technique, reliable source location and damage mechanism characterisation must be ac- complished. A novel approach based on a chirplet atomic decomposition, time-frequency energy distribution and dispersion analysis, where the failure-emitted signals are separated from extraneous noise and the detected modes are analysed according to their dispersive behaviour and angular dependence characteristics is presented in this section. Dispersion relations are obtained by the use of the higher order plate theory proposed in Chapter 3, and then used in conjunction with the previous methodologies for mode identification and localisation. A statistical onset-picker (AIC) is used to estimate the wave arrival and dispersion analysis is then conducted on the decomposed signal in conjunction with quadratic time-frequency analysis so that the frequency content is extracted and the en- ergy orientation examined in order to classify the detected modes of propagation in the recorded signals, and then select a common mode to the sensor network.
By recognizing and identifying the dominant modes of propagation in the received AE waveform then it would be possible to discriminate between damage types. For example, the antisymmetric wave modes with dominant out of plane motion will interact most strongly with damages lying parallel to the plane of the wave propagation such as delamin- ations, skin or core debonding and impact damage. The symmetric wave modes with dom- inant in-plane motion will be more strongly related with damages lying perpendicular to the plane of wave propagation such as matrix cracking, matrix splitting and core crushing [McGugan et al. 2006]. Thus, by carefully analysing the dispersive characteristics of the signals or by the application of advanced pattern recognition methods the identification of the failure modes with different type of damage mechanism could be accomplished [Paipetis and Aggelis 2012, Sause et al. 2012]. The focus on classification of AE events is based on the dispersive energy attributes of the wave packets constituting the waveform.
4.6.1. Improved Atomic Decomposition
A major difficulty in acoustic emission is the analysis of broad-banded signals and the discrimination of the modes contained in the recorded signals. The matching pursuit al- gorithm (MAP) is proposed for the atomic decomposition of AE waveforms focused on the ability of the method to classify between modes based on dispersive energy characteristics. The MAP was introduced by [Mallat and Zhang 1993] and has been successfully applied
in SHM by different researchers. It is an iterative algorithm that decomposes a signal into a linear combination of waveforms, so-called atoms, that are selected from a redundant database of atoms, named dictionary, having similar time and frequency characteristics to the original signal, in our case, the AE waveforms. The atom from the dictionary that locally better defines the signal is then selected for reconstruction. The first step of the algorithm is to create a redundant dictionary D of atoms g which are well localized in time and frequency, and possess unit energy. The second step is to find the best match from the dictionary (where < •,• > symbolises the inner product) in which the residual
r0(t) equals the sensor signal s(t) for the first iteration according to:
gi= argmax
g∈D |< ri−1,g >| . (4.10)
The third step is to compute the residual after subtracting the component along the best atom as shown in Eq. (4.11):
ri= ri−1− < ri−1,gi> gi. (4.11)
Finally, the second and third steps are repeated until a maximum number of iterations
n is met or a predefined energy threshold of the original signal energy is reached. The
signal can be finally reconstructed according to Eq.(4.12) as:
s=
n−1
∑
j=0
< rj,gj> gj+ rn. (4.12)
The proposed dictionary is composed of chirplet atoms which are well suited for the analysis of dispersive signals with no stationary time-frequency behaviour [Raghavan and Cesnik 2007]. The chirplet atom is defined as follows:
gk(t) = 1 π0.25√s k exp −1 2 (t −tk)2 s2k + i ωk(t −tk) +βk 2 (t −tk) 2 , (4.13)
where the controlling parameters sk, tk, ωk, and βk indicate the time extent, time centre,
angular frequency centre, and the linear frequency modulation rate, respectively. An example of chirplet atoms with different parameter values is shown in Figure 4.7.
In case of acoustic emission stress waves, the dictionary can be designed with knowledge regarding the spectral characteristics of the expected AE signals. To achieve maximum resolution in time shift, the time translations tk are selected depending on the sampling
interval. The angular frequency ωk should lie in between the frequency range of the anti- aliasing filters. The parameters sk and βk are optimized by finding the optimal values
that lead to a better match in the neighbourhood of the initial set of parameters. This strategy significantly improves the resolution of the decomposition without increasing the size of the dictionary. In this context, a small dictionary refers to a coarse discretization
0 0.05 0.1 0.15 0.2 -0.1 0 0.1 0 0.05 0.1 0.15 0.2 -0.2 0 0.2 0 0.05 0.1 0.15 0.2 -0.1 0 0.1 Atoms Amplitude [a.u.] 0 0.05 0.1 0.15 0.2 -0.2 0 0.2 Time [ms]
Figure 4.7. Example of different chirplet signals with different controlling parameters.
step of the parameters controlling the chirplet atoms contained in it. It is well-known that a smaller discretization interval of the control parameters produces a large number of functions which usually provide a better decomposition in terms of matching the signal. Nevertheless, as the size of the dictionary increases the computational effort also increases. Therefore, the goal of the proposed numerical implementation is to improve the decom- position performance without increasing the size of the dictionary. The dictionary can be further improved by taking into consideration the effects of dispersion in order to count with a dictionary of signals that will reflect the real wave propagation phenomenon. This can be done by means of spectral analysis since once the signal has been characterised, i.e. once the different modes have been calculated with the model proposed in Chapter 3 (wavenumbers, attenuation, excitability, etc.), the propagation and reconstruction of the signal becomes fairly easy. A general diagram depicting the different factors affecting a received time-domain acoustic emission wave once the propagation and attenuation of waves through the structure have taken place is depicted in Figure 4.8.
