Dynamic Properties of Stay Cables on The Penobscot Narrows Bridge

CHAPTER 5. PHASE 2 INVESTIGATION

TESTING

Phase 2 testing of the cable-stays on the Penobscot Narrows Bridge was conducted in September 2007, post-damper installation, and lasted one week. Due to favorable weather conditions and shorter decay periods provided by the new dampers (shown in figure 37 and figure 38), it was possible to perform more than double the number of test runs completed during phase 1. During phase 2, the following cables from the four fans were tested: 20A through 12A, 20C through 14C, 20B through 15B, and 20D through 13D (shown in figure 39). The number of test runs performed for each cable varied between seven to ten runs, with the majority of cables undergoing eight or nine runs.

Figure 37. Photo. Dampers.

Figure 38. Photo. Damper's attachment to cable and median slab.

Figure 39. Illustration. Arrangement of cables tested during phase 2.

ANALYSIS

Frequency Content

As in phase 1, spectral density analyses were used to determine the fundamental mode frequencies for each cable tested during phase 2. Once again, the spectral densities were calculated in Matlab^® using Welch's averaged modified periodogram method and plotted on a scale from 0 to 10 Hz. The mode frequencies were determined by tracing the peaks of the density plot. The data from phase 2 produced clear spectrums allowing frequencies up to the seventh mode to be recorded for each time-series. An example of the spectral density plot is show below in figure 40 for cable 19A. The full table of frequency values for the testing done in phase 2 can be found in appendix A.

Figure 40. Graph. Phase 2 spectral density plot of cable 19A, run 2.

Damping Analysis

The damping analysis for phase 2 testing utilized the same initial procedures outlined above for phase 1. Differences between the two phases of testing became apparent after bandpass filters were applied to the phase 2 data sets. The effect of the newly installed dampers was obvious as the logarithmic decay of the damping curves occurred much faster. Figure 41 shows the effects of the bandpass and time filters on the signal.

Figure 41. Graph. Phase 2 plots of the original time series (left), the bandpass filtered time
series (center), and the truncated time series (right).

Once again, best-fit lines of the positive and negative peaks were used to determine the appropriate length of the decay curve. The correlation between the best-fit line and the peaks was kept above .990 by adjusting the length of the sample of the decay curve used. The length of the decay curves ranged between 12 and 50 s for testing done on the first mode, and between 6 and 25 s for testing done on the second mode. Similar to phase 1, shorter cables had shorter decay periods, but unlike phase 1, data obtained between the two boxes were not identical. In general, cable data obtained from box 2 had shorter decay periods than data obtained from the same run from box 1. This was due to box 2's placement on the cable and the corresponding close proximity to the damper attachment.

After best-fit lines were established for each cable run, the damping ratios could again be calculated and then averaged for each individual cable. During phase 2 testing, each cable had a minimum of 7 runs, with the majority getting 8 to 10 runs. Again, the Student-t test was utilized to find a 90 percent confidence on the mean of the damping ratio for each cable for both the first and second modes. Graphs of the results are shown in figure 42 to figure 49, while table 4 contains a summary of the damping ratio, correlation, and frequencies for each cable. The figures and table show the averaged data from the two boxes.

Figure 42. Graph. First mode, 90 percent confidence interval on the mean, fan A.

Figure 43. Graph. First mode, 90 percent confidence interval on the mean, fan B.

Figure 44. Graph. First mode, 90 percent confidence interval on the mean, fan C.

Figure 45. Graph. First mode, 90 percent confidence interval on the mean, fan D.

Figure 46. Graph. Second mode, 90 percent confidence interval on the mean, fan A.

Figure 47. Graph. Second mode, 90 percent confidence interval on the mean, fan B.

Figure 48. Graph. Second mode, 90 percent confidence interval on the mean, fan C.

Figure 49. Graph. Second mode, 90 percent confidence interval on the mean, fan D.

Table 4 . Phase 2 summary of results data.

DISCUSSION

Frequency Content

Phase 2 testing also produced consistent results for frequency analysis. The main differences between the two phases were that over twice the number of cables was tested during the second phase and, more importantly, cables from all fans were tested. While in phase 1, cables were tested in only one outer and one inner fan (fans A and C, respectively), phase 2 tested both sets of outer and inner fans so they could be appropriately compared. Once data from all of the runs were averaged for each cable, the resulting average frequencies were entered into graphs so they could be compared across all cables for each mode. Figure 50 below shows the graph for the first mode comparing the frequencies obtained across each fan of cables during phase 2 testing. Take note that fans A and D are the cables outside the pylons, while fans B and C are the longer cables supporting the main bridge deck. The remaining graphs for modes 2 through 7 can be found in appendix B.

Figure 50. Graph. First mode frequencies from phase 2 testing.

