field evaluation of dampers
TRANSCRIPT
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CHAPTER 7
Field Evaluation of Dampers
In order to evaluate the effectiveness of fluid dampers for stay-cable vibration
mitigation and to facilitate the design of even more effective and economical systems, it
is important to perform quantitative assessments of damper performance under various
types of excitation. Toward this end, this chapter seeks to evaluate the effectiveness of
passive linear dampers installed on two stays on the Fred Hartman Bridge in Houston,
Texas, by comparing response statistics before and after the damper installation and by
investigating in detail the damper performance in a few selected records corresponding to
different types of excitation.
Fluid dampers, specified to have a linear force-velocity relationship, have been
installed for evaluation on two stays on the Fred Hartman Bridge, and at the time of
writing, data have been collected for more than two years after the damper installation to
evaluate the damper performance. The two stays on which the evaluation dampers have
been installed are indicated on the schematic drawing of the bridge in Figure 7.1 (AS16
and AS23); this figure also indicates the coordinate system used for reporting wind speed
and wind direction. Properties of the two stays on which evaluation dampers have been
installed are given in Table 7.1 along with the damper locations and the specified damper
coefficients.
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previously, the error in the predictions of the universal curve is most significant near the
optimal portion of the curve, and it is evident here that the asymptotic approximate value
differs most significantly from the exact value in mode 1, for which the damper is nearly
optimal. The error near the optimal portion of the curve becomes more significant in the
higher modes, but because the damper is far from optimal in the higher modes for these
stays, the universal curve gives quite good predictions of the damping ratios in the higher
modes. The Sc > 10 criterion discussed previously requires modal damping ratios of
0.53% for Stay AS16 and 0.64% for Stay AS23. It is evident from Figure 7.2 that,
according to the analytical predictions, this criterion is satisfied in the first nine modes for
Stay AS16 and in the first five modes for Stay AS23. Damping values for the stays have
not yet been directly estimated from measurements for comparison with the analytical
predictions, but work is currently in progress and results will be reported in future
publications (Delong Zuo, personal communication).
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
1 2 3 4 5 6 7 8 9 10
Mode Number,i
AS16: numericalAS16: asymptotic
AS23: numerical
AS23: asymptotic
i
Figure 7.2: Predicted Damping vs. Mode Number for Hartman Stays AS16 and AS23
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In order to assess the presence of the damping-induced frequency shifts predicted
by the analytical formulation, the natural frequencies of Stay AS23 before and after the
damper installation were estimated from the peak values of averaged acceleration power
spectra. Five 5-minute records of ambient vibration under low wind speeds were selected
both before and after the damper installation, and each set of five spectra was averaged to
obtain averaged undamped and damped spectra. The damping-induced frequency shift in
each of the identified modes was computed by taking the difference of the damped and
undamped frequencies. These measured frequency shifts were then normalized by the
fundamental frequency of the stay to obtain dimensionless frequency shifts, which are
plotted in Figure 7.3 for the first 12 modes. Also plotted with the measured values are
the dimensionless frequency shifts predicted by numerical solution of the eigenvalue
equation (3.6) (labeled Analytical), and the frequency shift i which would be
induced if the cable were fixed at the damper location (labeled Clamped). It is evident
that the measured values agree quite well with the analytical predictions, and the
frequency shifts are quite significant, especially in the higher modes.
0
0.1
0.2
0.3
0.4
0.5
1 2 3 4 5 6 7 8 9 10 11 12
Mode Number, i
Measured
Analytical
Clamped: i
1o
oii
f
ff
Figure 7.3: Frequency Shift vs. Mode Number: Estimated and Measured Data
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As was mentioned above, when the dimensionless frequency difference i in a
given mode reaches 0.5, new regimes of behavior are observed. In the case of Stay
AS23, the value of i associated with mode 13 is slightly greater than 0.5, and the
analytical formulation for the taut string with linear damper in Chapter 3 indicates that
the damping in this mode is supercritical, so that no oscillatory solution exists for this
mode, but a non-oscillatory decaying solution emerges instead; this behavior is observed
because the damper is located sufficiently near the antinode of mode 13 for Stay AS23.
