Dynamic Calibration of a Dynamometer
G. C. Foss
Boeing Defense & Space Group
Structural and Environmental Test Labs
E. D. Haugse
Boeing Defense & Space Group
Structural Development
D. W. Princehouse
Boeing Information & Support Systems
Research and Technology
Abstract.
T& USA of frequency response
functions to develop a dynamic calibration matrix for a
three component dynamometer is described. Derivation of the calibration will be presented and its application will be demonstrated using test data from a cutting force dynamometer. The inverted complex matrix
is wed to correct cutting force measurements for structural response and off-axis sensitivity. It will be shown,
for a variety of force inputs to the dynamometer table,
that the usable frequency range can be significantly extended using this technique.
Notation
FRF Frequency Response Function
Hz
Cycles per second
RPM Rwolutions per minute
1 Introduction
Dynamometer tables, used for measurement of cutting
forces, are standard tools in manufacturing research
labs. They are wed to determine optimum feed and
speed rates for manufacturing machinery such as lathes,
mills, and routers. One such dynamometer table is
shown in figure 1.
cutting and the removal of heat. Analysis of these prob
lems requires a knowledge of the periodic input forces
generated by cutting. Unfortunately, most dynamometers were designed for lower cutting speeds and have
structural modes in the frequency range of interest.
In typical operation a dynamometer
table is mounted
to the bed of a machine and a test specimen is bolted
to it. As material is removed from the specimen, piezw
electric load cells, internal to the dynamometer,
measure the instantaneous reaction force from each cut.
The measured forces are resolved into 3 orthogonal
forces and additionally, in some cases, 3 moments. This
paper addresses a 3 force dynamometer only.
As with most transducers, dynamometers
have natural frequencies which limit their useful range. If they
are mounted to a relatively soft machine, the machine
modes may limit the flat frequency response range
even further. Figure 2, for example, is the lowest fre
quency (approximately 930 Hz) mode of the figure 1
dynamometer, mounted to a milliig machine. The
largest square in the stick figure represents four points
on the bed of the machine itself. In this case the mode
involves y-axis motion of the entire machine, 88 well as
flexible motion of the dynamometer table.
A primary goal for manufacturing research is to increase material removal rates so as to improve prw
duction efficiency. This typically requires faster cutter
speeds. Twenty years ago 10000 RPM was considered
to be a very high cutting speed. Currently machinery is being delivered with capabilities of 50000 or even
100000 RPM.
Accurate characterization of cutting forces requires fre
quency response to at least an order of magnitude
greater than the rotational speed. This is necessary
to allow enough terms in a Fourier series to describe
complex periodic cutting forces. With faster cutting
speeds, it is therefore increasingly important to extend
the usable frequency range for cutting force measurements.
At these speeds structural modes, formerly well separated from forcing rates, are now excited to the degree
that they affect product surface finish and machine reliability. Also significant at high speed is the physics of
Natural frequencies for dynamometers of the type presented in this paper, range from 1000 to 4000 Hz. Vendors usually recommend one fifth of this as the usable
range of flat response (200 to 800 Hz). Given the above
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requirements, this type of dynamometer is suitable for
cutting speeds of up to 4800 RPM or less.
The following paragraphs document an effort to extend
the useful frequency range of a dynamometer table by
applying an FRF based calibration to its output force
data. The calibration considers:
1. Machine mounted frequency response, both magnitude and phase.
2.
Quasi-static
3. Cross
output
sensitivity.
This produces the desired result of operating on the
force table outputs to predict the input forces. Equation 3 corrects for unwanted dynamic charzuzteristics
of
the force table by using complex frequency response
functions between the inputs and outputs as a calibration matrix. All translational cross-axis
coupling terms
are included.
axis sensitivity.
While the following details show a specific
application
of the technique, it can be used, in general, to extend
the range or improve the accuracy of any transducer
or measurement system. While the technique is mentioned in reference [l], we have rarely seen all three of
the above corrections applied simultaneously in pramtice. This paper provides teat data that show the effectiveness of the technique.
2 Calibration Matrix
The first step in developing the calibration matrix is
to measure the frequency response functions between
three orthogonal input forces and the three orthogonal
table output forces. Section 3 describes how the three
input forces were applied to the table. Using the fre
quency response functions, the following relationship
between input and output is assumed:
f. =
fu =
F.TJJF. + &TJ=/F,
+ FZTJJF.
F.TJJ,
+ &,TJ.IF.
+ FZTJ,IF.
fz =
FzTJ./F=
+
+ FZTJJF.
FYTJJF.
3 Test Set-up and Instrumentation
An instrumented impact hammer was calibrated to an
NBS traceable standard and verified to have flat respouse within the frequency range of interest. The
hammer handle was pinned to a fork which clamped
to a frame constructed over the dynamometer table.
Thii allowed the hammer to be precisely aligned with
the x, y, and z axes and guaranteed repeatability for
several impacts. A four channel dynamic signal
analyzer (HP35670A)
wa8 used to compute frequency response functions between the hammer load cell and the
dynamometer
x, y, and e outputs. Ensemble averaged
tests were performed in each axis, measuring both the
in-axis
and cross-axis
sensitivities. This resulted in nine
frequency response functions. A separate calibration
was performed for each specimen tested.
4 Transfer Function Calibration
- Demonstration
(1)
or in matrix form:
Where, for example, TJv,Fz is the frequency response
from the input force in the x direction to the table
output force in the y direction. T is a three dimensional
complex array. It represents a 3x3 complex matrix at
each measured frequency point.
