1. Introduction
Structures made of conventional materials are being extensively replaced by fibre-reinforced polymer (FRP) composite materials in the aerospace, automotive, energy and other industries owing to their superior specific strength [
1], negligible thermal expansion, as well as fatigue and corrosion resistance compared to metals. The safety and reliability of structures, such as aircraft fuselages, helicopter blades, as well as wind turbine blades, is ensured by surveys using non-destructive testing (NDT) methods or planned maintenance. Many NDT approaches require manual inspection, whereas planned maintenance often assumes taking the structure out of operation, which increases downtime and increases costs. What is more is that sometimes the inspection is unnecessary due to lack of damage. In order to detect the onset and progression of existing structural damage in a timely manner and reduce the maintenance costs, effective structural health monitoring (SHM) solutions are crucial.
Output-only SHM involves the detection and possible characterization of damage by analysing only the response signals collected from sensors mounted on the structure. Structural excitation can be provided if conditions for operational modal analysis (OMA) are fulfilled [
2]. In practice, such is the case of ambient excitation of a stochastic nature, for example, that is caused by wind, sea waves or traffic loads. Specialized signal-processing algorithms can be applied to these output-only responses to extract essential information on the status of structural integrity. This is normally achieved through a statistical pattern recognition framework, where damage-sensitive features (DSFs) are extracted from the sensor measurements [
3]. The DSFs in OMA are typically modal parameters—resonant frequencies, damping ratios and mode shapes. Although several techniques of modal parameter identification exist, such as stochastic subspace identification [
4] and least-squares complex exponential [
5], etc., it is possible to extract only a limited amount of modal parameter observations and only from repeated measurements. For a statistical damage-detection approach using a significantly larger amount of data, traditional OMA modal parameter estimation techniques can be substituted by time-frequency analysis.
Time-frequency analysis methods, such as short-time Fourier transform, Wigner–Ville distribution and Hilbert–Huang transform and wavelet transform (WT) [
6], among others, have become established techniques for analysing transient signals, which are nonstationary in nature. In the case of OMA, free structural vibrations induced by ambient excitations are transient signals with a finite energy localized in time and frequency. Another merit of time-frequency techniques is that they have an ability to decompose a composite signal consisting of several modes of vibration (degrees of freedom) into individual modes [
7]. This is normally achieved by finding wavelet ridges—high energy curves in a time-frequency plane tracing that allow for system identification via the extraction of modal parameters. Staszewski was the first to demonstrate the WT can be used as a tool for structural modal parameter identification [
8]. Wavelet transform has been used in modal parameter estimation for real structures, such as long-span cable-stay bridges and suspension bridges using continuous wavelet transform in [
9]; a cable-stay bridge in Taiwan using wavelet packet transform in [
10]; a 600 m tall building in China using a combination of empirical wavelet transform and Hilbert transform in [
11]; and a pedestrian overpass in the USA using multisynchrosqueezing transform, a variation of wavelet transform that yields a more concentrated estimate of modal parameters at the cost of higher computational complexity in [
12]. Authors in [
13] proposed an output-only modal identification and structural damage detection technique based on time-frequency techniques, including wavelets. The above studies have focused on modal parameter estimation via wavelet transform. However, it has been demonstrated in [
14] that the time-frequency approach with continuous wavelet transform (CWT) allows for the extraction of numerous instances of modal parameters which, when employed in statistical pattern recognition schemes, are more beneficial, since more data are available. This leads to a larger dataset and, therefore, issues of model overfitting and underfitting can be solved.
After the extraction of DSFs, an anomaly detection algorithm is employed to identify outliers supposedly originating from damage or changes in environmental conditions [
15]. A popular class of methods of anomaly detection is based on dissimilarities between a reference structural state and a potentially anomalous state. The Mahalanobis distance (MD) metric has been successfully used for such purposes [
16,
17,
18], owing to its ability to detect outliers in a multidimensional feature space with an arbitrary number of extracted DSFs. However, for MD to be used, DSFs have to follow a normal distribution. On the other hand, kernel density estimation (KDE) is an approach used to identify the underlying probability density function without any assumptions regarding a probability distribution of data. KDE was used for structural damage detection in [
19,
20].
