1. Introduction

2. Reminder about PMCs and Their Damage Mechanisms

3. Vibration Methods

3.2. Linear Vibration Methods

3.2.1. Natural Frequencies

The natural frequencies of a structure are the frequencies where the structure will tend to vibrate, after applying an external force. Using these frequencies to excite the structure will introduce the effect of resonance which creates a higher amplitude of vibration. They are directly linked to the mass and stiffness of the structure, as well as its boundary condition. This is what pushed many researchers to use them as damage indicators. Indeed, the damage will create a loss of stiffness, which induces a decrease in natural frequencies (Figure 8). Moreover, this technique is one of the cheapest and easiest to perform. In a review published in 1996 [55], it is indicated that the first one to use natural frequency loss as a damage indicator was Lifshitz [56] in 1969.

In [57], the change of the first natural frequency in E-glass/epoxy laminated composite plates with 2 different dimensions (13x6cm² and 19x6cm²) and 3 stacking sequences ([90/90/90/90]°, [0/0/0/0]° and [0/90/0/90]°), with a cantilever configuration (Figure 9) was investigated. Damage was introduced by cutting the inner fibers in two locations: in the middle and at the root of the structure (which plies is unknown). The fiber cutting dimension was 2.1x1 cm² and 3.1x1cm². PZT transducers have been used to excite the plate but also to catch the output signal. An oscillating voltage with different amplitudes was transmitted to the actuator (from 40 to 80V) and the output was displayed on a sensing unit. The frequency was manually varied from 0 to 40 Hz until a sharp increase in the sensing voltage was observed. The frequency of this increase corresponded to the first natural frequency of the structure. The frequency decreased (1.5 to 4Hz for the smaller plate and 0.3 to 1Hz for the larger plate) when the structure was damaged for the [90/90/90/90] stacking sequence. However, this was not the case for the other stratifications, due to the high stiffness in the [0/90/0/90] sequence.

In [58,59], two kinds of inner piezoelectric transducers MFC and PFC have been used in glass/epoxy laminate and sandwich beams in a cantilever configuration. Different severities of delamination (10/20/30/40/50/60 % of the area of the structure) and crack (2/3/7/14/20/27/34 % of the length of the structure) were created in laminates (300x32x21mm³) and only delamination between core and skin (10/30/45/60/75% of the area of the structure) in the sandwich material (300x32x14mm3). The excitation was introduced by displacing the tip of the beam by 0.5 inches and the transducers caught the vibration of the structure. Here the change of the first two natural frequencies was investigated. The frequencies decreased when the severity of damage increased, the decrease is almost linear and slow, which means no clear distinction is possible before a large amount of damage (20% for the delamination and 15% for the crack of the laminates and 60% for the sandwiches), which means the small damage cannot be detected. The loss of natural frequency is higher in the laminated composite.

In [60], PZTs have been surface-bonded and used as actuators and sensors, on a graphite/epoxy [90/+45/-45/0]° plate (250x50x1mm³) clamped at one end. The natural frequencies of six modes of vibration were used for several types of damage: a 6.4 mm diameter hole in the center, matrix cracks induced by fatigue testing and mallet hit, and delamination (8%) made of Teflon strip and cut. A signal generator was used to transmit the voltage to the PZT and a LV was used to compare with the results from PZT. Clear distinctions about frequency loss for any type of damage, from LV or PZT, were not possible.

In [61,62], experimentation was conducted on a plate (305x244x2.16mm³) made of 8 plies of carbon/epoxy stack in [0]_s. To isolate the dynamical response, the plate was suspended with wires. An impact hammer made the structure vibrate and the signal from surface-bonded MFCs and accelerometers was caught, then it was processed in a data acquisition LMS system and displayed on a PC analyzer interface Testlab. Results were difficult to analyze and no real distinctions between undamaged and damaged structures (made from hole and impact – no information is given regarding the dimensions of damages) were observable. For this reason, the authors used different metrics that compared healthy and damaged signals (see FRF section).

