However, in the context of civil structural health monitoring, the apparent limitation of the wavelet transform is that the rational evaluation of the wavelet coefficient from structural vibration modes requires the mode shape measurement with a relatively high spatial resolution and reasonable accuracy. The mode shape spatial resolution can be further enhanced by performing a large number of modal tests, utilizing advanced vibration instrument such as non-contact scanning laser Doppler vibrometer Janeliukstis et al.

In this paper, the mode shapes are interpolated using a spline function with 20 interpolation nodes between each measuring point, resulting in a total number of sampling nodes or pseudo sensors. The main advantage gained by using wavelets is their multi-resolution characteristics allowing to zoom-in on any interval of time and space and the ability to perform a local analysis of a signal. The objective of this paper is to show that the procedures based on wavelet analysis of mode shapes offer a superior performance over traditional vibration-based damage assessment methods, especially for low levels of damage.

Different criteria based on statistical and probabilistic performance, namely the probability of detection, probability of false alarms and safety index, are evaluated as a function of damage level for an experimental beam and discussed in detail in the following sections. The damage detection criterion in this paper is formulated in the form of statistical test of hypothesis and based on comparing dynamic properties of a beam for increasing levels of damage.

The Continuous Wavelet Method is used to detect damage through the characterization of anomalies in the vibration modes of a structure or more specifically in this application, for a beam. The dynamic response of the beam is analyzed by comparing with a wavelet that is scaled and shifted along the length of the beam. The CWT of a one-dimensional function f x mode shape in this application is defined as Shahsavari et al. Wavelets with higher coefficients indicate a high correlation between the signal and the wavelet function.

If a significant change is detected in wavelet coefficients, it may be indicative of damage Cao and Qiao, However, CWT fails to clearly detect and locate damage in noisy conditions and for low levels of damage at multiple locations. In order to overcome these limitations, Shahsavari et al. The authors' main contributions for noise removal from experimental data set and damage localization are summarized as follows: a Statistical pattern recognition techniques are first used to improve the efficiency of CWT for data with low signal to noise ratios, b Principal Component Analysis PCA is then applied to wavelet coefficients obtained at a set of locations along the beam for two successive sets of measurements.

This technique improves the identification of patterns of variation that are most correlated with damage and filters out noise present in measurements. The scores associated with the dominant components of the PCA are found to be highly correlated with incremental damage; whereas the scores associated with higher components are assumed to be correlated with existing noise in wavelet coefficients and, hence, are discarded from further investigations, c statistical tests of hypothesis are used on the selected PCA scores to detect statistically significant changes in the distribution of scores to conclude to damage detection.

Tests are performed on location parameters of the PCA scores derived from wavelet coefficients with the t -test for the equality of score means values and the Mann-Whitney U -test for the equality of medians scores, d when a statistically significant incremental damage is detected, a Likelihood Ratio LR test is then performed to determine the most likely location of damage. The wavelet coefficients and PCA scores used for performing the reliability study of this paper are based on the above-mentioned procedure formulated in detail by Shahsavari et al.

The tests are conducted under controlled conditions with a steel beam where boundary conditions and damage levels at two locations can be varied Shahsavari, The test specimen is an I-beam Wx37 with three sections connected by bolts and plates. Figure 1 shows the setup in the initial assembled state. The full-damage state consists of a 0. Levels of damage are simulated by adding or removing sets of bolted connections at the two locations which are at 0.

Figure 1. Test beam configuration Shahsavari et al. Figure 2. Sixteen equally spaced accelerometers installed on the top flange of the beam are used for measurements.

## Comprehensive Fringe Pattern Processing Using Continuous Wavelet Transform

Random impulses are applied with a hammer with a mass sufficiently large to engage the first few modes of vibration. Series of measurements are recorded for each damage state to account for the variability of the mode shapes due to experimental errors and noise. The test setups used in this experiment consists of introducing incremental damage at only one of the two locations at a time.

The locations at 0. In all cases, the beam is considered to be in an initially damaged state E 0 at the locations of the connections but with only one location where damage is increased incrementally.

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For our purposes, the damage level dl is defined as the percentage change in the moment of inertia I—I 0 relative to the moment of inertia in the initial state I 0. The experimental program consists of dynamic measurements performed at five levels of incremental damage E 0 , E 1 , E 2 , E 3 , and E 4 at two locations along a beam. The change in dynamic properties is used to derive predictive equations for the dynamic properties as a function of damage level. The prediction equations are used to derive the probability of detection and the probability of false alarms as a function of the damage level.

Alvandi and Cremona compare four vibration-based damage identification techniques to detect damage on a simply supported beam. The probability of detection and of false alarm are analyzed for different levels of damage and noise and with different detection threshold levels. The authors showed that a high threshold produces fewer false alarms but has a low probability of detection, while a low threshold not only increases the probability of detection but also the number of false alarms. In addition, by considering data from elements adjacent to the damage location, the probability of detection is increased while reducing the probability of false alarms.

In conclusion, a higher damage level has a higher probability of detection since the corresponding damage index is higher and more likely above the threshold levels.

The damage detection criterion investigated in this paper is derived by comparing the dynamic parameters of a system for increasing levels of damage. Since there is uncertainty and variability associated with measurements of dynamic parameters, the criterion is formulated in the form of a statistical test of hypothesis.