In modes 14 and higher, the damper is located past the first antinode, and solution of the
eigenvalue equation indicates that the frequency of damped oscillation is less than the
undamped frequency in these modes. Although not presented herein, the frequency shifts
measured in the higher modes agree reasonably well with the analytical predictions.
The clamping ratio ci associated with each mode can be computed by normalizing
the dimensionless frequency shifts plotted in Figure 7.3 by the dimensionless frequency
difference i. The value of the nondimensional damper parameter associated with
each mode can be readily computed from the definition in (3.32) using the stay and
damper properties in Table 7.1. Figure 7.4 shows the resulting plot of versus ci; a
distinct data point is plotted here for each of the first 12 modes, larger values of
corresponding to higher mode numbers. Also plotted with the measured values are the
analytical values determined from numerical solution of (3.6) and the curve
corresponding to the asymptotic approximation in (3.31). The measured values agree
reasonably well with the analytical predictions, and a clear trend of increasing clamping
ratio with mode number is evident. The measured frequency shifts thus confirm that this
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damper, which was optimized for mode 1, is effectively more rigid in the higher modes
and is tending to lock the cable at the damper location in these modes.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Measured
Analytical
Asymptotic
ci
Figure 7.4: Non-Dimensional Damping Parameter vs. Clamping Ratio
7.2 Global Damper Performance
In order to evaluate the effectiveness of the trial dampers installed on the two
stays, response statistics are compared from data collected before and after the damper
installation. The data presented herein are obtained from queries issued to the database
described earlier. In order to minimize the inclusion of records with noise-corrupted or
saturated data, all queries include a filter that retains only records with skewness with a
magnitude less than 2.0 and kurtosis less than 5.0 (recall that a normal distribution has a
skewness of zero and a kurtosis of 3.0). (A slightly more aggressive skewness filter of
magnitude less than 1.5 was used for the load cell data). While this scheme obviously
might exclude some data records that are of interest, experience has shown this approach
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to be a reasonable method of eliminating problematic data while retaining the majority of
features being sought.
In the figures included herein, representative values of mean wind speed were
computed using the same scheme described previously in Chapter 2. Wind direction is
measured in degrees clockwise from the bridge axis, with zero degrees corresponding to
wind approximately from the North, directly along the bridge axis, as indicated in Figure
7.1. Acceleration data are reported from transducers installed on the stays usually about
6 m vertically above deck level, and for the purposes of this investigation and for the data
reported herein, the accelerations are reported at the transducer location.
The global performance of the dampers installed on stays AS16 and AS23 is
summarized in Figure 7.5 and Figure 7.6, respectively, which illustrate the following
Characterization of oscillations (in-plane) occurring before installation of the
damper as a function of wind speed and direction.
Characterization of oscillations (in-plane) occurring after the installation of the
damper as a function of wind speed and direction.
Measurement of damper force as a function of wind speed and direction.
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Figure 7.5: Stay Vibration and Damper Force Characteristics: Stay AS16
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Figure 7.6: Stay Vibration and Damper Force Characteristics: Stay AS23
Before the damper is installed, both stays show similar characteristics, and
patterns that are consistent with the field observations that were described earlier. A high
density of points is seen near the abscissa, which generally corresponds to vortex-induced
vibration in a variety of modes and low-level buffeting response of the stay under random
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excitations. Both figures clearly indicate the characteristic signature of wind-rain
oscillation as the multiple points over a wide range of wind speeds that are of high
amplitude i.e., root-mean-square (RMS) accelerations greater than 0.5g. One-minute
mean wind speeds at deck level reached 15 m/s before the dampers were installed, and
almost 18 m/s in the period after installation. This latter value corresponds to a one-
minute average wind speed of 27 m/s (60 mi/h) at the top of the tower, recorded during a
thunderstorm. The dependence on wind direction is also clear, with AS16 showing its
peak responses between 90 and 160 degrees, and AS23 over a narrower range between 90
and 135 degrees. The primary goal of a mitigation system is to reduce significantly or
eliminate these large-amplitude events, while respecting the fact that the stays and bridge
form dynamic systems and will always exhibit some level of dynamic response.
The corresponding figures after the installation of the dampers suggest the
following:
Amplitudes are significantly reduced across all recorded wind speeds (up to 18
m/s at deck level) with maximum RMS acceleration amplitudes of around 0.5g.