The effects of damage on natural frequencies of the first three modes have been investigated, on glass/epoxy laminate (200x30x8mm³) [63] and sandwiches (250x40x26mm³) [64,65] with PVC foam core and glass/epoxy skin, on a cantilever configuration. The damages were debonding between the core and skin for the sandwich (from 20 to 180 mm of debonding by 20 mm step) and fatigue induced for the laminate. The structure was excited with a shaker and a power amplifier, and the vibration was sensed with an accelerometer. The excitation signal was a swept-sine around bending mode frequency with an amplitude of 50 mV. The natural frequency decreased with damage severity and the higher mode seemed more influenced by this severity (Figure 10). In these references, the change of natural frequency was studied and investigated, especially, the loss factor and the nonlinear elastic and dissipative parameters. These magnitudes will be introduced later.

Through these articles, detecting damages from the variation of natural frequencies seems efficient only when damage is already consequent. However, when damage is small (< 20%), this technique showed some limits. Table 3 gives additional bibliographical references using this method. Nowadays, it is less and less used, because of its high sensitivity to environmental effects, like temperature or boundary conditions.

3.2.2. Mode Shapes

The mode shapes correspond to the displacement of a structure for a particular natural frequency (a particular mode). The simplest example is the vibration of a fixed string, for each mode, there will be nodes (fixed part) and antinodes (maximum displacement) (Figure 11).

The first thing is to obtain the mode shapes of the structure, so it is necessary to perform a modal analysis that takes the vibration measurements at several locations. This means having a network of sensors, moving excitation location and sensor position [39,72], or scanning the structure with a LV.

Several techniques are existed using modal shapes to detect and localize damage: the most classical ones are the Mode Shape Displacement difference (MSD) or the use of assurance criteria such as Modal Assurance Criterion (MAC) [73] and criteria derived from it, such as Co-Ordinate MAC (COMAC) and others …[74]. Damage indexes derived from these criteria can also be used such as the Mode Shape Damage Index (MSDI).

The MAC [75] is the main statistical tool regarding mode shapes, it compares two sets of modal vectors using this formula:

$$MAC\left(\left\{\phi r\right\},\left\{\phi s\right\}\right)=\frac{{\left|{\left\{{\phi }_r\right\}}^T\left\{{\phi }_s\right\}\right|}^2}{\left({\left\{{\phi }_r\right\}}^T\left\{{\phi }_s\right\}\right)\left({\left\{{\phi }_s\right\}}^T\left\{{\phi }_r\right\}\right)}$$

(1)

where φ_s and φ_r are the modal vectors. MAC will be close to one if there exists a linear relationship between 2 modes (if the vectors have a similar way of moving), otherwise, it will be close to zero (Figure 12). In other words, MAC can detect which modes will be affected by the damage. A MAC can have similar modal vectors even for an intact structure, this is why reference of MAC from the intact structure is necessary to detect the variation. The MAC is also commonly used to see the coherence between experimental and numerical vibration analysis.

In [76], the modal parameters of a hemp fiber/epoxy resin plate (150x150x5mm³) in a free-free condition were used to detect six damage scenarios: cut of a layer with different sizes and positions (20x20, 15x15, 10x10mm² in the center and 20x10mm² on the edge), cut of 2 layers (10x10mm²) and 3 layers (10x10mm²). The classical impact hammer/accelerometer combination was used to perform the modal analysis (Figure 13). A grid of 6x6 was created on the plate to acquire 36 data points through the roving hammer method [72]. The MAC and COMAC confusion matrixes between intact plates were created, in order to have a reference of both matrixes without damage. Then, it was made between the intact and the six damaged plates and both detected damage occurrence in the structure. Moreover, the COMAC has higher sensitivity to small damage than MAC.