Building on the concept introduced in [
14], the aim of the present study was to develop a structural damage detection algorithm using continuous wavelet transform as an alternative modal parameter estimation technique. The use of CWT enables a statistical approach to damage detection using KDE. The underlying tasks of the study were to estimate the probability density function of the extracted modal parameters and calculate the probability centroids for modal features. Finally, Euclidean distances between centroids at a reference point and various states of damage were to be calculated, explored as a potential damage indicator and compared with a Mahalanobis distance in terms of accuracy.
2. Modal Identification
The current study is based on the concepts described in detail in [
14]. Wavelet ridges are hidden constituent elements of a finite-energy signal revealed through wavelet decomposition of a said signal. Ridges contain all of the essential information on structural modal parameters of a structure whose impulse response is available. Wavelet phase can be used to extract numerous observations of resonant frequencies and damping ratios for each mode of structural vibration. These instances comprise a dataset that is representative of a structural condition in the current state. Each new structural state, for example, the occurrence of damage is associated with changes in modal parameter values. This concept can be utilized in machine-learning-aided structural damage detection and, in a broader sense, structural health monitoring (SHM).
A quick recap of the methodology from [
14] is given as follows:
CWT on the recorded response signals
is carried out using analytical Morlet function and storing complex-valued CWT coefficients in a matrix form:
where
is the vector of scale factors,
is the vector of translation parameters, and
is the signal length.
Wavelet ridges (in terms of scale parameters ) of each response signal are found by, firstly, finding the and parameters (denoted by and ) corresponding to the maximum value of modulus of CWT coefficients and, secondly, testing the ridge condition at a fixed parameter (time instant when vibration amplitude is maximum).
The conversion from scale parameter to frequency is performed through the following relation:
where
is the central frequency of wavelet function. It is essentially a pseudofrequency or a frequency that the wavelet function would have if it was a harmonic function.
Damping ratios are extracted in the following substeps:
The moduli of CWT coefficients are extracted along the wavelet ridge, and its natural logarithm is calculated. It is denoted as .
By plotting
versus the time axis, a straight line is obtained for most of the time span of response because the modulus of CWT coefficients decays exponentially with time. This straight line is fit with a linear function, and the slope parameter is extracted. This slope parameter is equal to
where
is the damping ratio, and
is the undamped natural frequency.
The relationship between the damped and undamped natural frequencies is well-known from structural dynamics as
. Hence, in practice, the damping ratio is obtained as
3. Damage Detection Algorithm
SHM systems estimate the health state of structures during operation to ensure their safety and economic efficiency. Such structures can be industrial, transport or energy equipment with structural elements, for example, wind turbines or wind generators and their elements—tower and blades. An SHM system’s sensors network will be connected to the structure, read the vibration data and send it to the workstation (computer). Then, an operator estimates the modal parameters of the structure. The resulting parameters together with the possible external operational factors, such as static loads, temperature or speed of rotation (blades), are input to a modal passport [
21,
22], where the past measurements are stored. Modal parameters in a reference or intact structural state together with their deviations due to operational factors form a “signature” of a structure that is unique to this structure or for structures of this type. As next step, a specialized algorithm analyses the modal passport and recognizes a particular structure. By analysing modal parameter changes with damage, one can infer on the severity of the damage, which allows for further planning of the agenda of structural serviceability—repair, replacement or resuming operation if damage is not significant.
The damage detection algorithm proposed for an SHM system originates from an anomaly detection field. It has three distinct phases, namely, Phase I: signal collection, Phase II: feature extraction, and Phase III: statistical control, as shown in
Figure 1.
