Definition of the LDI Index - Dynamic Characteristics of Cracked Beam and The Damage

CHAPTER 4 Dynamic Characteristics of Cracked Beam and The Damage

4.1 Definition of the LDI Index

The previous researchers, Pandey et al. [21] introduced the application of curvature mode shape for the detection of damage location. In this research, we authors defined an index for crack location detection, named as LDI (Location Detect Index). For Euler-Bernoulli beam, the strain energy(Ui) of an intact beam with respect to mode shape-i (φ_i) can be expressed as:

where EI, l were the section rigidity and the length of beam. For an infinitesimal length dx located at xj along beam's axis, the strain energy of length dx can be

( )

² The authors defined the energy fraction with respect to total energy of entire beam Ui asF_{i j},

ij ij

/

F = u U

(4.3)

0^l

F

_{i j}

= 1.0

∫

(4.4)

For the same operation, for a cracked beam, we have：

( )

² mode shape-i and energy fraction of cracked beam respectively.

Let δκ_{i j} as the temporary feature for location detection and it can be expressed as follows：

* ij

F

δκ = −

(4.9) by the normalization operation, we have the location's discrimination feature LDI as follows：

The authors calculate each of the discrete point of the curvature related feature by central difference and then plot it along beam axis to complete the LDI curve.

For sample beam G, H, and I, all its crack location happened at 24.3mm but with different crack depths, 0.6cm, 0.8cm and 1.0cm, by examining the LDI index on the simulation result regarding mode 1, 2 and 3 shown in Fig. 12(a), (b) and (c) and EMA result for mode 1, 2 and 3 in Fig. 13(a), (b) and (c). We found the LDI index works well and it is consistent between simulation and EMA reach the below conclusions:

(1) Only at the location of crack will caused significant change on LDI by a sharp peak.

(2) The deeper crack depth will made the peak of LDI curve sharper.

(3) In practical EMA, due to limited impact location (sensors) applied, the LDI index will lose its accuracy to indicate the crack location due to larger measurement spacing, there will be discussion in session 5.4. (Refer to Fig. 13(c) mode 3 of EMA)

(4) In practical EMA, the authors also found that the intensity of LDI is stronger in low mode, in beam-I case almost double amplitude for mode 1 and 2 (Refer to Fig. 13(a) and (b) of EMA)

(5) There will no curvature change when crack located on the anti-node of modal curvature.

(a) Mode 1

(b) Mode 2

LDI LDI

(a) mode-1

(b) mode-2

LDI LDI LDI

4.2 Definition of FCI curve

Modal frequency will changed due to crack existence on specific location and depth. Refer to Fig. 14, the EMA result is represented by point symbols, and line symbols are for simulation result for comparison purpose. The line, also called FCI (Frequency Change Index) curve, from simulation was made by a frequency change due to a constant crack depth with different crack location which is traveling along beam length.

The observation from Fig. 14 can be concluded as following:

(1) The crack will made modal frequency changed, we observed that the deeper crack depth made the larger amplitude on FCI curve

(2) The FCI curve from frequency change is in accordance with modal curvature shape, there will no frequency change on the anti-node of modal curvature shape

(3) The frequency change possesses symmetry property among spatial distribution, so we need to find crack location before identify its severity to avoid finding the fault crack location on the symmetry side.

(4) By judging from the changes of each modal frequency and set 2%

tolerance limit in general engineering, the authors defined it is a shallow crack beam with the crack depth is least than 1/4 of depth. A sample beam (beam-N) in session 5.5 is designed to test for the effectiveness of proposed method.

The specific point on each FCI curve represents a damage state (certain crack depth and location) of cracked beam. The FCI index is a significant feature for finding crack severity. The authors then defined the frequency change as a specific point on FCI curve. It can be expressed as:

,int ,

where f_j_,int_act and f_{j damaged}_, are the frequency of mode-j for intact and damaged

(a) mode-1

4.3 FCI for Depth Identification of Single Crack Beam for Property Invariant System with Noise-free Measurement

When we have the noise-free measured frequency from EMA for a property invariant structure system, the table lookup process was adopted for damage severity estimation. After the crack location identified by LDI index, we can identify the unknown crack depth by applying linear interpolation between two FCI curves.

