Acoustics 2002 - Innovation in Acoustics and Vibration
Annual Conference of the Australian Acoustical Society
13-15 November 2002, Adelaide, Australia
FREE-FREE DYNAMICS OF SOME VIBRATION ISOLATORS
James A. Forrest
Maritime Platforms Division (MPD), Defence Science & Technology Organisation (DSTO)
PO Box 4331, Melbourne VIC 3001, Australia. Email: james.forrest@dsto.defence.gov.au
1. INTRODUCTION
This paper discusses the experimentally measured free-free dynamics of some simple
vibration-isolator assemblies. If these dynamics are known, the behaviour of the vibration
isolators can be incorporated into a mathematical model of a machine mounted onto a
foundation via those isolators. Frequency-response functions (FRFs) for two single-stage
isolators are presented, the isolators comprising two steel endplates and one or four rubber
elements respectively. The FRFs were measured with the isolators suspended on a light string
and excited by a modal hammer. From these, mode shapes are deduced and natural
frequencies and modal damping are estimated. The actual isolator dynamics prove to be more
complicated than initially assumed.
The measurements described were undertaken to provide data for comparison to mathematical
models being developed by Defence Research & Development Canada (DRDC). The shortterm aim is to extend this work to a two-stage isolator. The longer-term aim is to obtain
accurate models for realistic vibration-isolator assemblies which can be used in larger models
of machinery-excited structural vibration transmission in ships and submarines. An overview
of foundation design issues, including isolator impedance, is given in Tso et al.1 The dynamic
characteristics of rubber depend primarily on static pre-load, temperature and the frequency
and amplitude of vibration. These issues with others are discussed by Lindley2, temperature
effects by Kari et al.3 and static load effects by Dickens4. Warley5 gives a method for
selecting rubber materials for a particular isolator performance. The dynamics of an isolator
may be expressed as a blocked impedance as in Verheij6, or as its four-pole parameters as
described by Snowdon7 and Norwood and Dickens8. A method and testing machine to
determine the four-pole parameters is described by Dickens and Norwood9. While all these
approaches are able to include the effects of various parameters, they deal either with the
basic material properties of rubber or with one-dimensional dynamics of vibration isolators in
the axial direction. A free-free modal analysis does not allow for any pre-load for example,
but does make it possible to easily examine dynamics in all directions and check for coupling.
The following sections describe the isolators, the measurement approach and the results
obtained thus far for the two single-stage isolators.
2. VIBRATION-ISOLATOR MODELS AND OTHER EQUIPMENT
The two single-stage isolators for which FRFs were measured are shown in Figures 1(a) and
1(b). These are referred to as the “small” and “medium” isolator respectively. Figure 1(c)
shows the two-stage isolator of three plates and eight rubber elements, to which the current
work will be extended. The three principal directions X, Y and Z are shown Figure 1. These
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Figure 1: Dimensions (mm) and construction of the (a) small, (b) medium and (c) two-stage vibration isolator
models. The two-stage isolator is essentially the medium isolator with another layer added. The end plates are
steel and the rubber elements are shown shaded.
are also referred to as the “lateral”, “transverse” and “axial” directions respectively, based on
the two shear and one compressive directions of translation of the rubber elements.
Each plate has three threaded holes in each edge and one threaded hole in each outward face.
For the results presented here, the single-stage isolators were suspended from a string
threaded through a hollow screw screwed into one of the outward faces. The mass of each
steel plate for the small isolator is 640 gram and for each plate in the larger isolators 2687
gram. Each rubber element is 31 gram and the suspension screw is 9 gram.
The suspended isolator model was excited by a PCB modally tuned impulse force hammer,
model 086C05, of sensitivity 0.24 mV/N, with a teflon tip. Responses were measured with
two PCB model 353B18 accelerometers of sensitivity 10 mV/g and mass 2.3 gram each.
They were secured with wax. Data was acquired using an HP 3566A FFT analyser connected
to a controlling PC. The measured data was processed using the Matlab software package.