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Tests can be performed to compare the full distributions of the dynamic parameters or some specific statistics such as the median or mean value. Note that the same principle could be applied to the full distribution or other statistics. The alternative hypothesis is that the average values are not equal in a scenario of incremental damage since all other conditions notably environmental conditions were kept constant in the laboratory. In this application, the t -test for the comparison of means is used the assumption of samples from normal distributions is verified.

For given samples from two populations, the p -value is the lowest level of significance that would provide the rejection limit of the null hypothesis. The null hypothesis is rejected when the p -value is smaller than the significance level of the test. The power of the statistical test is defined as the probability of rejecting the null hypothesis i. The test used in this application is for the equality of means for two random populations that follow the normal distribution when the variance is unknown but equal.

All these assumptions are validated for the data set used in this application. Given that variance is unknown but equal, a pooled estimate of the common variance is obtained as,. By definition, the power of probability of detection increases monotonically to 1. Figure 3. For our purposes, the safety index of the beam is defined for positive pure bending as a function of the level of damage, and provides a homogenous criteria for defects located anywhere along the length of the beam. For the purposes of the reliability analysis, the elastic moment capacity M R of the beam is used to characterize its resistance.

It is assumed that the safety index is equal to 3.

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The mean value of the moment capacity is obtained by assuming a bias factor of 1. The coefficients of variation of the moment capacity and of the moment demand M D are assumed to be 0. The average of the moment demand is specified to match the target safety index of 3. Table 1. Safety Index and percentage of damage as a function of the probability of detection and the probability of false alarms—fundamental frequency. Table 2.

## Two-Dimensional Wavelets and their Relatives

Safety Index and percentage of damage as a function of the probability of detection and the probability of false alarms—wavelets. The measurement of the natural frequency of a beam is a simple and widely used method for damage detection that requires a minimal amount of equipment. The change in frequency depends on the location of damage and its severity e. Various combinations of depth and location of cracks can provide similar results and the analysis of different modes of vibration can be used to provide a pattern that converges to the likely location of damage. The experimental protocol consists in taking several measurements at each damage level to account for the variability in the measurement of dynamic properties.

Extreme outliers are identified by using box-and-whisker plots prior to the statistical analysis Figures 4A , 5A. The measured frequencies are assumed to follow a normal distribution with a decreasing mean as a function of the level of damage. The results also indicate a greater reduction in frequencies for damage located at 0. The change in frequency at low levels of incremental damage is not as pronounced as for higher levels and hints at potential ambiguities in detecting low levels of incremental damage.

The damage level is quantified in this case as the percentage change in the moment of inertia of the beam compared to the initial condition state E 0. The frequencies as a function of damage for the first three levels E 0 , E 1 , and E 2 show a linear trend with constant variance when damage is located at 0.

The trend is non-linear for damage located at 0. The frequency as a function of the change in the moment of inertia is almost linear for low damage levels and increases rapidly for higher damage levels.

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The trend is more pronounced for the 0. Figure 4. Damage located at 0. A Whisker plot of the frequency of the first mode of vibration at 5 damage levels, B Linear regression of frequency as a function of damage level Damage levels E 0 , E 1 , and E 2. Figure 5. Since the main objective of SHM is the early detection of damage, a predictive model in the form of a linear regression is derived for the fundamental frequency as a function of the damage level using the data for states E 0 , E 1 , and E 2 for damage at 0.

The regression models provide estimates of the fundamental frequency as a function of damage level for intermediate levels of damage between state E 0 and E 1 Figures 4B , 5B. The difference between the two confidence intervals is indicative of the uncertainty associated with experimental measurements of fundamental frequency using impacts and the need for several repeated measurements for each damage level.

Also shown is the safety index as a function of damage level. The initial safety level was selected as 3. The two regressions indicate that the damage location has an effect on the fundamental frequency of the damaged beam with a larger decrease for damage located closer to the midspan of the beam. The performance of changes in the fundamental frequency for detecting early stages of damage is evaluated by performing tests of hypothesis on the mean value of the fundamental frequency between the initial state and increasing states of damage.

The distribution of natural frequencies as a function of damage level is found to satisfy the assumption of a normal distribution.

The mean and standard deviation in the initial state are obtained experimentally while the mean and standard deviation for intermediate levels of damage are obtained from the regression model. The uncertainty on the mean in the damaged state is computed for a set of 25 repeat impact measurements to reproduce the experimental protocol.

The t -test is performed with the pooled estimate of the variance since the variance of observations can be assumed constant at low levels of damage. The latter is evaluated for a given intermediate damage state dl by using the distribution for a prediction and averaging for a given number of repeat measurements e. The probability of detection, of false alarms and the safety index as a function of the level of damage are jointly shown in Figure 6 for damage located at 0.

The three levels for false detection 5, 1, and 0. The safety can be used as a structural performance measure to define the most appropriate detection protocol but also to compare the relative performance of competing SHM procedures. For example, SHM procedures can be evaluated on the basis of level of damage that can be detected with a high degree of certainty.

Tables 1 , 2 indicates the level of damage and safety index corresponding to given probabilities of detection and false alarms. The results indicate that damage can be more easily detected for damage located at 0. Figure 6.