Similar to the first phase of testing, the data obtained from phase 2 produced clear power spectrum densities, allowing the capture of mode frequencies up to the seventh harmonic. The spectral density graphs were slightly clearer in the higher frequencies, allowing a value for each mode to be extracted for over 97 percent of the cable runs performed.

Damping Ratios

The installation of dampers on the cable-stays had a noticeable impact on the damping ratio values, causing the damping ratios to increase by at least a factor of 5, and sometimes as high as 15. In general, first mode damping ratios for phase 1 testing were between 0.10 to 0.39 percent, where in phase 2 they ranged between 1.22 to 2.21 percent.

The damping ratios obtained during phase 2 testing were generally more consistent than their phase 1 counterparts, but still had some discrepancies that need to be addressed.

Since two accelerometers are used during testing to provide redundancy in the data, mainly to address the possibility of one being mounted in the proximity of a node at higher frequency, it was found that the two boxes rarely produced identical damping ratio data. For the majority of cases, the damping ratio obtained from box 2 was higher, for some cables up to 30 percent higher. For first mode data, 28 of 30 cables had higher damping ratios from box 2 data, while for the second mode data this ratio fell to 21 of 30 cables. Since the accelerometer in box 2 was mounted on the cable closer to the anchorage and the damper, it experienced higher levels of damping than the accelerometer located closer to the cable's mid-span.

There is a greater correlation in phase 2 data between cable length and damping ratio. As the cable length decreases, the damping ratio almost always increases. There are a few exceptions visible within the plots for each fan of cables, but there is definitely a stronger correlation than in similar plots from phase 1. Conversely, there is no correlation between cable length and the size of the confidence interval. The Student-t test gives a range of values from which between there is a 90 percent confidence that the mean of the damping ratio is located. For phase 2 testing, this range varied from 0.05 to 0.40 for the damping ratio, comparable to 2.5 to 16 percent of the mean value, and it varied in no pattern related to cable length or size. These correlations and lack thereof held true for both the first and second modes.

Another obvious result of the addition of dampers was that the length of the decay period was shortened significantly. For first mode decay, the period lasted on average 28.5 s for box 1 and 20.6 s for box 2, where the period was measured as the longest amount of time a best-fit line could overlay the data points with a correlation greater than 0.9900. Some periods were measured as high as 50.2 s. For second mode decay, the periods were much shorter, averaging 16.3 s for box 1 and 15.4 s for box 2.

In order to help determine the period length that would correspond to damping ratio with the highest correlation, multiple values for cable 20A were tested and plotted against each other to see if there were any advantages to any specific method. First, frequencies were chosen for the low and high bandpass values, which remained constant for each set of tests. Next, the peaks were analyzed using overlapping and increasing period lengths. The lengths would start at 15 s and be staggered by 5 s intervals across the duration of the decay. Then the period length would increase to 20 s and the method repeated until the period length roughly matched the entire decay, which usually happened between 35 and 45 s. The results of these tests are varied. The damping ratios obtained using the shorter periods had great variance proving that the decay curve is not linear and where the ratio is calculated will affect the final damping value obtained. Sometimes the damping ratio would start out low and get higher as the interval was delayed, other times the opposite happened, and yet other times the damping ratio would increase then decrease. Using shorter interval periods did not ensure higher correlation either, as sometimes the correlation would actually be lower than one calculated over a longer period.

This analysis was instrumental in deciding that a decay period would be chosen by the longest possible sample time span without dropping below a desired correlation threshold usually set around 0.995 or higher.

Scruton Number Analysis

Damping values obtained during phase 2 testing were entered into the equation in figure 34 in order to calculate the resulting Scruton number. The results varied between 7 and 12, which is significantly above the target value of 5 for cables with aerodynamic surface treatments. Table 5 shows the full range of Scruton numbers for the cable-stays tested.

Table 5 . Phase 2 Scruton number.

Aerodynamic Damping

After inconclusive results regarding the effects of aerodynamic damping for phase 1 data, the effect was again studied, this time for the data obtained during phase 2 testing. Wind speed and direction data were plugged into the equation in figure 35, Macdonald's formula for finding an equivalent wind speed, and the results were tabulated and then plotted. Since phase 2 contained both a higher average number of tests completed for each cable and more cables tested in total, the hope was to find better evidence of the aerodynamic damping effect.

The results from the formula were plotted against the average damping values, where a regression line could connect the data points and the corresponding crossing on the y-axis would be the remaining structural damping. However, once again, virtually every plot showed little correlation between equivalent wind speed and damping, and no best-fit lines could be accurately determined. Despite the higher number of data points, the data did not trend in any consistent direction. Almost 90 percent of R²-values of the regression lines for each cable were below 0.3, indicating very poor correlation. Further studies using the formula should be conducted focusing on fewer cables and a much broader variation of wind speeds and directions.