These FCI curves were above and below the EMA's on the specific crack location from simulation database. The authors increase the resolution of database to 1/16 of beam depth to avoid the calculation complexity, a linear interpolating then can be applied for unknown crack depth (βx) was shown as below and illustrated by Fig.

15. that were also calculated by Eq. (4.11) on the above frequency( f_u) and below frequency( fl ) compared with the EMA measured frequencies( fx ), βu and βl were crack depths with respect to Δf_u and Δf_l accordingly. By inserting Eq. (4.11) into Eq.

(4.12), we also have Eq. (4.13) as follows:

*( )

Figure 15. Crack depth assessment for property invariant beam structure

4.4 Influences of the Variation of Stiffness and Mass Density on LDI and FCI index

In order to clarify the effect of material property variation on LDI and FCI index, the authors prepared a simulated cracked beam (Beam-I, crack located 243mm, depth 10 mm) with a series combinations of different levels of mass density and Young's modulus variance that ranged from ±80% to ±120% of their mean value [15] [16]. It should emphasize that hear mentioned the stiffness variation is comes from environmental factor and other than the stiffness reduction due to the crack existence.

To review the result of Fig. 16, we can conclude that due to the stiffness and mass variation affecting the structure in a uniform way for the entire beam structure, the mode shape changed insignificantly on the variations, and the algorithm for crack location identification was held for the property variant systems. The LDI index can still indicate clearly for the crack location among various variation scenarios. The LDI index works well and robust for systems with uniform material property variations.

From the observation of Fig. 17, we found that the FCI index changed approximately ±20% when compared to invariant system. Hence, we should take into account the influences of property variations when applying FCI index for severity assessment. In the research, the authors represent these effects by statistical FCI databases, which were generated by LHS sampling in Monte Carlo simulation on beam with certain damage states incorporated with different level variances of mass density and Young's modulus.

(a) mode-1

(b) mode-2

LDI LDI

(a) mode-1

(b) mode-2

(b) mode-3

Figure 17. FCI index due to property variations (Beam-I, by simulation)

4.5 FCI for Depth Identification of Multiple Cracks Beam for Property Invariant System with Noise-free Measurement

The LDI index curve can be applied to multiple cracks case directly.

However, with compared to single crack case, the authors should do a little modification for depth identification of multiple cracks case due to the frequencies change were affected by all of the cracks in the beam.

When we have the crack locations from LDI index curve, we should extend the single crack FCI curve shown in Fig. 15 to build up a set of FCI contour curve as shown in Fig. 18 which was based on known crack locations, and each of the contour curve from specific normal mode represented the frequency change due to multiple cracks’ existence. To overlap these two contour curves and then the crack depths were identified by the intersection as shown in Fig. 19. A 2-crack beam example will be discussed in session 5.6 for demonstration.

We should noticed that when the crack number is more than two, its crack location can be assessed by the same LDI process, for the FCI database, we need to apply suitable mathematical tool to determine all the depths simultaneously. The artificial neural network could be an effective tool to achieve this purpose and need further study in advance, it will contain the training sample preparation, the sample training process and then used as the reference database for multiple crack depth assessment.

(a) Contour Line of Mode I

(b) Contour Line of Mode J

Figure 18. FCI Contour Lines due to multiple cracks existence

Figure.19. Determination of Crack Depths by the Intersection of Two FCI Contour Lines

4.6 Estimation of Crack Depth Probability for Variant Systems with Noised Measure Frequency

The severity identification process discussed in session 4.2, 4.3 and 4.5 were for the property invariant structure system and noise-free measured modal frequency. When the system mass density and stiffness were varied, the FCI simulation databases need to be extended. Basically, in the invariant system, for specific damage state of structure, its frequency change was a certain value only, it will map to a certain and confirmed point on FCI curve as shown in Fig. 15 or Fig.

19. But for a property variant system, for a specific damage state, the property variations will cause the change of frequency varied, then the corresponding point on FCI curve will be "smeared" as shown in Fig. 20. Usually we use a distribution function to describe the smearing phenomenon, for example, by the Gaussian distribution, and the noise polluted measured modal frequency could also described in a Gaussian distribution manner as shown in Fig. 21.