The dependence on wind direction has been altered significantly, with the
previously preferred range now largely unapparent, and the largest of the
responses now nearer to a 90-degree angle of incidence. The characteristics of
selected records corresponding to these locations will be discussed in more detail
later.
The third pair of figures shows the RMS damper force for the two installed
devices as a function again of wind speed and direction. For stay AS16, the
pattern of forces resembles closely the pre-damper acceleration figures (both in
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terms of wind speed and direction), suggesting that the dampers are being
engaged by the stays to suppress proclivity towards wind-rain vibration.
Relatively high-magnitude forces are also seen in the complementary range of
225 to 270 degrees, which represent wind directions also corresponding to the
declining direction of the stay (but now from the southwest rather than southeast.)
For stay AS23, the behavior is similar, though perhaps not as clear as in the
previous case. Much of the high force data for this stay is now clustered around
an incident angle of 90 degrees, although some up to 110 degrees are evident. An
interesting and unusual cluster of points labeled Record C will be discussed
subsequently. Note that in all the records, the highest amplitude RMS force
recorded is approximately 5.6 kN (1125 lb) a relatively modest level of load.
7.3 Analysis of Individual Records
While comprehensive investigation of all individual data records is challenging
due to the quantity of data, it is instructive and interesting to study some of these specific
records in some detail in order to better understand what the dampers are doing, or trying
to do. After some review of these datasets, three such representative records have been
selected and presented herein. These records are labeled A, B, and C in Figure 7.5 and
Figure 7.6; Records A and B are selected from stay AS16, and record C from stay AS23.
The reasons for the selection, as well as an interpretation of each record is provided
below. Each figure includes a ten-second time history of acceleration and damper force,
as well as a PSD of each over a 10-Hertz range.
Record A (Figure 7.7) represented one of the highest amplitude RMS acceleration
events after the installation of the damper on stay AS16. It corresponded to a wind speed
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of about 3.4 m/s and a wind direction of 60 degrees. The RMS force is moderate: about 2
kN RMS. This record is interpreted as traditional vortex-induced vibration of the stay,
with both acceleration and force presenting strongly single-mode responses at a
frequency of about 6.5 Hz. (This corresponds to the fifth mode of vibration of this
dampedstay. As noted above, the damper has a stiffening effect on the stay system that
is mode dependent; the frequency of the fifth mode of the undamped stay is
approximately 6.2 Hz.) An analysis of the Strouhal relationship for this wind speed and
frequency suggests a Strouhal number consistent with 0.2: the value commonly assumed
for a circular cylinder. Due to the relatively high frequency of the mode, displacement
amplitudes associated with this motion are small even with the peak acceleration of 1g
(about 6 mm at the transducer location.)
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Stay AS16 Acceleration Time History (Record A)
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
Time (s)
Acceleration(g)
Stay AS16 Acceleration PSD (Record A)
1.0E-07
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
0 2 4 6 8 10
Frequency (Hz)
PSDof
Acceleration
Stay AS16 Damper Force Time History (Record A)
-10
-8
-6
-4
-20
2
4
6
8
10
0 2 4 6 8 10
Time (s)
Force
(kN)
Stay AS16 Damper Force PSD (Record A)
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-021.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
0 2 4 6 8 10
Frequency (Hz)
PSDofDam
perForce
Figure 7.7: Time History and PSD of Stay Acceleration and Damper Force
(Record A AS16)
Record B (Figure 7.8) represented another of the highest amplitude RMS
acceleration events after the installation of the damper on stay AS16, but corresponded to
a wind speed of about 8.5 m/s and a wind direction of 90 degrees. The RMS force is one
of the largest recorded for this stay: about 4.8 kN RMS. The acceleration and force plots
both clearly contain two main frequency components, one at about 2.6 Hz and the other
at 5.3 Hz, corresponding to the second and fourth modes of the damped stay,
respectively. The force data are dominated by the second-mode contribution. A number
of other modes are evident in both spectra, but note that the log scale tends to exaggerate
their relative contributions. The characteristics of this record suggest that it is a case of
wind-rain vibration that is being suppressed by the damper, and analysis of the rain
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bucket data confirms the presence of moderate rainfall at the time of this record (a
rainfall rate of approximately 25 mm/hr was estimated). Note also the strong asymmetric
appearance of the acceleration record (and to a lesser extent the force record.) Indeed
this acceleration record has a mean of zero and is in fact not asymmetric; the addition of
the appropriately phased second and fourth modes gives this appearance. It should be
noted, however, that the two components conspire to produce peak accelerations of about
1g and a peak load in the damper of a little over 9 kN (2000 lb). These values are
certainly higher than what one would estimate assuming mono-frequency response.