In [77], COMAC was used to detect and localize damage in a helicopter composite blade made of glass fiber-reinforced plastic with a honeycomb (Figure 14). The structure was suspended to obtain a free-free condition for modal analysis. The actuator was an electrodynamic shaker implemented with a burst random signal and the sensors were accelerometers positioned on 55 localizations. “Fake” damage was introduced by adding an extra mass of 300 g at one location, the goal of the study was to localize this mass. The modal analysis was performed for both intact and damaged blades. The COMAC detected the presence of damage very close to the exact position.

Two derivations of MAC: PrMAC and WECoMAC have been used in [78], on composite plates, made of carbon/epoxy. Damage was introduced from a drop-weight impact test and then compression after the impact test has been made, in order to determine the compressive residual strength of the structure. The mode shapes were obtained from the roving hammer test at 25 different locations and damage indicators were calculated at each damage step, and PrMAC showed the highest sensitivity regarding damage detection.

To summarize, using changes in modes shape to detect and localize damage has been used by many researchers, however, there exist many techniques and damage indicators, and the classical ones are MAC and COMAC. In [74], a large list of such criteria derived from MAC is given. This technique shows good results in many studies, however, it needs a lot of sensors or heavy measurement processes to acquire the modal deformation accurately. There exist several vibration methods derived from the mode shapes, such as the mode shapes curvature and the modal strain energy, this is why this is common to see articles using several and comparing them.

Table 4. Other articles about the experimental use of mode shapes for SHM.

Table 4. Other articles about the experimental use of mode shapes for SHM.

Method	Structure	Fixation	Input	Output	Damage	Description	Ref
MSC	T-stiffened panels of carbon/epoxy	Elastic bands	Shaker (sinusoidal wave (0 – 10 kHz))	LV	Delamination (PTFE Film) Porosity	Change in mode shape displacement between damaged and intact structures was not the best indicator of damage presence	[79]
MSC and mode shape slope change	Carbon/epoxy laminated plate	Suspended with 2 cotton strings	Impulse hammer	Accelerometer	Surface and penetrated crack	The experimental results were not able to detect crack location using both methods (Mode Shape Displacement and Slope Change)	[80,81]
Gapped-Smoothing Method (GSM)	E-glass/epoxy laminated beam	Cantilever configuration	PZT (Continuous-sweep sine 140V) Power amplifier	LV (PSV 400 SLV) and PVDF	3 delaminations (Teflon insertion), impact damage and saw-cut	GSM is able to localize damage through LV or a network of PVDF	[82]

3.2.3. Mode Shape Curvature

The mode shape curvature is the second derivative of the mode shapes. It can be calculated from the central difference method as [83,84]:

$${\phi }_{ij}^{\mathrm{”}}=\frac{{\phi }_{i+1}-2{\phi }_{ij}+{\phi }_{i-1}}{h^2}$$

(2)

where φ_ij is the jth mode shape of the measurement point i and h is the distance between points i+1 and i-1.

The first thing usually made is to determine the difference of mode shape curvatures (named as Mode Shape Curvature criterion (MSC)), between intact and damaged structures:

$$MSC=∆\phi "=\left|{\phi }_d^{"}-{\phi }_i^{"}\right|$$

(3)

The largest value will give the location of the damage (Figure 15). As for the mode shapes method, the main disadvantage is the number of sensors needed to obtain the most accurate results.

Damage indicators have been derived from the MSC. The classical thing is to normalize the MSC (NMSC) and then the Normalized Curvature Damage Factor (NCDF) [85]:

$$NMSC={\left[1+\frac{∆{\phi }^{"}}{max\left(∆{\phi }^{"}\right)-min(∆{\phi }^{"})}\right]}^2$$

(4)

$${NCDF}_i=\sum _{j=1}^{n}NMSC$$

(5)

There exist a lot of possible damage indexes such as Modal Curvature Change Rate (MCI) and Curvature Mode Difference (CMD).

Modal curvatures have been used in [86], in order to detect damage in glass/epoxy plates in a cantilever configuration. The modal analysis was performed with 10x5 localization for the PCB impact hammer and a PCB accelerometer was fixed on the end of the plate. The damage was a crack made from a 1mm thick blade, positioned at 10% of the length of the structure close to the fixed end. NCDF detected well the damage position.