Since the FCI curve possesses a probability distribution characteristic in variant system, the results of identification will also display in a presence of probability distribution. As shown in Fig. 20 and 21, in statistical damage database, every point on FCI curve was accompanied with a Gaussian distribution, when we applied the measured frequency by EMA in probability distribution to find the unknown crack depths, we found that the probability distribution with mean value fx

was overlapped with several Gaussian distribution curves which represented for different damage states (crack depth). Each of overlapping represented the probability on these damage states. Hence, for a single noise polluted measured frequency by EMA, we will have several possible crack depths with its probability.

By collecting all the probabilities along various crack depths, the identified results will present by a probability distribution curve.

Figure 20. Crack Depth Assessment for Property Variant System

Figure 21. Crack Depth Assessment for Property Variant System subjected to noised measurement

Due to the measured frequency was noise polluted that we may represent the measured modal frequency by a probability distribution function. Assume that the material property variation was independent with measurement noise. Each probability (Pi) for damage state-i (crack depth) can be calculated by the following equation: the probability distribution function of modal frequency in simulation database for damage state-i (crack depth), and the p^EMA(f_x)was the probability distribution function of measured frequency with noise, the upper bound and lower bound frequency fb, fa should be determined by confidence level and the statistical t-test [12] that we should discuss later in this session. Both of the ( )p f and_i p^EMA(f_x)were defined by the Gaussian distribution function G(f) as below.

1 1

The statistical t-test was used to assess statistical significance of damage-sensitive features of EMA with the data in the simulated damage database.

As stated above in this session, the upper and lower bound frequency of probability function in Eq. (4.14) should be determined by confidence level and the statistical t-test. As described in reference [12], assigning 2 samples in population size n1 and n2 with sample meanX and₁ X and standard deviation S₂ 1 and S2, a test statistic Z

Where n1, n2 should be large enough to invoke the central limit theory to satisfied the normal distribution assumption and α was an arbitrary constant and assumed to be 0.0 in this research. The authors then set up the hypothesis to test the statistical significance by Eq. (4.17) equation. By solving for Eq. (4.16), we can then state that there was approximately 99% confidence level of truth if |Z| 3.0.≦

After we have assigned the confidence level to 99%, the upper and lower bound frequency of Eq.(4.14) can be determined by measure the distance between the mean of EMA data and simulated database that should not exceed three times of root sum squared of the standard deviations of EMA data and from simulated database's.

We may notice that the all the discussion above adopted the figure in the single-crack case; however, the algorithm described was suitable both for multiple cracks and single crack example. Except that the statistical FCI was function of 1 crack depth for single crack, for the multiple cracks the statistical FCI was function of many depths on the specific locations identified.

4.7 Procedures for Crack Detection and Identification

Three major steps in the process flowchart in chapter 1 as shown in Fig. 1 and the procedures are described as follows.

4.7.1 Cracks Location Detection

When we had prepared the modal frequency and mode shape of damaged beam from EMA and the modal frequencies and mode shapes of intact beam from simulation, by analyzing the peak response of LDI, we can identify crack location by Eq.(4.10).

4.7.2 Generate Simulated Statistical FCI Database

Since we had obtained the crack locations of the damaged beam, to build up the damage severity database should be followed. For the property variant system, we need to describe the property variation in the form of mean and standard deviation of Gaussian distribution. By using the LHS sampling technique [15], we shall have a minimum but useful samples that incorporated with various E*(stiffness variation), ρ*(mass density variation) and ξi*(specific crack depth).

Where E* and ρ* were a specific variation value of stiffness and mass density randomly selected by LHS sampling, and the ξi* denoted the specific depth of cracks from a series of possible cracks' depths. By assigning each set that composed of ξi* with E* and ρ* for finite element normal mode analysis repeatedly, we could generate the simulated statistical FCI databases represented by its modal frequency and variation among various cracks' depths. The above process is also mentioned as the Monte Carlo simulation.

The same procedures were used for the property invariant system to generate simulated FCI database, except that for the deterministic system, there

deterministic normal mode analysis for one set of specific cracks' depth(ξi*) among the possible cracks' depths to build the database.

4.7.3 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

Figure 22. Crack location detect by LDI curve

(a) Mode 1

(b) Mode 2

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

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