III. EXPERIMENTAL VALIDATION
3. Validation Using Real-world Bridge Vibration Data
After developing the analytical tool for harvester stability assessment and deriving and performing rudimentary validation on the PHR technique for estimating a harvester’s power generation, the next logical step was to test the practicality of these tools. Explicitly, as these tools were designed to be used for improving vibration energy harvesting from bridges, they first required validation using actual bridge vibration data. Thus, the final experimental validation step is the following demonstration of the effectiveness of these tools using real vibration data collected from bridges.
First samples of such data were collected from the Bob Sheehan Memorial Bridge in Nashville, Tennessee (36°11'10.8"N 86°38'00.8"W) shown on the left in Figure 8. This bridge was formerly open to traffic, but is now part of the Stones River Greenway trail system. This bridge provided a unique opportunity to collect both the free response (via impact testing) and forced response (via vehicle excitation) data in a completely controllable fashion. The free response was collected by using three accelerometers to record the bridge’s vibratory response to impacts from a modal sledge distributed over twenty-one impact points spanning the bridge (shown on the right in Figure 8). The purpose of collecting free response data was to determine the bridge’s primary modes. The first nine modes, as well as rough depictions of their shapes, are shown in Figure 9.
In order to assure that the above nine frequencies do in fact constitute different bridge modes, the modal assurance criterion (MAC) [36] was used to assess their linear independence from each other.
The criterion is essentially a method for gauging linear independence between vibration modes. Figure 10 shows a plot of the MAC values for fourteen possible modes, with mode numbers 1-9 designating the nine modes shown in Figure 9. Values close to one indicate that the modes are linearly dependent. This plot indicates that the nine modes identified in Figure 9 show a high degree of linear independence. Modes represented by numbers above 9 in the figure either did not possess the same degree of linear independence (10-14) or were too small in amplitude to be significant. This method for determining the bridge’s primary modes was used to confirm that the forced excitation response of the bridge contained significant frequency components coinciding with frequencies determined to be modes of the bridge.
19
Figure 8: (A) Bridge used for experimental testing; (B) Schematic indicating impact points (x) and sensor locations (o) used during impact testing.
Figure 9: The first nine modes and their respective shapes of the Bob Sheehan Memorial Bridge 0 200 0
500 1000
1500 2000
x (in) y (in)
3.1668 Hz 5.0002 Hz 5.5002 Hz
6.3335 Hz 7.0002 Hz 7.5002 Hz
9.0003 Hz 10.3337 Hz 11.8337 Hz
20
Figure 10: Modal assurance criterion values indicating the linear independence of the modes.
To obtain the bridge’s forced response, a 2007 Chevrolet Tahoe (total weight including passengers ≈ 5900 lbs (2680 kg)) was driven across the bridge at a near constant speed of 20 miles per hour (32.2 kph).
The response of the bridge was measured using the accelerometer located at the bridge’s midpoint. Figures 11 and 12 show two collected time histories of the acceleration data that are representative of the collected data sets; the figures also show the respective DFTs of the data sets. Note that the peaks in the DFTs are either at, or very close to the determined primary modes of the bridge.
Figure 11: Bridge acceleration time series data set A (top) and its DFT (bottom)
0 2 4 6 8 10
-0.2 -0.1 0 0.1 0.2
Time [sec]
Amplitude [m/s2 ]
0 5 10 15 20 25
0 0.005 0.01 0.015
Frequency [Hz]
Amplitude [m/s2 ]
21
Figure 12: Bridge acceleration time series data set B (top) and its DFT (bottom)
Data sets A and B were used as excitation for the experimental harvester described at the beginning of Section III to test both, the stability assessment method based on the Nyquist Stability Criterion and the PHR power estimation technique. The testing procedure was analogous to that used for multifrequency excitation testing discussed in Section 2 on p. 14. The sole two differences were the excitation which was used and the selection of loads to be tested.
Using the procedure described in the Stability Considerations section on p. 7, when the above load is substituted as into (4), the resulting transfer function has three unstable open loop poles. The Nyquist plot of (4) shown on p. 69 does not encircle the critical point 1 0 ∙ , thereby predicting the existence of three unstable loop poles in the harvester’s equivalent closed-loop transfer function (3). This prediction of unstable behavior using the stability assessment method was confirmed when a servo amplifier was used to attempt to enforce the behavior required by (17).
Table 3 displays the loads that were tested using the physical bridge data collected off of the Bob Sheehan Memorial Bridge as excitation.
Note from the table that active loads were included at this stage of experimental validation. MPTT represent the active load prescribed by the maximum power transfer theorem. According to this theorem, the maximum amount of power is transferred from a power source to a load when that load’s impedance is equal to the complex conjugate of the source impedance. Given the harvester’s source impedance shown in Figure 2 on p. 7, the load which satisfies MPTT is given by (17)
0 2 4 6 8 10
-0.2 -0.1 0 0.1 0.2
Time [sec]
Amplitude [m/s2 ]
0 5 10 15 20 25
0 0.005 0.01 0.015
Frequency [Hz]
Amplitude [m/s2 ]
22
∗ . (17)
Using the procedure described in the Stability Considerations section on p. 7, when the above load is substituted as into (4), the resulting transfer function has three unstable open loop poles. The Nyquist plot of (4) shown on p. 69 does not encircle the critical point 1 0 ∙ , thereby predicting the existence of three unstable loop poles in the harvester’s equivalent closed-loop transfer function (3). This prediction of unstable behavior using the stability assessment method was confirmed when a servo amplifier was used to attempt to enforce the behavior required by (17).