Consideration of the peaks as well as RMS values should generally be undertaken for this
reason. The damper is certainly able to effectively control/reduce the large-amplitude
behavior observed before the installation, and again through the provision of modest
levels of force.
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Stay AS16 Acceleration Time History (Record B)
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
Time (s)
Acceleration(g)
Stay AS16 Acceleration PSD (Record B)
1.0E-07
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
0 2 4 6 8 10
Frequency (Hz)
PSDof
Acceleration
Stay AS16 Damper Force Time History (Record B)
-10
-8
-6
-4
-20
2
4
6
8
10
0 2 4 6 8 10
Time (s)
Force
(kN)
Stay AS16 Damper Force PSD (Record B)
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-021.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
0 2 4 6 8 10
Frequency (Hz)
PSDofDam
perForce
Figure 7.8: Time History and PSD of Stay Acceleration and Damper Force
(Record B AS16)
Record C (Figure 7.9) represented the highest amplitude RMS force recorded to
date on stay AS23, even though the acceleration measured on the stay was small. This
corresponded to a wind speed of about 10 m/s and a wind direction of 60 degrees. The
RMS force is the largest recorded for this stay: about 5.7 kN RMS. This record is very
unusual and interesting for a number of reasons. Note that the acceleration amplitude is
relatively small about 0.25g and clearly contains a broad range of frequency
components. The peak force magnitude, however, is close to 8 kN (1800 lb) and is
dominated (strongly) by the fundamental mode of the stay at 0.66 Hz. Again, the multi-
frequency contributions plotted at log scale exaggerate the higher mode contributions; the
most significant higher mode contribution is more than two orders of magnitude lower
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than that of the fundamental mode. This is believed to be an example of deck-stay
interaction that the damper is responding to suppress. First, it is unusual to see such a
strong component of force at the fundamental mode of the stay, as preceding data have
suggested. Ozkan et al. (2001) discuss a situation where oscillation of a stay was clearly
preceded and presumably precipitated by oscillation of the deck of the structure in the
fifth vertical mode of vibration. As discussed therein, the analysis suggested that in
that case stay AS24 had a fundamental frequency very close to the third symmetric
vertical mode of the deck, which evidently drove the stay in its fundamental mode. A
vertical deck frequency was also identified at 0.67 Hz, which in this case is quite
likely driving (or attempting to drive) stay AS23 in its fundamental mode. Unfortunately,
the deck accelerometers were non-functional at the time this record was made, so this
hypothesis cannot be confirmed in this case by reference to deck data. Again, the damper
seemed to perform well under these circumstances.
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Stay AS23 Acceleration Time History (Record C)
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
Time (s)
Acceleration(g)
Stay AS23 Acceleration PSD (Record C)
1.0E-07
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
0 2 4 6 8 10
Frequency (Hz)
PSDof
Acceleration
Stay AS23 Damper Force Time History (Record C)
-10
-8
-6
-4
-20
2
4
6
8
10
0 2 4 6 8 10
Time (s)
Force
(kN)
Stay AS23 -- Record C
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
0 2 4 6 8 10
Frequency (Hz)
PSDofDam
perForce
Figure 7.9: Time History and PSD of Stay Acceleration and Damper Force
(Record C AS23)
7.4 General Remarks
The preceding section demonstrates that under a wide range of field parameters
that the damper solution appears to provide a reasonable and acceptable solution to the
stay cable vibration phenomenon. Despite these promising results, a few comments and
cautions are in order.
The ranges of wind speeds evaluated, even over a three-year period, are limited.
It has been suggested that high-wind-speed phenomena such as galloping might
occur in inclined stay cables, and that a damper solution would be incapable of
providing sufficient capacity to suppress such phenomena. While we have not