3.1. Phase I—Signal Collection
Vibration signals as a response to structural excitation are measured with sensors connected to measurement channels to . In the case of numerous instances of impact excitation ( excitation of the structure during the measurement session), any individual free-vibration decay profile is isolated from the whole signal. Afterwards, , these individual vibration profiles, are averaged to obtain averaged vibration responses.
3.2. Phase II—Feature Extraction
In Phase II, a time-frequency analysis of the averaged responses is performed using CWT. CWT analysis involves the identification of wavelet ridges from the ridge condition in Equation (2) in the time domain signal analysed. Wavelet ridges represent the oscillatory modes, which comprise the components of vibration decay signals. Subsequently, this ridge information from the ridges identified at
is used to identify the resonant frequencies
from wavelet phase (Equation (4)), and the decay profile of CWT coefficients in time domain is used to extract the damping ratios
(Equation (6)). CWT analysis involves operating with wavelet scale parameters. In order to convert the scale parameter to frequency values, a wavelet scale-to-frequency conversion is realized through Equation (3) and is illustrated in
Figure 2 for the analytical Morlet wavelet function. In this study, Morlet mother wavelet is used since it exhibits a high correlation with the time domain vibration data. This mother wavelet has been used for structural damage detection [
23,
24]. The identified modal parameter value pairs form features
and
that are organized into a two-column matrix
, where rows correspond to observations and columns correspond to features. This feature matrix is used in the next phase of the anomaly detection algorithm proposed.
3.3. Phase III—Statistical Control
Phase III is concerned with performing statistical control of the feature values. The first stage is the feature-value filtering, which is carried out by using the interquartile range (IQR) rule. The goal is to remove the outliers from the features. Unlike the threshold set to mean plus/minus two or three standard deviations, the IQR approach for outlier removal is appropriate for data that does not necessarily follow a normal distribution. The frequency values for the ridges identified at
are filtered according to
where
is the interquartile range,
is the first quartile, and
is the third quartile of the filtered resonant frequencies from the previous step. The filtered resonant frequency and damping ratio values are stored as two-column vectors
.
The next step involves exploring the underlying probability distribution of the filtered features. For this purpose, the kernel density estimate (KDE) is computed. The reason is that a kernel distribution representation of the probability density function (PDF) of the data does not make any assumptions on the underlying distribution. The KDE is defined by a smoothing function and a bandwidth value that controls the smoothness of the resulting density curve. The kernel density estimator of the data at hand (
) is given by
where
is the sample size,
is the kernel-smoothing function governing the shape of the curve used to generate the PDF estimate, and
is the bandwidth. In this study, the obtained vectors of filtered frequency and damping ratio values are used as the data
, and normal density is used as a kernel smoothing function since the feature values approximately follow Gaussian distribution. It is important to choose the optimum bandwidth parameter since it regulates the degree of smoothing. In this work, bandwidth optimization was carried out by the following procedure illustrated in
Figure 3:
Perform a cross-validation partition on the data to create 10 folds where one fold is used for testing and 9 folds are for training. Perform 10 iterations of such a partition, where a different fold is used for testing in each iteration.
Define a range of bandwidth parameters to test.
In each training fold and the single testing fold, compute the KDE according to Equation (8) for each value of the bandwidth parameter. Then, compute an error
between the KDEs of the testing and each training set according to
where
is the number of folds.
Calculate the cross-validation error as a mean-squared-error of the errors in Equation (9) across all folds for each value
according to
Find the optimum bandwidth parameter by calculating the minimum of these cross-validation errors across all bandwidth values
Afterwards, the KDEs with these optimized bandwidth parameters are calculated for reference and all monitoring cases. Then, a centroid value of the KDE for both features is calculated according to
where data
in this study are the filtered feature values. Thus, both quantities,
and
, are calculated for reference and all monitoring cases. Next, the centroid values of both features are organized into a row vector for each case at hand, and the Euclidean distances between centroids at reference and each monitoring case are calculated as
where
,
, and
since the centroid vector contains two values.