3. METHOD
To simulate the free-free state, the isolator models were suspended axially from a thin string,
as shown in Figure 2(a). This suspension modifies the zero-frequency rigid-body modes that
would otherwise be seen for a truly free-free set of boundary conditions. For example, the
rigid-body translations in the lateral (X) and transverse (Y) directions are replaced by the
pendulum motion of Figure 2(b); others are shown in Figure 2. Nevertheless, the frequencies
associated with these modified rigid-body modes were found to be very low, typically a few
Hz. These modes therefore have negligible effect on the higher modes that are of interest.
It was initially assumed for the small isolator with its one rubber element that there would be
six basic non-rigid-body modes. These modes are shown in Figure 3: a translational and a
rotational mode for each of the three principal directions. This assumes that the steel plates
are rigid and that the rubber element can be construed as a set of uncoupled springs aligned in
the different directions. It was assumed that the medium isolator would exhibit a similar set
of modes.
Given these six modes, it should be possible to arrange two accelerometers such that they
detect motion due to only one translation and one rotation. This is illustrated in Figure 4(a)
for picking up translation in the lateral (X) direction and rotation about the axial (Z) axis, and
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Acoustics 2002 - Innovation in Acoustics and Vibration
Figure 2: The suspension (a) of the isolator on a string to give conditions similar to free-free. Various modified
rigid-body modes (b)-(d).
Figure 3: The six assumed basic non-rigid-body modes based on motion in the directions of the three principal
axes, shown for the small isolator.
(a)
(c)
(b)
Figure 4: The two-accelerometer approach for (a) shear modes of the rubber element and (b) compressive
modes of the rubber element, and (c) the comprehensive approach for axial modes, shown for the small isolator.
in Figure 4(b) for picking up axial (Z) translation and transverse (Y) rotation. To separate the
modes, a sum of the two accelerations would give the translation, while a difference would
give the rotation.
The approach of Figure 4(b) for the modes producing displacement in the axial direction was
not successful. The accelerometers for compressive modes are always aligned axially, so will
pick up more than two modes at once if others are excited, unlike for the shear modes. Also,
the simple compressive modes postulated in Figure 3 are not the only ones which appeared in
the compressive measurements. Thus a more comprehensive approach was required, given in
Figure 4(c). The top plate was designated A, the bottom B, and all the corners were
numbered, with A1 directly above B1 and so on. One accelerometer was attached to A1 and
the other to B1 upside down. Then impulse forces were applied to each corner of A in turn.
By reciprocity, the responses at A1 to forces at each corner of A are equivalent to the
responses at each corner of A to a force at A1. Similarly, by symmetry, reciprocity and a
negation (for the upside-down positioning), the responses at B1 to forces at each corner of A
are equivalent to the responses at the four corners of B to a force applied at A1. This fully
characterises the motion of the isolator excited by a force applied at one corner.
A separated mode can be treated as the response of a simple harmonic oscillator to determine
the modal quantities. The position of its maximum magnitude gives the natural frequency of
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Acoustics 2002 - Innovation in Acoustics and Vibration
the mode. Strictly speaking this is the damped natural frequency ωd, but if the modal
damping ratio ζ is small, it is very close to the undamped natural frequency ωn because
ω d = 1 − 2ζ 2 ω n
(1)
The damping ratio ζ can be estimated from the mode’s halfpower bandwidth ∆ωn, which is
the width of the modal peak on the magnitude plot at 1 2 of the peak’s maximum height, or
3 dB down from the maximum. This is shown graphically on the results in the next section.
The relationship for finding the damping ratio is, from Newland10
∆ω n ω n ≈ 2ζ
(2)
where angular frequency ω could be replaced by 2πf with the frequency f in Hz. These
relationships can also be applied to individual modal peaks in a multi-mode response if the
modal overlap is low, so that adjacent modes do not influence one another significantly.
4. RESULTS FOR SHEAR MODES
Figure 5 shows examples of the frequency-response functions (FRFs), H1 and H2, obtained for
the small isolator in the transverse direction following the approach of Figure 4(a). That is, the
measurements are of transfer FRFs from one end of the isolator to the other. H1 is the FRF
for the accelerometer in line with the off-centre applied force. Two main peaks can be seen,
with some variation below 20 Hz due to the modified rigid-body modes.
