Identify Cracks Depth

For the variant system, by assigning confidence level to approximate 99%, then the statistical significance of damage level was examined by t-test, the upper and lower bound of integration in Eq. (4.14) then determined. Since we have built the simulated statistical FCI databases for property variant system, we can map the noised measured frequency by EMA to the data of simulated databases that were both represented in Gaussian distribution form, then the cracks' depth were assessed by its probability.

CHAPTER 5

Demonstration Examples

The damaged beam (Beam-I) was used to demonstrate the assessment of single crack beam with different measurement resolution. A single shallow depth cracked beam (Beam-N) was also used to test for the capability of the proposed method. Multiple cracks example was represented by a 2-crack beam (Beam-M), the assessment process was demonstrated as follows：

5.1 Crack Location Detect of Beam-I

From the finite element normal mode analysis and EMA data, we have the three lowest mode shapes of damaged beam and intact beam. The crack location can be detected by applying the LDI by Eq. (4.10) in session 3.1. By reviewing the results in Fig. 22(a), we found that the crack was located at 245mm by the peak of LDI index curve of mode 1 and 3; by mode 2 the crack was located at 235mm.

When compared to the real crack location 243mm, the averaged absolute error was 1.64%.

5.2 Crack Depth Identification of Beam-I for Property Non-variant System

As discussed in chapter 4.7 and the procedures shown in Fig. 1, for a property non-variant system, we had built the simulated FCI database according to the crack location 245mm that was determined in the previous session. Then the unknown crack depth can be identified by Eq. (4.13). Since we had the EMA measured frequencies that were 193.43Hz, 602.38Hz and 1210.66Hz for the lowest three modes, by the interpolating process as shown in Fig. 23, we have crack depth 10.02mm, 10.26mm and 9.66mm for the three lowest modes respectively. The errors were +0.2%, +2.6% and –3.4% for the three modes and

(a) Beam-I, Meas. Resol. 10mm

(b) Beam-I, Meas. Resol. 50mm

(c) Beam-N, Meas. Resol. 10mm

Figure 22. Crack location detect by LDI curve

(a) Mode 1

(b) Mode 2

(c) Mode 3

5.3 Crack Depth Identification of Beam-I for Property Variant System

For a property variant system, refer to procedures described in session 4.7 and flowchart shown in Fig. 1, we need to build the simulated statistical FCI database according to the crack location found in session 5.1. In order to build the statistics database, we assumed the variations of mass density & Young's modulus were ±2%, ±5% and ±10% of its mean value. By sensitivity analysis, 300 samples were used for LHS sampling. The typical data from Monte Carlo simulation results on certain crack depth with different level of variations for different modes were shown in Table 4. Each set of mean with standard deviation represented a probability distribution on a point (crack depth) of statistical FCI curve as shown in Fig. 20 and 21.

Although the experiments controlled in the laboratory, there was still variability in the experimental data. For the study of noised measured frequency effects, the authors assumed that we have noise on the measured frequencies. The variations of noised frequency were assumed as ±2%, ±5% and ±10% of measured frequency incorporated with ±2%, ±5% and ±10% material variations. In Eq. (4.14) we could assess probability on specific severity by mapping the measured frequency to each of the probability distribution curves of statistical FCI database. By changing to different depths in sequence, we have probabilities at all depths.

The interpreted probability distributions of crack depth on material variations and varied noise level measured frequency were shown in Fig. 24. We have observed that from mode 1 results (1st row in Fig. 24), for 2% and 5% material variation, the maximum probability of crack depth all occurred at 10.0mm for 0%, 2%, 5% and 10% measured frequency noise, for 10% material variation, the maximum probability of crack depth occurred at 10.25mm for 0%, 2%, 5% and 10%

noise in measured frequency. From mode 2 data (2nd row in Fig. 24), only results for 2% and 5% material variation with measured frequency noise least than 2% can

from mode 3 (3rd row in Fig. 24), only the results of the material variation and noised measure frequency both least than 2% can be identified, the maximum probability of crack depth occurred at 9.75mm.

By reviewing Fig. 24 again, we found data in first mode; the crack depth has the distinct peak and the narrowest spreading on its probability distribution. This means that the lowest mode has less scattering on severity identification. We have also found that both the larger material variation and the larger noise level of measured frequency will made the probability distribution wider on severity and the reliability will be decreasing on the depths which had been identified.