Figure 3 represents the three diagonal and six off-axis
coupling terms of the force table frequency response
function matrix for a particular application. Thii is
a graphical representation of equation 1, showing the
magnitude of the frequency response functions. The
plotted data were developed by hammer testing as described in section 3. Figure 4 shows the magnitude
of the inverse transformation ([TJ-’ from equation 3).
Figures 3 and 4 are useful tools to help visualize how
the calibration matrix is affecting force measurements
and the significance of off-axis terms.
The next step is to invert the 3x3 matrix, T, and multiply both sides of equation 2 by [T’)-‘:
Figures 5 and 6 are examples using the calibration matrix, shown in figure 4, to operate on table output spec-
,
(2)
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trums for known input forces in the table x and z directions. Input and output spectrums are compared, with
and without calibration matrix transformation. Figure 7 shows a similar comparison, in the time domain,
for a square wave input, again using the figure 4 calibration transformation.
5 Milling Machine Example
The FRF calibration technique wa8 wed in a milling
test at the Boeing Company. A large, lo-inch diameter, single tooth, end mill made a “slotting” cut in an
aluminum specimen bolted to the dynamometer table.
In this type of cut, the tooth is engaged in the material
over a full 180-degree swath. During a particular teat,
the cutter path intersected two bolt holes in the middle
of the cut; these injected two short sharp transitions to
zero force and excited the mechanical resonances of the
dynamometer acmes a wide spectrum. Cutting through
the bolt holes was not an intended teat, but the resulting data can be used to demonstrate the efficacy of the
calibration technique.
Figure 8 shows the raw measured x-axis force and 8580
ciated theoretical driving force, versus time. The x-axis
force is basically the negative half of a cosine. It begins when the tooth first engages the metal (at 0 degree
rotation), reaches a negative peak at about 90 degrees
(where the tooth is fully engaged in the metal), and
returns to zero when the tooth leaves the metal at 180
degrees. The driving forces are zero for the next halfrotation. Intersections with the bolt holes is modeled
as a simple drop in cutting force for the duration that
the cutter passes through the holes. In addition there is
a surge in x-force when the cutter first enters the material. The standard deviation of the difference between
the raw x-force and theoretical driving force is 32.7 lb.
Note the prominent transient response when the cutter
intersects the bolt holes at about 1.883 seconds. The
frequency of thii response is approximately 1800 Ha, a
result of the series of mechanical resonance shown in
the T/=/F= transfer function. Figure 9 shows the effect
of a low-pass filter with a 3000 Hz cutoff. Both the
theoretical prediction and test data were subjected to
the same 3000 Hz filter. The 1800 Ha signal ia still
prominent. Filtering alone has reduced the standard
deviation to 18.2 lb. Figure 10 shows the result of using
the calibration matrix and 3000 Hz filter. The 1800 Hz
response is further reduced and the calibrated signal is
considerably more accurate in the vicinity of the bolt
holes. The standard deviation reduces to 9.7 lb.
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6 Summary
A technique using frequency response functions, to dynamically calibrate a three component cutting force dynamometer, was developed and evaluated. The results
presented in sections 4 and 5 show that accuracy can
be improved and the usable frequency range can be significantly extended using this technique.
In practice, the effect of the technique is dependent
on the character of the frequency response functions
used to develop the calibration matrix. Figures 5
and 6 illustrate the effect of damping on the technique. Figure 5 shows a series of low damped modes
between 1500 and 1900 Hz. The technique is limited
in the range where the frequency response functions
have steep slopes (low damping). By contrast, figure 6 has two modes at 1100 and 2000 Hz which are
much more damped and the compensation easily re
rncwes them from the dynamometer
output. For the
dynamometer application addressed in this paper, it
was advantageous to we the calibration technique for
all of the test cases that were evaluated.
References
[I] K. G. McConnell, Vibmtion Testing Theory and
Pmctice, John Wiley and Sons, Inc., 1995, Page
201 and following.
Figure 1: Force Dynamometer Table.
*t
q
Figure 2: Dynamometer Table Fundamental Mode (930 He).
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Figure 3: FFlF Matrix from Input Forces to Table Output Forces.
Figure 4: Calibration Matrix from Table Output Forces to Input Forces.
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No
.g
,g
s
e
;
z
.g
w
1.4
1.2
1.0
0.8
0.6
Tronsformotion.
“sing
Hclmmer
x
i-’
0
1000
2000
Frequency ( H Z )
Figure 5: Calibration Effect for X-Direction Hammer Input Force - Frequency Spectrum.
Figure 6: Calibration Effect for Z-Direction Hammer Input Force - Frequency Spectrum.
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3000
I
’
8,
I
’
*
’
I
100
”
I
”
’
Theoretical -
I.BB2
1.836
lime, sec.
Figure 7: Calibration Effect for X-Direction Square
Wave Input Force - Time History.
I
”
.
I
’
”
I
Theoretical-pnfiltered
Figure 9: Measured Force, Compared to Prediction,
Both Filtered at 3000 Hz.
”
-
Compensated
1.882
1 .a04
1.886
lime. sec.
---.
1.680
u
Figure 8: Raw Measured Force Data (32765 HZ Sample
Bate), Compared to Prediction.
lime. sec.
Figure IO: Force Filtered at 3000 Hz and Compensated,
Compared to 3000 Hz Filtered Prediction.
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