Table 3: Electrical Loads Used During Experimentation
Load Type Load
Passive, resistive 52.7 Ω Passive, resistive 23.6 Ω Passive, resistive and capacitive 52.7 Ω and 240 μF Passive, resistive and capacitive 23.6 Ω and 240 μF
Active MPTT Active ACTIVE
The theoretical and experimental load voltage responses for when the harvester was being excited by the experimental version of bridge acceleration time series data set A (from Figure 11) is shown in Figure 13.
A detailed step-by-step explanation of how the stability assessment was carried out to predict that would cause the harvester dynamics to become unstable, while , shown in (18)
would not do so can be found in Section 6.1 on pp. 68-71.
60
20 . (18)
23
Figure 13: Load voltage response to excitation when is chosen according to MPTT
In addition to validating the stability assessment tool, the time series data sets shown in Figures 11 and 12 were also used to test the accuracy of PHR in estimating power generation for excitation representative of realistic bridge vibrations
Figure 14 depicts the PHR-predicted average power generation values plotted against their resultant error quantities for all of the experimental runs the excitation of which was based on the acceleration time series data set A from Figure 11. Figure 15 plots the corresponding results for experimental runs that used excitation based on the acceleration time series data set B from Figure 12
A more thorough explanation of these results and the experimental conditions/procedure for obtaining them can be found in Section 7 on p. 74. The key observation that should be made from Figures 14 and 15 presently is that the magnitude of the displayed error values is small. According these figures, the maximum difference between the PHR-predicted average power generation and that which was experimentally obtained while using realistic bridge data never exceeded 6 % for any of the tested loads, including
.
The small error values achieved from testing all loads and using physical bridge data lent more credibility to the practical usefulness of the PHR method. However, despite the fact that the data used for experimental excitation was obtained from an actual bridge, it was considered an imperfect representation of the type of vibratory behavior a bridge would exhibit during typical use. The controlled nature of the excitation via the use of a single vehicle traveling at a predetermined speed was great from an experimental repeatability standpoint, but could hardly be claimed as the most usual type of excitation bridges are usually exposed to.
0 0.5 1 1.5
-15 -10 -5 0 5 10 15
Time [sec]
Load Voltage [V]
Experimental Theoretical
24
Figure 14: Average power generation predicted by PHR and its percentage difference from experimentally measured power generation for excitation based on data set A
Figure 15: Average power generation predicted by PHR and its percentage difference from experimentally measured power generation for excitation based on data set B
For this reason additional forced response data was collected from another bridge – the I-40/I-65 Broadway overpass, Nashville, Tennessee (36°09'20.9" N 86°47'21.4" W). Typical traffic conditions served as the excitation source inducing bridge vibrations during this data collection. A sample of the collected time series acceleration data is shown in Figure 16 along with its respective DFT.
5 10 15 20 25 30 35 40 45 50
-1 0 1 2 3 4 5 6
PHR-predicted Power Generation [mW]
Error [%]
52.7 Physical 52.7 Emulated 52.7 + 240 F 23.6 23.6 + 240 F ZACTIVE(s)
5 10 15 20 25 30 35 40 45
-1 0 1 2 3 4 5 6
PHR-predicted Power Generation [mW]
Error [%]
52.7 Physical 52.7 Emulated 52.7 + 240 F 23.6 23.6 + 240 F ZACTIVE(s)
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Figure 16: 25 minutes of typical traffic-induced time series acceleration data (top) and its respective DFT (bottom)
Several datasets of varying time lengths were collected to account for traffic light cycles. Some datasets were taken during constant traffic flow, while others provided more “average traffic flow” DFTs as in addition to including consistent excitation, they also included lengths of time during which the traffic light severely limited vehicle movement along the overpass. All of the collected dataset were used to create excitation representative of typical bridge vibrations for the experimental harvester. The harvester’s experimentally measured average power delivery, as well as the power delivery predicted using the PHR estimation method, are displayed in Table 4. The specific details of the testing, including descriptions of the excitation and the employed electrical load, can be found in Section 0-1 on p.80.
Once again, the magnitudes of the errors between the PHR-predicted and experimentally measured average power generation values are quite small, never exceeding 5 %.
Given the amount and breadth of the experimental validation, the PHR method for average power generation was deemed to be an accurate analytical tool. This conclusion led to an important question: if PHR is capable of predicting power output based on expected excitation, harvester dynamics, and electric load, can it be used as a design tool for creating a more efficient energy harvester?
0 250 500 750 1000 1250 1500
-0.5 0 0.5
Time [s]
Amplitude [m/s2]
0 5 10 15 20 25 30
0 1 2 3 4x 10-3
Frequency [Hz]
Accel. Amplitude [m/s2]
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Table 4: Average power generation predicted by PHR and experimentally measured average power generation
Dataset
#
Duration [min]
Excitation Scaling
Factor
Experimentally Measured [mW]
PHR-predicted [mW]
Error [%]
1 1 20
2.52 2.49 -1.19
2.51 2.50 -0.40
2.56 2.51 -1.95
2 1 20
3.24 3.32 2.47
3.15 3.29 4.44
3.14 3.29 4.78
3 5 25
4.66 4.62 -0.86
4.74 4.66 -1.69
4.74 4.68 -1.27
4 5 20
2.04 2.03 -0.49
2.06 2.03 -1.46
2.06 2.03 -1.46
5 5 15
1.35 1.34 -0.67
1.362 1.360 -0.15
1.346 1.347 0.07
6 5 25
3.87 3.88 0.26
3.89 3.88 -0.23
3.90 3.88 -0.49
7 5 25
3.43 3.47 1.17
3.44 3.46 0.61
3.43 3.46 0.85
3-7 25 15
1.41 1.35 -4.05
1.38 1.32 -4.62
1.38 1.32 -4.70