Table 4. Monte Carlo Simulation Results among Various Crack Depth (Beam-I)

( in % of mean) Frequency Standard

Deviation Frequency Standard

(a)2% material variation, mode 1

(b)5% material variation,

mode 1 (c)10% material variation, mode 1

(d)2% material variation,

mode 2 (e)5% material variation,

mode 2 (f)10% material variation, mode 2

(i)2% material variation, mode 3

(j)5% material variation, mode 3

(k)10% material variation, mode 3

Figure 24. Probability distribution of Beam-I among varied crack depth (Meas. Resol. 10mm)

5.4 The Measurement Resolution Effects on Assessment Results

The above example was based on the assumption that we have the appropriate number of measurements; the authors took 10mm as the spacing between measurement points near the crack zone and 50mm~75mm on others, 18-impact locations in total. In this section, in order to test for the effectiveness of the proposed algorithm, the measurement spacing was extended to 50mm~75mm for the entire beam, 12-impact locations totally. According to the result of EMA and finite element normal mode analysis, the crack location can be detected as shown in Fig. 22(b). From the peak of LDI curve of mode 1 and 2 we found that the crack was located at 225 mm (-7.4% error). From mode 3 we have the crack located at 125 mm (-48.6% error). Mode 3 data lost its accuracy and it cannot be used for further identification on depth. With compared to the results of 18-impact measurement resolution (1.64% error), we have less accurate on crack location due to the larger measurement spacing.

For the property invariant system, took mode 1 and 2 results (crack location 225mm) as the basis to generate FCI database. With the same procedures described in the above example, by the interpolating process as shown in Fig. 25, the crack depth was identified by FCI, we have 10.26mm (+2.60% error), 8.50mm (-15.0% error) in crack depth for the first and second mode respectively. With compared to the results of above 18-impact measurement resolution example, its averaged absolute error of the lowest three modes (2.07%), we have less accurate results on crack location. Besides, due to the FCI database was based on crack location 225mm, the location was very closed to one of the node of curvature mode shape 3; hence we have poor result when applied mode 3 data for crack assessment.

For the property variant system with noised measurement, the statistical FCI database was also based on mode 1 and 2 results. For various property variation and different level measurement noise, the crack depth was determined by the

highest probability. The assessed probability distribution was shown in Fig. 26.

From mode 1 results (1st row in Fig. 26), for 2%, 5%, 10% and material variation, the maximum probability of crack depth all occurred at 10.25mm for all level measurement noise. From mode 2 data (2nd row in Fig. 26), only results for 2%

and 5% material variation with measurement noise least than 2% can be identified for crack depth, its maximum probability occurred at 8.50mm. Results from mode 3 (3rd row in Fig. 26), there was no clear indication for crack depth due to the crack location used for statistical FCI database was very closed to one of the node of curvature mode shape 3. With compared to the results of 18-impact measurement resolution example, 0.83% error for mode 1 and 2.5% error for mode 2 and 3, we have less accurate results on crack depth identified.

By reviewing Fig. 26 again, we have the same conclusion as the example of 18-impact measurement resolution that the result from first mode has the distinct peak and the narrowest spreading on its probability distribution on severity. The higher mode used in assessment, the larger material variation or the larger noise level of measured frequency will made the error larger, the probability lower and the distribution wider.

(a) Mode 1

(b) Mode 2

(c) Mode 3

Figure 25. Crack Depth Identification by FCI

(Beam-I, Meas. Resol. 50mm, Crk. Loc. 243mm, Depth 10mm )

(a)2% material variation,

mode 1 (b)5% material variation, mode 1

(c)10% material variation, mode 1

(d)2% material variation,

mode 2 (e)5% material variation,

mode 2 (f)10% material variation, mode 2

(i)2% material variation, mode 3

(j)5% material variation,

mode 3 (k)10% material variation, mode 3

Figure 26. Probability distribution of Beam-I among varied crack depth (Meas. Resol. 50mm)

5.5 The Shallow Crack Depth Example (Beam-N Case)

It is always a challenge work to identify a small depth crack on structure. A sample beam, named as Beam-N, was designed to complete the study. For the convenient to compare with the Beam-I example, a cracked beam was manufactured by wire-cut with the same crack location 243mm but a smaller crack depth 3mm.