Simple addition and subtraction of the two FRFs does not separate the modes, because the
modes are present in various proportions. One approach which does work is to look at the
imaginary parts of the FRFs and apply the quadrature peak-pick method as described in
Hewlett-Packard’s application note11. This relies on the fact that there is a 90° phase change
at resonance, so that the imaginary part goes from zero to some peak value at the natural
frequency. At the same time, the real part goes to zero. This means the total magnitude of the
mode at resonance is represented by the imaginary part of its FRF. Given FRFs for different
parts of a structure, the imaginary parts at a given resonance are thus a measure of the modal
coefficients for that mode, i.e. they give the mode shape. All this is valid for well-spaced
modes; with high modal overlap, the imaginary maxima and real zeros are obscured.
The imaginary parts for the measurement of the shear modes are shown in Figures 6 and 7.
Separation of the two modes appearing in each case can be achieved by noting that if the
isolators are linear, the FRFs H1 and H2 must be a linear combination of the modes such that
H2
mag. (dB re g/N)
mag. (dB re g/N)
stt: H1
0
−20
−40
0
50
100
150
frequency (Hz)
200
0
−20
−40
0
50
100
150
frequency (Hz)
200
Figure 5: The two transfer FRFs obtained for the small isolator with the two accelerometers aligned in the
transverse (Y) direction. H1 is the FRF derived from accelerometer 1 and H2 is the FRF from accelerometer 2.
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Acoustics 2002 - Innovation in Acoustics and Vibration
H 1 = aY1 + bY2
(3)
H 2 = cY1 + dY2
where Y1 is the lower and Y2 the higher of the two underlying modes and a, b, c and d are the
modal coefficients denoting each mode’s contribution to the FRFs. If the modes are assumed
to be normalised to unity, the coefficients can be taken to be the signed magnitudes of the
peaks from the plots of FRF imaginary parts. From equations (3), it can be seen that the two
modes are separated by the operations
(d b) H 1 − H 2 = (ad b − c)Y1
(4)
(c a ) H 1 − H 2 = (bc a − d )Y2
The values for the coefficients are given in Table 1, obtained by graphically selecting the
imaginary parts’ peaks when plotted on the computer screen. The value for b for the medium
transverse case was estimated from the other three coefficients using the result that the new
imaginary part of the first mode is equal to (ad b − c) when the first of equations (4) is
applied to obtain Y1. Since it is an estimate, it appears in brackets.
The fact that the peak at about 39 Hz for the small isolator (69 Hz for the medium) appears in
both the lateral and transverse cases shows that this is the axial rotation mode. This means
that the other peak in each case belongs to the relevant translation mode. This is also borne
out by the signs of the modal coefficients. For the lower, rotational, natural frequency f1, a
and c have opposite signs, meaning the two adjacent corners where accelerometers 1 and 2
were attached are moving in opposite directions. Similarly, b and d have the same sign, so the
two corners are moving in the same direction for the higher mode of frequency f2.
However, the magnitudes of these coefficients are not the same, which means the modes are
neither purely rotational nor purely translational. This indicates that there is coupling
between the axial torsion and shear translation of the single rubber element in the case of the
sll: H1
H2
imag. pt. (g/N)
imag. pt. (g/N)
(a)
0
−0.5
−1
0
50
100
150
frequency (Hz)
0.5
0
−0.5
200
0
50
stt: H1
0
−0.5
−1
0
50
100
150
frequency (Hz)
200
H2
imag. pt. (g/N)
imag. pt. (g/N)
(b)
100
150
frequency (Hz)
200
0.5
0
−0.5
0
50
100
150
frequency (Hz)
200
Figure 6: Imaginary parts of the two FRFs obtained for the measurement on the small isolator in the (a) lateral
(X) direction and (b) transverse (Y) direction.
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Acoustics 2002 - Innovation in Acoustics and Vibration
mll: H1
H2
imag. pt. (g/N)
imag. pt. (g/N)
(a)
0
−0.2
−0.4
0
50
100
150
frequency (Hz)
0.2
0
−0.2
200
0
50
mtt: H1
0
−0.2
−0.4
0
50
100
150
frequency (Hz)
200
H2
imag. pt. (g/N)
imag. pt. (g/N)
(b)
100
150
frequency (Hz)
200
0.2
0
−0.2
0
50
100
150
frequency (Hz)
200
Figure 7: Imaginary parts of the two FRFs obtained for the measurement on the medium isolator in the (a)
lateral (X) direction and (b) transverse (Y) direction..