In [87,88], the modal shape curvature was tested on a carbon/epoxy beam (241.3x25.4x1.76mm³) in a cantilever configuration, damaged from different sources: an impact, a saw-cut notch and three different types of delamination (Figure 16). The modal analysis was performed using PZT or impact hammer as actuators and 16 PVDF as sensors, all surface-bonded (Figure 17). Both NCDF and the DI were used to localize damage, these methods were efficient for 3 damage scenarios: the saw cut, the impact and the delamination far from the fixed end, but the conclusion that more sensors are necessary for the 2 other damage scenarios was made: the two delaminations closed to the fixed end.

Table 5. Other articles about the experimental use of mode shapes curvature for SHM.

Table 5. Other articles about the experimental use of mode shapes curvature for SHM.

Method	Structure	Fixation	Input	Output	Damage	Description	Ref
MSC	T-stiffened panels of carbon/epoxy	Elastic bands	Shaker (sinusoidal wave (0 – 10 kHz))	LV	Delamination (PTFE Film) Porosity	MSC showed that it can detect delamination accurately, but porosity was not detectable	[79]
MCI and CMD	Carbon/epoxy laminated beam	Cantilever configuration	Modal hammer	Accelerometer	Delamination (Teflon patch)	CMD showed higher sensitivity than MCI to localize damage	[89]
MSC	Carbon/epoxy beams	Suspended on a frame	Shaker (periodic chirp) Impulse hammer	LV	Delamination and crack	The damage was identified efficiently by the MSC	[90]

3.2.4. Modal Strain Energy

Modal Strain Energy (MSE) is the energy that is stored in the structure for each mode of vibration. According to [91,92], the MSE methods are more sensitive to SHM than the natural frequencies and mode shapes. Many articles used this method on offshore platform structures.

In [93], the ways of calculating MSE for a beam and a plate are given. A review of MSE-based methods for SHM is made on [94]. Four groups were introduced: Damage Indexes (DI) such as Stubbs’ DI (SDI) or MSE Decomposition (MSED), Modal Strain Energy Chance (MSEC), Cross-Modal Strain energy (CMSE), and others. In this review, the methods, their calculations and an experimental study using them were presented. Moreover, a lot of other methods derived from MSE can be found in the literature.

Several articles [95,96,97] used the MSE method on several composite structures, for example in [95] on a carbon/epoxy plain-woven laminate (310x222x2.2mm³), the damage was a surface crack (16x1mm²) made from a knife located in the middle of the small dotted area in Figure 18. First, a tensile test was made to determine the mechanical properties of the material. The modal analysis was performed with an impact hammer and an accelerometer fixed on the plate (Figure 18). The total (U_k) and sub-region (delimited by the grid – U_k,ij) strain energies induced by mode shapes for healthy and damaged plates were calculated from plate formulation. From these equations, the fractional energies (F_k,ij) for healthy and damaged (*) plates, and the damage index (β_ij) and its normalization (Z_ij) for each sub-region, were calculated as follows:

$$F_{k,ij}=\frac{U_{k,ij}}{U_k}$$

(6)

$${\beta }_{ij}=\frac{\sum _{k=1}^m{F}_{k,ij}^{*}}{\sum _{k=1}^m{F}_{k,ij}};Z_{ij}=\frac{{\beta }_{ij}-\overline{{\beta }_{ij}}}{{\sigma }_{ij}}$$

(7)

where $\overline{{\beta }_{ij}}$ and ${\sigma }_{ij}$ are the mean and standard deviation. This damage index was able to localize it accurately after using at least 3 modes of vibration, by creating a map of DI on the (x,y) position.

The same helicopter blade [77,98] as with the COMAC method previously discussed (Figure 14), has been used, the “fake” damage was a mass adding. The modal analysis was performed with an impact hammer and shaker, with a burst random signal, the sensors were 55 accelerometers. The method used to calculate strain energy is based on a Euler-Bernoulli beam and used the modal shape curvature. These calculations are similar to the previous one of the plates. The main goal is to calculate a damage index. The modal strain energy methods showed higher sensitivity than COMAC and were able to detect and localize damage, with a high value on a XY map.