According to the finite element normal mode analysis result and EMA data (by 10 mm measurement resolution near the crack, 18-impact example), the crack location can be detected as shown inFig. 22(c). From the peak of LDI curve of mode 1 and 2, we found that the crack was located at 225mm (-7.41% error) and from mode 3 we have the crack located at 245mm(+0.82% error). The averaged absolute error was 5.21%, with compared to the results of Beam-I we have larger error for a small crack depth beam example in crack location detection.

For the identification of shallow depth cracked beam in property invariant system, the authors took the average of mode 1 and 2 results (averaged crack location 232mm) as the basis to generate FCI database. With the same procedures as Beam-I, we had the EMA measured frequencies 231.78Hz, 639.68Hz and 1248.47Hz for the lowest three modes, by the interpolating process as shown in Fig.

27, the crack depth was identified by FCI as 3.09mm (+3.0% error), 2.60mm (-13.3% error), and 4.63mm(+54.3% error) for mode 1, 2 and mode 3 respectively, only mode 1 result was acceptable in accuracy. However, the small denominator (crack depth 3.0mm) made the large relative error. If we take a look at its absolute error, 0.09mm, 0.40mm, 1.63mm for mode 1, 2 and 3, with compared to the result of Beam-I, 0.02mm, 0.26 mm and 0.34mm; there were in the same error level except for mode 3. But, due to the large denominator (crack depth 10.0mm) we will have smaller absolute error for Beam-I.

For the crack depth identification of property variant system with noised frequency measured, the statistical FCI database was based on mode 1 and 2 results (crack location 232mm). The results of probability distribution assessed were shown in Fig. 28. We have observed that only the crack depth can be identified, by mode 1 under 2% material variations with no measurement noise, and the crack depth identified as 3.20mm (+6.67% error) by the highest probability. With compared to the results of Beam-I (10mm crack depth); its crack depth was 0.83%

error for mode 1 and 2.5% error for mode 2 and 3, we have less accurate results on crack depth identified. We have found that first mode result of Beam-N do not have a sharp peak and a narrow spreading on its probability distribution. The probability distribution for mode 1 has the same shape as the higher mode with higher measurement noise in Fig 24 and 26.

(a) Mode 1

(b) Mode 2

(c) Mode 3

Figure 27. Crack Depth Identification by FCI

(Beam-N, Meas. Resol. 10mm, Crk. Loc. 243 mm, Depth 3mm)

(a)2% material variation, mode 1

(b)5% material variation,

mode 1 (c)10% material variation, mode 1

(d)2% material variation,

mode 2 (e)5% material variation, mode 2

(f)10% material variation, mode 2

(i)2% material variation,

mode 3 (j)5% material variation,

mode 3 (k)10% material variation, mode 3

5.6 The Multiple Cracks Example (Beam-M Case)

The assessment of above examples was all based on single crack configuration. Multiple cracks may exist in structure systems. In this section, the multiple cracks sample (named as Beam-M) was adopted to test for effectiveness of the proposed algorithm. For the convenient to compare with Beam-I, the authors made an extra crack on Beam-I, it was also manufactured by wild-cut at a new crack location 131mm with the crack depth 8mm. The Beam-M then has 2 cracks on its configuration that the first crack located at 131mm, depth 8mm plus the second crack located at 243mm, depth 10mm.

To review the results of EMA and finite element normal mode analysis, the cracks' location can be detected as shown in Fig. 29, from the peak of LDI index curve of the mode1 and 3; we found the cracks were located at 135mm(+3.05%

error) and 245mm(+0.82% error), and there were different sensitivities on the peak of LDI for various modes. The averaged absolute error of multiple cracks was equal to +1.94%. With compared to Beam-I case (1.64% error), we have the same error level on crack location detection.

After we applying the identified multiple cracks location by LDI for FCI database generation (cracks location 135mm and 245mm). From EMA we have the first three modal frequencies 191.25Hz, 574.83Hz and 1079.46Hz of Beam-M and also from finite element analysis for no damage beam (Beam-S), its modal frequencies were 234.19Hz, 642.53Hz and 1251.1Hz. We have the frequencies change 18.34%, 10.54%, 13.72% for the three modes respectively. We need 2 sets of FCI curves to identify the depths due to the frequency change dominated by multiple cracks simultaneously.A little modification with compared to single crack case, we should first plot the contour lines of frequency change on each mode as shown in Fig. 30. Then by the intersection operation of two contour lines from different mode as shown in Fig. 31, the crack depths were identified as 8.00mm (0.0% error) and 9.90mm (+1.00% error) by the first and second mode or by the

first and third mode. The averaged absolute error for the depths of multiple cracks was equal to 0.50%. With compared to the results of Beam-I (2.07%), we also have the same error level on depth identification for multiple cracks case for the invariant system.