Once the Euclidean distance between the reference and all monitoring cases is calculated, a threshold value is established according to the following scheme:
- 6.
Consider all available structures of the same type at their reference state.
- 7.
Perform a modal parameters estimation to form feature vectors and calculate their centroid values for each structure.
- 8.
Calculate the Euclidean distance between centroid values in all possible combinations of structure pairs.
- 9.
Calculate the median value of these Euclidean distances and confidence bounds as
where
is the number of Euclidean distance samples,
is the quantile corresponding to median (50% of data), and
is the critical value dependent on a chosen confidence level. For confidence level 0.95,
. The threshold of the Euclidean distances is set as a lower confidence bound at
.
4. Experimental Campaign
The algorithm proposed is validated on the modal parameters extracted from five glass-fibre-reinforced polymer composite specimens with a cylindrical shape manufactured in the scope of an SHM system prototype research project. Cylindrical structures mimic the structural components of serial production, such as a helicopter tail boom, for which the current anomaly detection algorithm is intended.
4.1. Specimens
Testing objects are the structures in the form of cylinders fabricated from a composite material with the flanges made of plywood rings. The specimens are made of 300 g/m
2 fibreglass fabric with a fibre orientation of 45° and LG 385 epoxy resin (HG 385 hardener). The weight of the specimen with the upper and lower flanges is 4.37 kg. Photo of a specimen is shown in
Figure 4a. The specimen design includes (see
Figure 4b):
Item 1—composite cylinder made of fiberglass and epoxy resin;
Items 2 and 3 —top and bottom annular flanges for cylinder fixation made of laminated plywood (30 mm thickness), respectively;
Item 4—a network of 48 piezoelectric strain sensors;
Item 5—wires connecting the sensors;
Item 6—4 D-SUB type connectors at the places for connector fastening.
The dimensions of the specimen are as follows: nominal diameter of 300 mm, nominal length of 710 mm (with the flanges of 773 mm) and wall thickness of 1.45 ± 0.05 mm.
4.2. Measurement Subsystem
Each specimen includes its own sensor network, which outputs the signals during testing to the measuring system, providing a registration and storage of the signals.
The sensor network of each specimen comprises 48 polyvinylidene fluoride piezo film sensors connected to four conductor terminals and wiring harnesses with connectors. These films are flexible, lightweight and have piezoelectric properties [
3]. Due to these properties, a strain in the film causes a change in stress. The piezo film is located between two printed silver electrodes, forming a capacitor-like structure. The dimensions of the sensors are roughly 45 mm × 20 mm × 0.05 mm, electrical capacity of 1.3 nF, operating temperature from −40 °C to 60 °C. The view of the sensor prepared for gluing on the specimen is shown in
Figure 5a. The wires used for the SHM system prototype are the small, lacquered copper wires with a diameter of 0.25 mm. These wires are glued to the sensors with a special two-component epoxy glue, after which the sensor is covered with a nonconductive insulating tape. The electrical conductors connecting the sensors with the terminals (
Figure 5b) are laid in the form of bundles in the longitudinal and circumferential directions and fixed with adhesive tape. On each terminal, 12 similar (conditionally signal) conductors are assembled on 12 contacts and 12 conditionally negative conductors on one common contact. Bundles of wires are soldered to the terminal contacts. The sensors are installed on the specimens in accordance with the premade markings, as shown in
Figure 5c. At this stage, the sensor network is covered with a protective composite layer, and the specimen is glued into the annular grooves of the flanges.
