserted between
the holder and tool to account for what was presumed to be a
nonrigid connection 共even for the shrink fit test case studied here兲.
The spring and damper values were then determined using a nonlinear least-squares best fit 关15兴. The least-squares algorithm was
initiated using connection parameters obtained from a visual fit
and continued until the frequency-dependent residual between the
predicted and measured H11 results was less than 1 ⫻ 10−15 m / N.
The four parameter values were constrained to be zero or greater,
but no other restrictions were applied. The average values for the
connection parameters 共see Table 8兲 were then used to make predictions for various overhang lengths. The predictions were carried out using Eq. 共18兲, where
K=
Table 5 Solid
parameters
冋
kx + icx
0
0
k + ic
body
facemill
„28
册
.
inserts…
substructure
Substructure
I
II
III
IV
do 共mm兲
L 共mm兲
共kg/ m3兲
E 共N / m2兲
126.2
55.0
130.3
18.3
80.0
62.7
69.9
18.3
7800
2 ⫻ 1011
0.0015
Fig. 10 Measured and predicted H11 results for 20-insert
endmill
Table 4 Solid
parameters
body
endmill
„20
inserts…
substructure
Substructure
I
II
III
do 共mm兲
L 共mm兲
共kg/ m3兲
E 共N / m2兲
99.8
85.6
80.1
94.9
7800
2 ⫻ 1011
0.0015
69.9
16.8
786 / Vol. 127, NOVEMBER 2005
Fig. 12 Standard holder direct receptances for two nominally
identical, geared spindles „CAT-50 holder-spindle interface….
Line 1a „solid… shows the average of five measurement sets
completed without removing the holder from spindle 1; line 1b
„dotted… gives the average of three more spindle 1 measurements after removing and replacing the holder; line 2 „dashed…
shows the average of five spindle 2 measurements
Transactions of the ASME
Fig. 13 Measured and predicted H11 results for 16-insert facemill. Results are shown for predictions from spindle 1 „dashed…
and spindle 2 „dotted… standard holder measurements. Measurement recorded using spindle 1
G11共兲 =
冋
H11 L11
N11 P11
册
= GS11共兲 − GS14共兲关GS44共兲
+ GS55共兲 + K−1兴GS41共兲
共18兲
Figure 15 shows the measured and predicted results for four
different overhang lengths. These lengths were selected to provide
results: 共1兲 near the 1200 Hz interaction frequency shown in Fig.
14 共76.2 mm兲; 共2兲 between the interactions at 800 and 1200 Hz
共94.0 mm兲; 共3兲 near the 800 Hz interaction 共106.7 mm兲; and 共4兲 to
the left of the 800 Hz interaction 共132.1 mm兲. Reasonable agreement is observed in all cases. To determine the impact of the
residual disagreement, however, stability lobes were constructed
using the 94.0 mm overhang case for both the measured and predicted tool point receptances 关38兴. A 50% radial immersion upmilling cut using a four-flute cutter with cutting force coefficients
of 800 N / mm2 and 0.3 was assumed for demonstration purposes.
This result is provided in Fig. 16. Although there is a shift toward
lower speeds for the lobes computed using the predicted receptance 共due to the underprediction of the natural frequency兲, the
diagram does not exhibit extreme sensitivity to this frequency
error. Based on this result, while the use of finite connection stiffness values, i.e., a nonzero K−1 matrix in Eq. 共18兲, may improve
the receptance prediction accuracy, a rigid connection appears to
be adequate to guide the selection of stable cutting conditions
provided points near the stability boundaries are not chosen.
Table 6 Solid
parameters
body
Fig. 14 Measurement results for thermal shrink fit tool holdertool blank case study. „Top panel… 30 different carbide blanks
were sequentially inserted and the tool point receptance recorded. „Bottom panel… The spindle displacement-to-force receptance identified using the standard holder
facemill
„16
inserts…
substructure
Substructure
I
II
III
do 共mm兲
L 共mm兲
共kg/ m3兲
E 共N / m2兲
279.4
27.2
63.5
88.9
7800
2 ⫻ 1011
0.0015
69.9
15.9
6
Conclusions
Tool point dynamics prediction using the second generation
RCSA method was demonstrated. The improved method includes
the following features: 共1兲 separation of the spindle-holder-tool
assembly into three substructures—the spindle-holder base, extended holder, and tool; 共2兲 experimental identification of the
spindle-holder base subassembly translational and rotational receptances using a finite difference approach; 共3兲 analytical and
finite element modeling of the holder and tool substructure receptances; and 共4兲 rigid coupling of the spindle-holder base subassembly to the extended holder and rigid or flexible/damped coupling of the tool to this result to determine the tool point response.
Experimental validation of the method was provided for multiple
spindle-holder-tool setups.
Acknowledgments
This work was partially supported by the National Science
Foundation 共Grant No. DMI-0238019兲, the Office of Naval Research 共2003 Young Investigator Program兲, the Naval Surface
Warfare Center—Carderock Division, and BWXT Y-12. Any
opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of these agencies. The authors also wish
to acknowledge contributions to the development of the RCSA
method by Dr. M. Davies, University of North Carolina-Charlotte,
Charlotte, NC, and Dr. T. Burns, National Institute of Standards
and Technology, Gaithersburg, MD. They also acknowledge Mr.
R. Ketron, Caterpillar, Inc., Aurora, IL, Ms. J. Dyer, Eastside High
School, Gainesville, FL, Mr. Duke Hughes, BWXT Y-12, Oak
Ridge, TN, and Dr. P. Jacobs, BWXT Y-12, for their assistance in
collecting portions of the data used in this study.