For the variant system with noised measurement, the statistical FCI database was also based on the cracks located at 135mm and 245mm that were detected by peak LDI. Various property variation and different level measurement noise applied, the crack depths were determined by the highest probability. As mention above, due the frequency change was dominated by multiple cracks simultaneously, we will determine all the depths at the same time. The probability of identified crack depths was represented by its brightness; the higher the probability, the brightened it was and vice versa. A red point in the figure indicated the highest probability on the depths. The assessed probability distributions of depths were shown in Fig. 32, 33 and 34 for mode 1, 2 and 3 respectively. The identified depths and its averaged absolutely error were list in Table 5.

Results from mode 1 as shown in the 1st column (2% material variation) and the 2nd column ( 5% material variation) in Fig. 32, its maximum probability of cracks' depth occurred at 8.50 mm and 9.90 mm for all levels of measurement noise. The maximum probability of cracks' depth occurred at 7.57 mm and 10.15 mm for all levels of measurement noise for 10% material variation as shown in the 3rd column in Fig. 32. In the 1st row of Table 5, we have found that their averaged absolute errors were ranged from 3.44% to 3.63% for mode 1. With compared to Beam-I single crack case (0.83% error for mode 1), we have acceptable error on the cracks' depth identified.

From mode 2 results, for 2% material variation (1st column in Fig. 33), the maximum probability of cracks' depth occurred at 7.50mm and 10.77mm for 0%

and 2% measurement noise, the cracks' depth occurred at 8.25mm and 9.47mm

33), the maximum probability of cracks' depth all occurred at 8.25mm and 9.47mm for all level measurement noise. For 10% material variation (3rd column in Fig. 33), the maximum probability of cracks' depth all occurred at 8.25mm and 10.33mm for all level measurement noise. In the 2nd row of Table 5, we found that their averaged absolute errors were ranged from 3.21% to 6.98% for mode 2. With compared to Beam-I single crack case (2.50% error for mode 2), we also have acceptable error on the crack depths identified.

From mode 3 results, for 2% material variation (1st column in Fig. 34), the maximum probability of cracks' depth occurred at 7.75mm and 10.33mm for all level measurement noise. For 5% material variation (2nd column in Fig. 34), the maximum probability of cracks' depth occurred at 8.50mm and 9.03mm for 0%, 2%, and 5% measurement noise, and occurred at 8.25mm and 9.47mm for 10%

measurement noise. For 10% material variation (3rd column in Fig. 34), the maximum probability of cracks' depth all occurred at 7.00mm and 11.63mm for all level measurement noise. In the 3rd row of Table 5, we found that their averaged absolute errors were ranged from 3.21% to 14.40% for mode 3. With compared to Beam-I single crack case (2.50% error for mode 3), we have the larger error on the cracks' depth identified.

Observation from Fig. 32, 33 and 34, we have the same conclusion that the result from the lower mode, the lower material variant and the lower noise in measurement, we will have the brightened (sharpest) peak and the narrowest spreading on its probability distribution on cracks' depth identification.

Table 5. Peak Probability among Material Variation with Different Level Measurement Noise Material Variation

(Mass Density & Young's Modulus)

σ= ±2% μ σ= ±5% μ σ= ±10 % μ

Figure 29. Multiple Cracks Location Detect by LDI

(Crk-1 loc.131mm, Crk-1 Dep. 8mm, Crk-2 loc.243mm, Crk-2 Dep. 10mm)

(a) mode 1 (18.34%)

(b) Mode 2 (10.54%) (c) Mode 3 (13.72%)

(a) by Mode 1 and 2

(b) by Mode 1 and 3

(a) 2% Mat. Var.;0% Meas.Noise (b) 5% Mat. Var.;0% Meas.Noise (c) 10% Mat. Var.;0% Meas.Noise