Table 6. Other articles about the experimental use of modal strain energy for SHM.

Table 6. Other articles about the experimental use of modal strain energy for SHM.

Method	Structure	Fixation	Input	Output	Damage	Description	Ref
MSEC	Carbon/epoxy laminated plate	Suspended with 2 cotton strings	Impulse hammer	Accelerometer	Surface and penetrated crack	MSEC was able to locate both types of damage	[80]
MSE and Differential Quadrature (DQM)	Carbon/epoxy laminated plate	Suspended with 2 cotton strings	Impulse hammer	Accelerometer	Surface and penetrated crack	For MSE peaks are present around the location of damage, but some also occur at other positions, this is why the DQM is implemented and very good results were obtained with it	[81]
MSEC	E-glass/epoxy laminated beam	Cantilever configuration	PZT (sweep sine 140V)Power amplifier	LV (PSV 400 SLV) and PVDF	3 delaminations (Teflon insertion), impact damage and saw-cut	Strain energy detected well the location of damage, except for special cases with some doubt caused by other high-value	[82]
MSECR	E-glass/epoxy skin and PVC foam for sandwich beams	Cantilever configuration	Electrodynamic shaker	Accelerometer	Debonding + remove face (Teflon sheet)	MSECR correctly located damage for single and multiple damage cases	[99]

3.2.5. Modal Flexibility

The modal flexibility method (MFM or MF) is another kind of damage detection technique, which seems more sensitive than others and needs less experimental data. The modal flexibility is defined as a matrix:

$$\left[F\right]=\sum _{i=1}^n\frac{1}{f_i^2}{\left\{X\right\}}_i{\left\{X\right\}}_i^T$$

(8)

where f_i is the i-th natural frequency and {X}_i is the mass normalized mode shape vectors. What is interesting to observe using MFM is the flexibility difference between undamaged and damaged structures:

$$\left[∆F\right]=\left|\left[F_D\right]-\left[F_U\right]\right|$$

(9)

The highest value of variation will give the location of the damage (Figure 19).

In [100], the MF was also compared to other methods previously seen (Figure 20). In this experiment, the MF seemed the most accurate SHM method.

In [101], the MF and a derivate damage indicator (MDI) have been used on a carbon fiber bi-directional corrugated sandwich panel, with the dimensions 200x200x15mm³. A grid of 9x9 points was created on the surface, in order to obtain the vibration at different locations and localize the damages. They created 2 debonding (1.2% of total area) and 3 core damage (0.3% of total volume of the core) cases, with single and multiple damage cases. The vibration analysis was performed by suspending the plate and with two accelerometers and an impact hammer, an LMS signal analyzer stored all the output and input data of the first four modes of vibration. The MF only was not able to clearly distinguish the damage location, as several peaks with similar amplitude were observable. These peaks also occurred in the MDI, but the peak closed to the damaged area had a higher value, which avoids the fake damage detection of the MF. They concluded that the new MDI is efficient in detecting single and multiple damages in composite structures.

3.2.6. Damping

Damping corresponds to the energy of dissipation of material. In [102], a review of the use of damping for SHM is provided. According to this article, the change in damping has higher sensitivity than natural frequencies or mode shapes to detect damage, but research is still in progress. The simplest analysis of damping is the difference in damping between intact and damaged structures, usually, it will be higher when the structure is damaged and it shows higher sensibility than the change in natural frequency (Figure 21). Damage identification using damping can either be linear or nonlinear.