Test
small lateral
small transverse
med. lateral
med. transverse
f1 (Hz)
39.0
39.0
69.1
69.3
Results from H1
a
f2 (Hz)
-0.743
100.0
-1.016
92.7
-0.397
76.6
-0.387
(72.8)
b
-0.405
-0.334
-0.145
(-0.203)
f1 (Hz)
38.8
39.0
68.9
68.9
Results from H2
c
f2 (Hz)
0.316
99.9
0.580
92.6
0.193
77.2
0.078
72.8
d
-0.172
-0.167
-0.112
-0.031
Table 1: Natural frequencies and modal coefficients obtained from the imaginary parts of the relevant FRFs.
small isolator. Therefore the rubber element cannot be considered as a set of uncoupled
springs as originally assumed. In the medium isolator, the disposition of the four rubber
elements implies that shearing of each individual element would have to be involved in an
overall rotational mode in any case, but the coefficients show that this also is not pure
rotation.
The separated modes obtained by applying equations (4) with the modal coefficients from
Table 1 to the FRFs in Figures 6 and 7 are shown in Figures 8 and 9. What are labelled as Y1
and Y2 on the graphs are actually the expressions on the right-hand side of equations (4), i.e.
Y1 and Y2 multiplied by combinations of modal coefficients. Shown on the magnitude plots
are the –3 dB lines on each peak which define the halfpower bandwidth. The natural
frequencies, bandwidths and corresponding modal damping ratios calculated from (2) are
listed in Table 2. There is about 15% difference between the two values of damping ratio
given for the axial rotation in both cases, an indication of the damping values’ uncertainty.
While the separation is quite good, the curves show traces of the mode which was being
eliminated. This demonstrates the approximate nature of the quadrature peak-picking method
to obtain the modal coefficients. Apart from errors in determining the imaginary peak
magnitudes (especially for significantly overlapping modes), it is possible that the modal
coefficients are in fact complex rather than real, since the modes are damped. Better results
would require the use of more sophisticated curve-fitting approaches.
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sll
0.5
imag. pt. (g/N)
0
Y1
Y2
−20
−40
−60
0
50
100
150
frequency (Hz)
0
−0.5
−1
200
0
50
100
150
frequency (Hz)
200
0
50
100
150
frequency (Hz)
200
stt
0.5
imag. pt. (g/N)
(b)
magnitude (dB re g/N)
(a)
magnitude (dB re g/N)
Acoustics 2002 - Innovation in Acoustics and Vibration
Y1
Y2
0
−20
−40
0
50
100
150
frequency (Hz)
0
−0.5
−1
200
mll
0.2
imag. pt. (g/N)
0
Y1
Y2
−20
−40
−60
0
50
100
150
frequency (Hz)
0
−0.2
−0.4
200
0
50
100
150
frequency (Hz)
200
0
50
100
150
frequency (Hz)
200
mtt
0.2
imag. pt. (g/N)
(b)
magnitude (dB re g/N)
(a)
magnitude (dB re g/N)
Figure 8: Magnitudes and imaginary parts of the separated modes for the small isolator in the (a) lateral (X)
direction and (b) transverse (Y) direction. Halfpower bandwidth indicated by lines on magnitude peaks.
Y1
Y2
−20
−40
−60
0
50
100
150
frequency (Hz)
200
0
−0.2
−0.4
Figure 9: Magnitudes and imaginary parts of the separated modes for the medium isolator in the (a) lateral (X)
direction and (b) transverse (Y) direction. Halfpower bandwidth indicated by lines on magnitude peaks.
small
isolator
medium isolator
mode
axial rotation
transverse transl.
lateral transl.
axial rotation
transverse transl.
lateral transl.
fn (Hz)
38.9
92.7
100.0
69.0
72.9
76.9
∆f (Hz)
3.58, 3.15
9.27
9.04
5.45, 4.73
4.70
6.29
ζ
0.046, 0.040
0.050
0.045
0.039, 0.034
0.032
0.041
Table 2: Modes, natural frequencies, half-power bandwidths and corresponding modal damping factors for the
small and medium isolator’s shear modes. The natural frequency is the average of the values for that mode from
Table 1. There are two bandwidths / factors for the rotation modes because they appear twice for each isolator.