(d)2% Mat. Var.;2% Meas.Noise (e)5% Mat. Var.;2% Meas.Noise (f)10% Mat. Var.;2% Meas.Noise

(i)2% Mat. Var.;5% Meas.Noise (j)5% Mat. Var.;5% Meas.Noise (k)10% Mat. Var.;5% Meas.Noise

(i)2% Mat. Var.;10% Meas.Noise (i)5% Mat. Var.;10% Meas.Noise (i)10% Mat. Var.;10% Meas.Noise

Figure 32. Probability distribution of crack depths by mode 1 (Crk.-1 Loc. 243mm, Dep. 8mm, Crk.-2 Loc. 243mm, Dep. 10mm)

(a) 2% Mat. Var.;0% Meas.Noise (b) 5% Mat. Var.;0% Meas.Noise (c) 10% Mat. Var.;0% Meas.Noise

(d)2% Mat. Var.;2% Meas.Noise (e)5% Mat. Var.;2% Meas.Noise (f)10% Mat. Var.;2% Meas.Noise

(i)2% Mat. Var.;5% Meas.Noise (j)5% Mat. Var.;5% Meas.Noise (k)10% Mat. Var.;5% Meas.Noise

(i)2% Mat. Var.;10% Meas.Noise (i)5% Mat. Var.;10% Meas.Noise (i)10% Mat. Var.;10% Meas.Noise

Figure 33. Probability distribution of crack depths by mode 2 (Crk.-1 Loc. 243mm, Dep. 8mm, Crk.-2 Loc. 243mm, Dep. 10mm)

(a) 2% Mat. Var.;0% Meas.Noise (b) 5% Mat. Var.;0% Meas.Noise (c) 10% Mat. Var.;0% Meas.Noise

(d)2% Mat. Var.;2% Meas.Noise (e)5% Mat. Var.;2% Meas.Noise (f)10% Mat. Var.;2% Meas.Noise

(i)2% Mat. Var.;5% Meas.Noise (j)5% Mat. Var.;5% Meas.Noise (k)10% Mat. Var.;5% Meas.Noise

(i)2% Mat. Var.;10% Meas.Noise (i)5% Mat. Var.;10% Meas.Noise (i)10% Mat. Var.;10% Meas.Noise

Figure 34. Probability distribution of crack depths by Mode 3 (Crk.-1 Loc. 243mm, Dep. 8mm, Crk.-2 Loc. 243mm, Dep. 10mm)

CHAPTER 6

Conclusions and Discussions

In the presented research, we have the conclusions and discussion as follows:

(1) A damage assessment algorithm was developed by introducing the Monte Carlo statistical process and the modeling of material variations, measurement noise by Gaussian model. The algorithm has been verified by single and multiple cracks in a uniform mass density and Young's modulus variations system which is incorporated with different level noise in modal frequency measured. The effects on measurement resolution and the shallow depth crack characteristic were also investigated.

(2) Due to the material variation of the beams were varied uniformly across the entire beam, the LDI index for crack location detection was hold for both the property variant and invariant system. For the middle depth single crack beam (Beam-I, 10/16 depth of structure) we have the averaged absolute error 1.64% on the basis of measurement resolution 1/60 in beam length near the crack zone and 5/60~7.5/60 on the others, 18-impact locations in total.

(3) The crack depth was determined by the FCI index or the statistical FCI database. For the middle depth single crack beam (Beam-I) on appropriate measurement spacing, we have the averaged absolute error 2.07% for the invariant system and 1.94% error for variant system with material variation, measurement noise least than 10%.

(4) For practical applications, the ‘fixed response’ method of EMA should be change to ‘fixed impact’ to save labor work. It would be adequate to use a non-uniform spacing between measurement points and apply 1%~2%

To increase the spacing from 1/60(Beam-I) to 5/60 (Beam-N) in beam length on the crack zone, the error will be increased to 4.52 times in crack location detection, to 4.25 times in crack depth for property invariant system, and to 4.51 times in depth for property variant system.

(5) For the shallow crack depth single crack example (Beam-N, 3/16 depth of structure), the LDI index works well for location detection. With compared to the result of middle depth single cracked beam (Beam-I, 10/16 depth of

(5) For the shallow crack depth single crack example (Beam-N, 3/16 depth of structure), the LDI index works well for location detection. With compared to the result of middle depth single cracked beam (Beam-I, 10/16 depth of