Establishment of Full-Field, Full-Order Dynamic Model of Cable Vibration by Video Motion Manipulations
11.3 Experimental Setup and Results .1 Laboratory Setup
The structure of interest was a 1.75 m long braided, stainless steel cable with a diameter of 2.38 mm (3/32 in.) held in place within a Unistrut frame. One end of the cable was attached to a load cell which outputted voltage readings to a multimeter.
The other end of the cable was connected to a turnbuckle, allowing the tension in the cable to be adjusted. Three separate data collection systems were used: (1) a load cell connected to a high-sensitivity multimeter; (2) PCB accelerometers with a National Instruments data acquisition unit; and (3) a Sony digital video camera set up facing the cable to capture a video of the vibrating structure. The video was set to collect at 480 fps with an ISO setting of 10k (Fig.11.1).
11.3.2 Automated Filtering
A video motion processing algorithm was used to ‘filter’ the original video’s pixel phases to extract the vibration video of the mode shape for each specific modal frequency, which is automatically specified by the taut string theory. This algorithm can extract the mode shape videos for each modal frequency up to the Nyquist frequency, which in this case is 240 fps.
11.3.3 Mode Shape Estimation
Another algorithm was used to break down the mode shape videos into mode shape vectors. These mode shape vectors can be used to create a mode shape matrix of the size n by m where n is the number of pixels along the length of the cable and m is the number of modal frequencies. The algorithm breaks down the mode shape video into many frames by using the canny edge detection algorithm. See Fig.11.2for the procedures.
Fig. 11.1 Experimental setup
11.3.4 Results
11.3.4.1 Compare Tension with Accelerometers
Since the video estimated frequencies play a central role in the construction of the frequency response function, two different methods are used to verify them. First, taut string theory provides a relationship between the tension in a cable and the cable’s natural frequencies, thus it is possible to find the cable tension from the video estimated frequencies. By taking advantage of this relation the video estimated frequencies can be compared to the accelerometer estimated frequencies and the cable tension as measured by the Futek load cell.
Figure11.3below compares the video estimated tensions with the accelerometer estimated tensions for both the impact and wind excitation data from 42 different trials. In both cases, the video estimated tensions match the accelerometer estimated tensions with a max error of 37 and 38 N for wind and impact data, respectively.
Fig. 11.2 Process for converting the filtered mode shape videos to mode shape vectors
400 350 300 250 200 150 100 50
400 350 Impact excitation data Ideal response
300 250 200
Tension from accelerometers (N) Video tension vs accel tension (impact excitation)
Tension from Video (N)
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0
400 350 300 250 200 150 100 50
400 350 Wind excitation data Ideal response
300 250 200
Tension from accelerometers (N) Video tension vs accel tension (wind excitation)
Tension from Video (N)
150 100 50 0
0
Fig. 11.3 Comparing tension estimated from video to tension estimated using acceleration data from PCB accelerometers
11.3.4.2 Compare Tension with Load Cell
Figure11.4 below compares the video estimated tensions with tension values estimated from the load cell for both the impact and wind excitation data from 90 different trials. Again, tension in the cable is calculated from the video estimated fundamental frequency and the relation given by taut string theory. Tension values from the load cell are determined after careful calibration of the load cell with weights spanning the entire range of 0.06–440 N. In both cases, the video estimated tensions match the load cell estimated tensions with a max error of 36 and 41 N for wind and impact data, respectively.
11.3.4.3 Linear Progression of Frequencies
From taut string theory the natural frequencies are expected to follow a linear relation and this fact is confirmed experimentally by plotting the progression of frequencies estimated from videos (Fig.11.5).
400 y = 0.97*⫻−0.81 y = 0.99*⫻−0.48
350 R2=0.974 R2=0.973
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400 350 Impact excitation data Linear fit Ideal response
Impact excitation data Linear fit Ideal response
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Tension from load cell (N) Tension from load cell (N)
Load cell tension vs video tension (impact excitation)
Tension from Video (N)
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0
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400 350 300 250 200
Video tension vs load cell tension (wind excitation)
Tension from Video (N)
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Fig. 11.4 Comparing tension estimated from video to tension estimated with Futek load cell (model LSB302) 250
200
150
100
50
0
1 2 3 4 1 1.5 2 2.5 3 3.5 4
Number of modes
Linear progression of cable frequencies (impact) Linear progression of cable frequencies (wind)
Frequency (Hz) Frequency (Hz)
Number of modes
5 6 7
140
100 120
80
60
40
20
Fig. 11.5 First modal frequencies from six videos with impact excitation and six videos with wind excitation
Fig. 11.6 Mode shape vectors for seven modal frequencies
11.3.4.4 Mode Shape Vector
The following figure shows the mode shape vectors of seven modal frequencies extracted by using the algorithms discussed in Sect. 11.3.3. These seven modal frequencies are the full field, full order dynamic model of a cable at a tension of 241 N.
The value carried in the mode shape vectors are the pixel displacement along the pixel length of the cable (Fig.11.6).
11.3.4.5 Comparison with High Speed Camera
A high speed camera with a high sampling rate (thus more modes) was used to compare the extracted modal frequency results measured from the standard camera (Sony). While there are advantages to extracting more vibration modes (Fig.
11.7), there were problems with using a very high-cost high speed camera. First, high speed cameras need extra light sources to perform correctly, which hampers practical use in the proposed manner. Second, the extracted modal frequency data from the standard camera did better, in most cases, than the high speed camera’s data (comparing to the accelerometers and load cell) because of the lighting issues from the first problem.
11.3.4.6 Establishing Dynamic Model
Given the smoothed mode shapes and the estimated natural frequencies, a full field and full order dynamic model of the cable can be established via
ŒH.!/ NNDŒ Nm
2i !21
mmΠTmN (11.4)
where the FRF matrix, [H(!)]NN, depends on a mode shape matrix []Nm and a diagonal matrix of the difference of squares between each natural frequency,i, and![8].
400 450 500
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Frequency (HZ)
0
0 2 4 6 8 10 12 14
Number of modes
Linear progression of cable frequencies with high speed camera (impact)
y = 33*⫻+ 0.48
High speed camera Linear fit
Fig. 11.7 First 14 frequencies from video with high speed camera of vibrating cable after impact