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5. RESULTS FOR COMPRESSIVE MODES
The imaginary parts of the corner FRFs of the small isolator for excitation at A1 are given in
Figure 10. These were measured in the manner of Figure 4(c) and applying the reciprocity
and symmetry reasoning given in Section 3. The peak values with their corresponding natural
frequencies are listed in Table 3. The 48.6 Hz mode is the simple lateral rotation (about X)
and the 134.9 Hz mode is the simple axial (Z) translation which were postulated in Figure 3.
However, the modes in between do not correspond to the transverse rotation (about Y) that
was expected. They are instead modes where the rotation of the end plates is about a face
diagonal instead of one of the principal axes. The 93.4 Hz mode comprises mainly of a
rocking motion of A1 and A3 and B2 and B4, opposite diagonals on the opposite faces.
Likewise, the 100.4 Hz mode comprises mainly of a rocking of A2 and A4 and B1 and B3.
These diagonal modes are sketched in Figure 11. It can be seen that the lower mode is more
like rotation about X with a narrow width of rubber, while the higher one is more like rotation
about Y, with a wide width of rubber and therefore greater stiffness.
Small: A1
B1
2
imag. pt. (g/N)
imag. pt. (g/N)
2
0
−2
0
50
100
150
0
−2
200
0
50
A2
imag. pt. (g/N)
imag. pt. (g/N)
0
50
100
150
0
50
A3
150
200
100
150
frequency (Hz)
200
B3
imag. pt. (g/N)
imag. pt. (g/N)
200
100
2
0
0
50
100
150
0
−2
200
0
50
A4
100
B4
2
imag. pt. (g/N)
2
imag. pt. (g/N)
150
0
−2
200
2
0
−2
200
2
0
−2
150
B2
2
−2
100
0
50
100
150
frequency (Hz)
200
0
−2
0
50
Figure 10: Imaginary parts of the small isolator’s corner FRFs in the axial (Z) direction, for an excitation
applied at corner A1.
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Position:
A1
A2
A3
A4
B1
B2
B3
B4
av. fn (Hz)
mode
f1 (Hz)
Im(H)
48.6
-1.28
48.8
-1.44
48.4
1.39
48.6
1.24
48.6
1.31
48.7
1.42
48.5
-1.39
48.6
-1.27
48.6
lateral (X) rotat’n
f2 (Hz)
Im(H)
93.0
-1.54
—
—
92.9
2.02
—
—
93.6
-0.53
94.7
-1.68
91.6
0.12
94.4
1.73
93.4
diagonal rotation
f3 (Hz)
Im(H)
—
—
99.7
1.42
?
+?
98.5
-1.20
?
-?
101.9
0.15
100.3
0.47
101.5
-0.07
100.4
diagonal rotation
f4 (Hz)
Im(H)
134.5
-0.74
135.5
-0.72
134.7
-0.63
134.9
-0.73
134.8
0.68
135.4
0.78
134.5
0.75
135.1
0.76
134.9
axial (Z) transl’n
Table 3: Natural frequencies and modal coefficients for the small isolator, obtained from the FRF imaginary
parts of Figure 10. The average natural frequency for each mode is given. A dash indicates the mode is not
visible and a question mark that its peak is not clear.