In modal analysis, the damping ratio is mostly calculated from the -3dB method (or half-power bandwidth) (Figure 23) in the frequency domain, but time domain and time-frequency domain methods are also found in the literature. In the time domain, the most frequent method used is the logarithmic decrement [103] and, in the time-frequency domain, the use of analyzing wavelets such as the Gabor wavelet is necessary [104]. There exist several quantities to define damping, you can find them in Figure 22. Damping to detect damage is also used with a Damage Index and it is often mixed with other methods such as modal shape, natural frequencies, modal strain energy, …

In [62], in addition to compare natural frequency, a comparison of the difference in damping between damaged and intact carbon/epoxy plates was made, in a free-free configuration from impact hammer excitation and two accelerometer sensors. Two damages were studied: one impact damage and one hole (unknown diameter) both in the center of the plate. No specific pattern in the difference in damping ratio was detectable.

In [63,64,65] the effects of damage on loss factors have also been investigated, on the laminated and sandwich composite with delamination and fiber cutting, it was shown that the loss factor increased with damage severity (Figure 24).

In [105,106], a method was developed, using damping mixed with Plane Shape Function (PSF) to localize damage, the name is Damping Damage Indicator (DaDI). It was tested on a woven carbon/phenolic plate damaged from impacts and quasi-static pressure. The structure was suspended with two nylon strings to obtain a free-free configuration. The DaDI needs modal strain shapes and modal damping factors, to localize damage. The excitation was made with an electromagnetic shaker and a multisite excitation force, the sensor was LV at four different points. The PSF for mode n and DaDI at coordinate (i,j) is calculated as:

$$PS{F}_{ij}^n=\frac{\left|x{\epsilon }_{ij}^n\right|}{\mathit{max}\left|x{\epsilon }_{ij}^n\right|}+\frac{\left|y{\epsilon }_{ij}^n\right|}{\mathit{max}\left|y{\epsilon }_{ij}^n\right|}{DaDI}_{ij}=\frac{\sum _{n=1}^N(PS{F}_{ij}^n∆{\eta }_n^D)}{\sum _{n=1}^N\left(PS{F}_{ij}^n\right)}$$

(10)

where $x{\epsilon }_{ij}^n$ and $y{\epsilon }_{ij}^n$ are the strain in both directions, $∆{\mathsf{\eta }}_n^D$ is the difference in damping ratio between damaged and intact structures. In this article, DaDI seems to well determine the location of both damages. Adding mass to the structure with transducers may decrease the accuracy of this method. In [107], similar experiments were made on carbon/epoxy plates. Several indexes have been used, such as Multiparameter Damage Indicator (MuDI) which is created from two others: the DaDI and the frequency damage indicator (FreDI), to localize the damage. A weighting function has been introduced to emphasize the role of FreDI or DaDI in the computation of MuDI, this weighting function showed the importance of the correct localization of damage.

3.2.7. Frequency Response Function Metrics

Comparing the FRF of intact and damaged structures can be a way to detect damage [108]. Several damage indicators based on FRF have been introduced in literature [61] such as the Frequency Domain Assurance Criterion (FDAC) [109] or the Frequency Response Assurance Criterion (FRAC) [110].

In [61,62,111], the metrics introduced were tested on a carbon/epoxy plate suspended in a free-free configuration, the sensors were accelerometers and MFC, and the actuator was an impact hammer. The plates were damaged by a central hole and an impact (dimensions are unknown). The change of natural frequencies was not sufficient to detect damage, but the metrics gave interesting results, it seemed that metrics using more than one sensor gave higher sensitivity.

3.2.8. Transmittance or Transmissibility Function (TF)

TF is a technique to determine the localization of the damage, it is built from the ratio of the vibration response of the structure at two separate locations and the location of the input force. It is often written as $T_{ij}^k$.

$$T_{ij}^k=\frac{x_i^k\left(\omega \right)}{x_j^k\left(\omega \right)}$$

(11)

where x is the displacement at i and j location, where the force is applied on k. TF can also be calculated from the velocity of the structure instead of displacement. Then detecting damage is made with a damage indicator on a frequency range [ω₁; ω₂], which compares the TF for healthy (h) and damaged (d) structures.