In essence, the input chirplet atom gk(t) is transformed to its spectrum Gk(ω) and the
Transducer Frequency Response Spatial Aperture Source Characteristics Frequency Content Source Orientation
Wave Propagation Model Phase velocity Group velocity Attenuation Excitability ω/k c k Re gr Im Noise Acoustic Noise Electronics Received AE Waveform
transformed solution is expressed at each frequency ωn and some position r in space as
follows:
uk(ωn) = P(ωn)Gk(ωn), (4.14)
whereP(ωn) is the analytically known transfer function of the problem. It is good to bear
in mind that the loop for PGk must be evaluated only up to the Nyquist frequency and
the remaining part is obtained by imposing that it must be the complex conjugate of the initial part, ensuring that the reconstructed time history is real only [Doyle 1997]. A flow diagram of the complete procedure for waveform reconstruction is presented in Figure 4.9. For the case discussed here, free wave propagation will be considered, i.e. no effects of boundaries are taken into account for the wave propagation problem. In other words, the propagation of a guided wave through an uninterrupted structure is analysed. Once the amplitude of a mode in a particular direction is known, it is straightforward to simulate its propagation. Taking the previous statements into consideration, the transfer function
P(ωn) can be defined as follows:
P(ω) =
∑
All modes
E(ω)√1 re
−ikRe(ω)re−ikIm(ω)r, (4.15)
where E(ω) is the modal excitability at the AE source as explained in Section 4.2, the
second term of the equation represents the beam spreading effect, the first exponential describes the propagation of the wave and the second exponential represents the material attenuation both given as exponential decays in signal amplitude with distance.
The dispersion knowledge gained with the proposed plate theory in combination with spectral analysis in the frequency domain and the understanding of the excitability func- tions can help in the development of a dictionary with optimal acoustic emission signals. However, if the dictionary is built with arbitrary signals with no relation to the physics of underlying signals, then the interpretations gained with the decomposition could lead to incorrect inferences for mode identification.
Do Loop at each Frequency uk( )ωn=P( ) ( )ωnGkωn Time Function: Chirplet Atom g tk( ) Fast Fourier Transform g tk( ) Gk( )ωn
Inverse Fast Fourier Transform
uk( )ωn u(t)
Time Function
u(t)
Figure 4.9. Flow diagram for waveform reconstruction: From input waveform to propagated,
0 0.75 1.5 2.25 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Frequency×Thickness [MHz×mm] Group V elocity [ ] km/s Ao So SHo
Figure 4.10. Fundamental modes of propagation: Group velocity dispersion curves.
4.6.2. Time-Frequency Analysis for Mode Identification
Once the matching pursuit decomposition has been accomplished, an energy distribution can be defined in the time-frequency plane without the interference terms obtained with conventional time-frequency representations (TFRs) [Mallat and Zhang 1993, Raghavan and Cesnik 2007]. As a result, this technique will provide a clearer picture of the energy distribution with well defined clusters of concentrated energy in comparison to traditional smoothed representations. Moreover, since the parameters of every matched atom are known, no special post-processing is required for the analysis. As shown in the previous chapter, the group velocityCgr is related to the velocity with which the envelope of a wave
packet propagates and it is equal to the energy velocity [Auld 1990]. Then, by analysing the dispersive characteristics of the recorded modes, it would be possible to identify them. The ability to measure Lamb mode dispersion from time-frequency analysis from acoustic waveforms is of great importance in this context [Prosser et al. 1999, Fucai et al. 2009]. It is known that the low frequency range is the most used in Lamb wave applications for structural health monitoring where just the fundamental S0 and A0modes of propagation are present and the influence of higher order modes of propagation is avoided in order to facilitate the analysis of the recorded signals. Figure 4.10 shows the group velocity dispersion curves for the case of a 1.5mm thick GFRP plate with material properties listed in Table 7.1 for the fundamental modes of propagation.
It can be seen that the behaviour of the SH0 and S0 modes is different from the A0 mode in both the low and high frequency zones. In the relatively low frequency range it can be seen that the higher the frequency of the A0 mode, the faster its group velocity. In an opposite manner for the S0 mode, the higher its frequency, the slower its group velocity. These energy characteristics of the modes are analysed in order to distinguish the recorded modes in a defined frequency range where their characteristics are noticeable, e.g. in the
Figure 4.11. Normalised TFR from Wigner-Ville distribution used in order to depict the energy
orientation of the fundamental modes of propagation.
relatively low frequency range the slopes of the dispersion curves are used to differentiate wave modes. From Figure 4.10 can be inferred that a wave package of the A0 mode at a relatively higher frequency arrives earlier than one at a lower frequency, whereas the situation for the S0 wave mode is the opposite of that of the A0 mode. From these observations made regarding the group velocity distribution, it can be concluded that atoms positive slope in the time-frequency representation are related to the S0mode, and conversely, atoms with a negative slope are related to the A0 mode. This effect of energy orientation is shown in Figure 4.11 for the AE signal depicted in Figure 4.6 where the high intensity of a point in the image represents a high amplitude in time and frequency. Once the onset time of the recorded signals is estimated, the atom with same time arrival characteristics is extracted and analysed for classification according to the interpretation presented above. As it is conventional in AE literature, special attention was not only paid to the A0 mode since the interest is placed for the analysis of the different fundamental modes of propagation contained in the signals and their correlation with possible damage mechanisms.