Table 7 Shrink fit holder case study substructure parameters
Substructure
I
di 共mm兲
do 共mm兲
L 共mm兲
共kg/ m3兲
E 共N / m2兲
19.1
Varied
II
III
IV
V
19.1
19.1
19.1
21
35.0
36.1
37.3
38.2
11.4
11.4
9.1
25.0
7800 共steel holder兲 14,500 共carbide tool blank兲
2 ⫻ 1011 共steel holder兲 5.85⫻ 1011 共carbide tool blank兲
0.0015
Journal of Manufacturing Science and Engineering
VI
6
38.5
17.0
NOVEMBER 2005, Vol. 127 / 787
Table 8 Connection parameters for shrink fit holder case
study
kx共N / m兲
k共N / rad兲
cx共N s / m兲
c共N s / rad兲
6.5⫻ 107
3.4⫻ 106
520
3540
Fig. 16 Example stability lobes „50% radial immersion upmilling cut using a four-flute cutter with cutting force coefficients of 800 N / mm2 and 0.3… developed using measured „solid
line… and predicted „dotted line… H11 results for 94.0 mm overhang length
Fig. 15 Measured and predicted H11 results for four different
overhang lengths „132.1, 106.7, 94.0, and 76.2 mm…. The overhang length for each of the four results is identified. Predictions were completed using the flexible/damped connection
„connection parameters are provided in Table 8…
l jj = − lkk =
− F1
EI共1 + i兲2F3
l jk = − lkj =
共A2兲
n jj = − nkk =
− F1
EI共1 + i兲2F3
n jk = − nkj =
Bishop and Johnson 关33兴 showed that the displacement and
rotation-to-force and moment receptances for uniform EulerBernoulli beams could be represented by simple closed-form expressions. For a cylindrical free-free beam with coordinates j and
k identified at each end, the frequency-dependent direct and cross
receptances are given by:
− F5
EI共1 + i兲3F3
h jk = hkj =
F8
EI共1 + i兲3F3
p jj = pkk =
F6
EI共1 + i兲F3
p jk = pkj =
F7
共A4兲
EI共1 + i兲F3
where E is the elastic modulus, I is the second area moment of
inertia, is the frequency-independent damping coefficient
共damping was not included in Bishop and Johnson, but has been
added as part of this analysis兲, and:
共A1兲
F1 = sin L sinh L
− F10
EI共1 + i兲2F3
共A3兲
Appendix: Beam Receptance Modeling
h jj = hkk =
F10
EI共1 + i兲2F3
4 =
2m
EI共1 + i兲L
共A5兲
F3 = cos L cosh L − 1 F5 = cos L sinh L − sin L cosh L
F6 = cos L sinh L + sin L cosh L
F7 = sin L + sinh L
F8 = sin L − sinh L
F10 = cos L − cosh L.
共A6兲
In Eq. 共A5兲, the cylindrical beam mass is given by
m=
共d2o − di2兲L
,
4
where do is the outer diameter, di is the inner diameter 共set equal to zero if the beam is not hollow兲, L is the length, and is the density;
the cylinder’s second area moment of inertia is
I=
共d4o − di4兲
;
64
and is the frequency 共in rad/s兲.
The Timoshenko beam model, which includes the effects of rotary inertia and shear, was implemented using finite elements 关34兴.
Each four degree-of-freedom 共rotation and displacement at both ends兲 free-free beam section was modeled using appropriate mass, M,
and stiffness, K, matrices 关39兴. The mass matrix was:
788 / Vol. 127, NOVEMBER 2005
Transactions of the ASME
M=
Al
共1 + 兲2
+
冤
13
35
+
7
10
+
2
3
9
11
3
11
+ 120 + 24 兲l
共 210
70 + 10 + 6
13
3
+ 40 + 24 兲l
共 1051 + 60 + 120
兲l2 共 420
2
2
2
2
13
35
+
7
10
+
2
3
冉冊
Al
rg
共1 + 兲2 l
2
冤
共 101 − 2 兲l
−
共 152 + 6 + 3 兲l2
2
6
5
− 共 10 −
1
6
5
Symmetric
2
兲l
兲
兲
兲
兲
2
3
40 + 24 l
1
2 2
140 + 60 + 120 l
11
2
11
210 + 120 + 24 l
1
2 2
105 + 60 + 120 l
13
−共
−共
共
Symmetric
6
5
− 共 420 +
共 101 − 2 兲l
− 共 30 +
1
共
共
6
+
2
6
兲 l2
兲l
兲 l2
1
− 10 − 2
2
2
15 + 6 + 3
冥
冥
where A is the cross-sectional area, l is the section length, rg is the radius of gyration, and is a shear deformation parameter given by
=
12EI共1 + 兲
,
k⬘GAl2
where
G=
E
2共1 + 兲
is the shear modulus 共 is Poisson’s ratio兲 and k⬘ is the shear coefficient which depends on the cross-section shape and 关40兴. The
stiffness matrix 共which included damping兲 was:
EI共1 + i兲
K= 3
l 共1 + 兲2
冤
k⬘AG2
+
4l共1 + 兲2
12
6l
− 12
6l
共4 + 2 + 2兲l2 − 6l 共2 − 2 − 2兲l2
12
冤
4
2l − 4
2l
l2 − 2l
l2
4
Symmetric
The element M and K matrices were then collected into the global
mass, M, and stiffness, K, matrices using Guyan reduction 关34兴
and the resulting equation of motion solved in the frequency domain. See Eq. 共A7兲, where n elements have been applied.
冤 冥冤 冥
x1
f1
m1
1
f2
x2
关− M2 + K兴 · 2 = m2
]
]
f n+1
xn+1
mn+1
n+1
共A7兲
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Transactions of the ASME