$$D_{ij}\left({\omega }_1;{\omega }_2\right)=\frac{{\int }_{{\omega }_1}^{{\omega }_2}\left|T_{ij}^h\left(\omega \right)-T_{ij}^d\left(\omega \right)\right|d\omega }{{\int }_{{\omega }_1}^{{\omega }_2}\left|T_{ij}^h\left(\omega \right)\right|d\omega }$$

(12)

A review of TF is given in [112]. This technique is mostly used numerically, but with a network of sensors or a LV this can be used experimentally. One issue of this method is the selection of the frequency range to obtain accurate results. As we can see in Figure 25, damage on paths 2-3 can be detected with the frequency range [0,39] but not with the range [0,300]. Usually, the upper value of the range is chosen as the first natural frequency of the structure. The position of the input force will also have a strong influence on the result.

To summarize, in the review, they said that the perfect localization of the damage is possible when the damage is located next to the input force localization, or by choosing the correct frequency band, which means the global vibration of the structure has to be known.

In [113], TF has been used on a cantilever fiberglass beam (112x1.18x0.635cm³) with a PZT actuator and accelerometers or PZT patches as sensors (Figure 26). The PZT actuator was excited with a random noise signal coming out of an amplifier. It was shown that using an impact hammer and shaker as actuators did not give nice repeatability of the experimentation, but the PZT actuator gave correct repeatability. Translational and curvature TF have been investigated. The damage was created by bonding a steel plate (5.08x5.08x0.635cm³) to the beam, between path 1-2. The translational and curvature TF showed efficiency in localizing moderate damage, the curvature one seemed more accurate while the frequency range was higher (10 – 20 kHz), and a very high-frequency range did not allow detecting damage and the curvature TF seemed more sensitive for small damage.

In [114], the modal analysis of a wind turbine blade made of fiberglass in a free-free boundary condition has been made. PZT patches have been used with a periodic chirp signal at 200V as an actuator and an LV as the sensor. The damage was introduced as “fake” damage by clamping a steel plate (8.5”x6.2”x0.75”) near one edge, closed to the path 11-15. The damage indicator of the TF indicated the correct location for the “fake” damage (Figure 27), this damage indicator should be near 0 for the healthy structure and healthy paths.

In [115], two models based on transmittance function were created and studied, the first one using only one transmittance model for the healthy state named “single transmittance model” (STM) and the second one using several transmittances models, the “multiple transmittance models” (MTM). Both models were based on a data-based response-only methodology using the autoregressive with exogenous (ARX). The study was using 29 composite beams made of carbon fiber in unidirectional or woven fabrics. The damage was introduced in the middle of two beams with an impact hammer at 5 and 15J. The modal analysis was performed in a cantilever configuration with a shaker as an actuator and two accelerometers. The STM-scheme detected most of the damage, but it gave some false alarms and failed to detect some damage, on the other hand, the MTM-scheme was able to detect all damage cases without false alarms.

3.2.9. Wavelet Transforms

In vibration analysis, the Wavelet Transform (WT) has been used a lot to process the vibration data coming from various sensors. The principle relies on dividing a time-series signal into a set of sub-signals (wavelets) and extracting data from these sub-signals. There exist different kinds of Wavelet such as Daubechies and Symlet. A review of WT is given in [116].

In [117], small delamination (0.11%, 0.167% and 0.22% of the total surface) of glass/epoxy plates (360x250x4mm³) with piezoelectric transducers (1 actuator and 4 sensors) were studied. The signals were decomposed for each sensor into 16 sub-signals with Daubechies Wavelet and a damage indicator was introduced, for intact and damaged structures, based on the energy of these sub-signals, then by setting a threshold of this damage indicator. The conclusion that using a threshold of 20% with WT helps to detect small delamination of an area superior to 0.12% of the total area of the structure. In Table 7, it is possible to see the damage indicator calculated for each sub-wavelets. If one of the sixteen sub-wavelet damage indicators is higher than the threshold, authors considered the delamination detected (grey cells).

4. Conclusions

Glossary

References

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