Figure 11: Schematic of the diagonal modes of the small isolator at the natural frequency of (a) 93.4 Hz and (b)
100.4 Hz. The rubber element is not shown. The arrows indicate the main motion only.
The imaginary parts of the corner FRFs for the medium isolator are plotted in Figure 12 and
are listed numerically with their natural frequencies in Table 4. Note, however, that only the
five most prominent peaks have been tabulated this time. In addition, there may be other
modes, but these are unclear. For example, A4 and B4 show a side bulge on the 115 Hz peak
at about 128 Hz, and A3 and B3 show a similar bulge at about 175 Hz. Since there is such a
high degree of apparent modal overlap, the imaginary parts listed in Table 5 are not
necessarily good estimates of the actual modal coefficients. Nevertheless, the 114.9 Hz mode
would seem to be a form of transverse rotation (about Y), and the 157.2 Hz mode a lateral
rotation (about X). The imaginary parts for the 74.5 Hz mode are all about the same
magnitude with the signs for plate A matching those of plate B, which suggests that this might
be a rigid-body type of rotation about Y, with the suspension string acting as a pivot. This
would have to be checked by changing the suspension point to an edge instead of the centre of
the plate face. The simple axial (Z) translation mode does not seem to be in the 200 Hz
frequency range analysed.
No modal damping ratios are given for either the small or the medium isolator. While it
would be possible to estimate some of these by the halfpower bandwidth method where the
modes are already well separated, as for the 48.6 Hz mode of the small isolator, high modal
overlap elsewhere prevents this. Since there are now many modes and much overlap, the
separation procedure is more complicated than the straightforward two-mode situations of the
previous section. A more comprehensive curve-fitting procedure would be needed to estimate
the damping for most of the compressive modes identified here. Such a procedure would also
allow more accurate determination of the mode shapes.
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Medium: A1
B1
1
imag. pt. (g/N)
imag. pt. (g/N)
1
0
−1
0
50
100
150
0
−1
200
0
50
A2
imag. pt. (g/N)
imag. pt. (g/N)
0
50
100
150
0
50
A3
150
200
100
150
frequency (Hz)
200
B3
imag. pt. (g/N)
imag. pt. (g/N)
200
100
1
0
0
50
100
150
0
−1
200
0
50
A4
100
B4
1
imag. pt. (g/N)
1
imag. pt. (g/N)
150
0
−1
200
1
0
−1
200
1
0
−1
150
B2
1
−1
100
0
50
100
150
frequency (Hz)
0
−1
200
0
50
Figure 12: Imaginary parts of the medium isolator’s corner FRFs in the axial (Z) direction, for an excitation
applied at corner A1.
Posit’n
A1
A2
A3
A4
B1
B2
B3
B4
fn (Hz)
mode
f1 (Hz)
64.6
64.9
64.8
65.5
64.7
64.7
65.1
65.4
65.0
Im(H)
-0.07
-0.04
0.15
0.07
-0.12
0.13
0.15
-0.12
?
f2 (Hz)
74.4
74.3
74.4
74.8
74.5
74.5
74.3
74.9
74.5
Im(H)
-0.17
0.14
0.16
-0.11
-0.19
0.13
0.15
-0.12
f3 (Hz)
—
103.7
—
—
—
103.5
—
—
103.6
?
Im(H)
-0.09
0.06
?
f4 (Hz) Im(H)
115.9
-0.91
113.3
0.03
115.0
0.43
115.6
-0.45
115.9
0.84
113.0
-0.04
115.0
-0.40
115.7
0.42
114.9
transv. (Y) rot’n
f5 (Hz) Im(H)
157.5
-0.11
157.4
-0.12
156.7
0.28
157.1
0.26
157.5
0.11
157.3
0.11
156.7
-0.27
157.3
-0.26
157.2
lateral (X) rot’n
Table 4: Natural frequencies and modal coefficients of the prominent modes for the medium isolator, obtained
from the FRF imaginary parts of Figure 12. The same format as Table 3 applies .
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6. CONCLUSIONS
The use of fairly simple modal analysis techniques has allowed the identification of several
modes with their natural frequencies for two single-stage vibration isolator models. In some
cases, modal damping ratios have also been able to be estimated with confidence. However,
the isolators exhibited more modes than originally assumed. A major reason for this is the
coupling of translational and rotational motion through the rubber elements, which would
therefore need to be included in any mathematical modelling of the isolators. The greater
number and more complicated nature of the modes means that there was considerable modal
overlap in many of the measured FRFs. A more sophisticated curve-fitting approach could be
used to determine the parameters of the modes in these cases.
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