In Simulation 3, the % of noise is kept to be same, i.e., 5%. Now, the number of measure- ment locations is reduced thereby increasing the distance between two measurement locations.
Now the distance between two measurement locations is double of the previous value. From Figure 6.35, there is a peak at the location of crack. Hence, the algorithm identified the pres- ence of crack when the distance between two measurement locations is increased. In the next section, the scheme is implemented for actual measurements on the shaft.
Figure 6.35: Crack probability functions for simulation C (9 measurement locations and 5%
noise).
For the verification of the MCDLA, presented in Section 6.5, a total of 21 measurement lo- cations are taken. Now the algorithm is attempted again with 11 measurement locations only.
Hence, distance between two measurement locations is doubled now.
tions at alternate measurement locations are taken out for the verification of the algorithm. Ex- citation frequencies for the algorithm are taken to be ωi = 10-26, 42-54 Hz at 2 Hz interval.
The natural frequency of the system is around 39 Hz. The single crack in Shaft-I is now near the 6th measurement location (i.e., near the mid-span of the shaft). The resulting crack proba- bility functions are given in Figure 6.36. There is a peak at 6th measurement location. Two small peaks are present at 3rd and 9th measurement locations although there is no crack at these locations.
Figure 6.36: Crack probability functions for the Shaft-I with a single crack near 6th meas- urement location (first set of measurements).
The crack probability functions for the second set of measurements over Shaft-I with a sin- gle crack near 6th measurement locations are given in Figure 6.37. Here, a single peak comes at the 6th measurement location. Hence, again the algorithm identifies the presence of crack over the shaft. Moreover, if we see the CPF plots for both set of measurements (Figure 6.36 and
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Figure 6.37); it is more clear that there is a crack in the shaft near the 6th measurement location.
And there is no crack in the shaft near the measurement locations 3 and 9.
Figure 6.37: Crack probability functions for the Shaft-I with a single crack near 6th meas- urement location (second set of measurements).
For the second shaft, the crack is now near the measurement location 7. The crack probabil- ity function for this case is given in Figure 6.38. There is a big peak at the crack location (7th measurement location) and one small peak at 9th measurement location.
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Figure 6.38: Crack probability functions for the Shaft-II with a single crack near 7th meas- urement location of the shaft (first set of measurements).
CPFs obtained from the second set of measurement for the same condition (Shaft-II with single crack at 7th measurement location) are given in Figure 6.39. Now there is a single peak in the CPF plots at the 7th measurement location. Hence, the algorithm is identifying the pres- ence of crack in the shaft properly. Again, comparing the CPF plots from the two set of meas- urements (Figure 6.38 and Figure 6.39), it can be concluded that there is a crack in the shaft near 7th measurement location.
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Figure 6.39: Crack probability functions for the Shaft-II with a single crack near 7th meas- urement location of the shaft (second set of measurements).
After observing the working of the MCDLA for the single cracked shaft, the algorithm is applied to Shaft-I with two cracks. With 11 measurement locations, now the two cracks are at 6th and 8th measurement locations. The crack probability functions obtained from the first and the second set of measurements are presented in Figure 6.40 and Figure 6.41, respectively. The crack near the middle of the shaft is identified in both the plots. There is no clear peak near the second crack location.
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Figure 6.40: Crack probability functions for the Shaft-I with two cracks. First crack near 6th measurement location and the second crack near the 8th measurement location (first set of measurements).
Figure 6.41: Crack probability functions for the Shaft-I with two cracks. First crack near 6th measurement location and the second crack near the 8th measurement location (second set of measurements).
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Now the Shaft-II is used for the verification of algorithm. Now the first crack and the se- cond crack are near to the 7th and 4th measurement locations, respectively. The crack probabil- ity functions obtained from the first and the second set of measurements are presented in Fig- ure 6.42 and Figure 6.43, respectively. The two cracks in the Shaft-II are identified in both the measurement sets.
Figure 6.42: Crack probability functions for the Shaft-II with two cracks. First crack is near the 7th measurement location and the second crack is near the 4th measurement location (first set of measurements).
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Figure 6.43: Crack probability functions for the Shaft-II with two cracks. First crack near 7th measurement location and the second crack near the 4th measurement location (second set of measurements).
The CPF plots, from Figure 6.23 to Figure 6.30 and CPF plots from Figure 6.36 to Figure 6.43 are obtained from the same measurements (i.e., the same forcing time history and the same response time history). Hence by comparing these two sets of CPF plots, it can be ascer- tained that the MCDLA is working better with the scheme. But definitely at the cost of reduced resolution with which the location of a crack is known along the shaft length. If the measure- ment noise is less, the algorithm can be used with more number of measurement locations and the location of a crack will be known with higher resolution.
From Figure 6.36 to Figure 6.43, it can be observed that the algorithm works well for all the cases except for Shaft-I with two cracks. The algorithm is not able to detect the presence of the second crack in Shaft-I, which is near the 8th measurement location. This may be attributed mainly to smaller size of the second crack in Shaft-I. Also, although the location of the second
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crack from the end support is same for both the shafts (i.e., Shaft-I and Shaft-II), distance be- tween the two cracks is less in case of Shaft-I.
Overall the MCDLA was tested for two shafts, namely, Shaft-I and Shaft-II. First a single crack was made in each of the two shafts and the algorithm was tested for single-cracked shafts. Thereafter another crack was introduced in the shafts and the algorithm was tested for double-cracked shafts. These four cases are, (i) Shaft-I with single crack (ii) Shaft-II with sin- gle crack (iii) Shaft-I with two cracks, and (IV) Shaft-II with two cracks. Results from two measurement set are presented for each of the four cases. The algorithm is found to be working for all the cases except for the case (iii). For case (iii) the algorithm is able to detect only one crack out of the two cracks present in the shaft in both the measurement sets. This is attributed to small size of the crack and its proximity to the end support. Hence, out of 8 results present- ed, the algorithm is working perfectly in six results and partially in remaining two results. This way the experimental verification of the algorithm has shown good repeatability.
The MCDLA is robust against measurement noise. As explained in Sections 6.5.1, if the measurement noise is more, distance between measurement locations should be more. Effect of increasing ‘the distance between measurements locations’ over ‘the working of the MCDLA’
is explained by numerical simulation in Section 6.6 and by experimentation in Section 6.7.
Some more characteristics of the algorithm need to be tested experimentally, such as: (1) performance of the algorithm for cracks near the end supports, (2) Minimum size of a crack that the algorithm can detect, and (3) performance of the algorithm for rotated cracks. Since the working of the algorithm is affected by the measurement noise, measurement techniques should be improved to reduce this noise. One feasible way for better measurement accuracy
would be to incorporate some mechanism for precision positioning of the sensors over the shaft.
In the present chapter, the MCDLA has been verified experimentally. The algorithm is test- ed for the single- as well as double-cracked shafts. A scheme is proposed to reduce the effect of measurement noise by increasing the distance between consecutive measurement locations.
The algorithm is able to detect the presence of cracks in the shaft.