4.3. Modal Testing
After the instrumentation installation, each specimen is fixed on a U-shaped modal testing stand, which, in turn, is mounted on a vibration isolation base (see
Figure 6). Structural excitation is performed by repeated impacts of the specimen with a plastic modal hammer in the radial and vertical directions for 120 s. This test procedure is repeated 3 times. The Brüel & Kjær (B&K, Singapore) system LAN-XI Type 3053 is used as the data-reading device. A total of four Type 3053 modules were used to measure 48 channels simultaneously. A portable computer with software from the manufacturer of measurement modules (B&K, Singapore), such as Pulse Labshop, was used for data collection, processing and management. The entire signal recording with a duration of 20 s and sampling frequency of 4096 Hz contained 4 to 6 free vibration decay responses corresponding to the 4 to 6 instances of impact excitation. The test cases realized in the current study involve a reference state and progressing damage. The damage comprises a circular hole drilled through the thickness of the specimens in the same location. The diameter of the hole is increased and has the following values: 4, 8, 16, 24, and 32 mm.
All test specimens manufactured were visually inspected to check for defects or damage, as well as their compliance with the specifications. Local deviations and small differences between the specimens were identified to be mainly due to the factors of hand-made technology. These factors include uneven filling of the specimens and flange joint, resin leaks on the specimen surface, air bubbles in the protective layer, undercut on the flange of specimen No. 5, slightly different location of the first hole on the flange for each specimen, resin pouring out at different locations on the inner and outer surfaces of specimens, and differences in wire layouts for sensors.
4.4. Test Cases
The test cases realized in the current study involve reference state and progressing damage, namely, a circular hole drilled through the thickness of the cylinders in the same location. Diameter of the hole is increased in five stages as follows—4, 8, 16, 24, and 32 mm. The location of the hole schematically is shown in
Figure 7a, while the close-up photo view of a 4 mm hole is shown in
Figure 7b.
6. Conclusions
In the current study, an anomaly detection algorithm based on output-only structural vibration responses of structural components was proposed. The algorithm detects changes of modal parameters caused by structural degradation, such as the progression of damage. Phase I deals with the acquisition of vibration response signals from the sensors mounted on the structure and subsequent signal averaging and fusion. Phase II uses the modal parameters as the features identified with continuous wavelet transform routine. Phase III is carried out for decision-making regarding the state of integrity of the structure in question based on the statistical descriptors of the extracted features. Here, the feature filtering scheme based on IQR Rule is adopted to remove outlier feature values. Feature filtering revealed that the distribution for some specimens is, in fact, bimodal, while for others it is skewed and not classical Gaussian. The probability density function of the filtered features is estimated using a kernel density estimate (KDE) to avoid the assumption that the underlying distribution is normal. The damage indicator is based on the Euclidean distance between the centroid of KDE for both features at reference state and any state of damage. The following can be concluded:
The Euclidean distance of the centroids of the modal features KDEs between the reference and damage states can be used to detect damage.
The damage indicator proposed shows an upward trend for damage progression, meaning that it is effective in detecting increasing severities of damage.
Some vibration modes are more sensitive to damage than others. Therefore, multiple vibration modes have to be identified in order to increase the reliability of the damage detection. For example, features originating from the vibration modes at 106 and 176 Hz have significantly higher deviations from the reference than the vibration mode at 185 Hz. Therefore, these vibration modes were more effective in damage detection. On the other hand, these vibration modes could not be identified for all damage cases, while the vibration mode at 185 Hz was present in all scenarios.
There is a significant scatter of the feature value deviations from reference among the test samples. For the most part, this is due to inconsistencies in the sample design and instrumentation, as mentioned in
Section 4.1. Specimens.
The damage indicator proposed was compared to the Mahalanobis distance metric for damage detection. Both methods yield comparable damage detection accuracy. Therefore, there is no reason to use a more computationally costly Mahalanobis distance approach.
Future research will be devoted to the consideration of an influence of environmental and operational factors on the structural characteristics and modal parameters for a practical SHM system. To reduce the influence of the ambient temperature, for example, some papers consider methods for suppressing this effect as interference [
27], in others, methods for constructing quantitative models, that accurately predict the modal frequency that corresponds to temperature change are proposed [
28]. A promising way to consider influence factors is to utilize the modal passport mentioned above, which is a method for collecting and processing all modal data of a structure while taking into account environmental and operational variances.