# Structural Health Monitoring of Walking Dragline Excavator Using Acoustic Emission
## 2 Materials and Methods
### AE Waveguide Characterization  
In the previous section, the source of AE is characterized. However, to detect AE signal impulses effectively against a background of noise, it is necessary to assess how AE parameters vary during propagation along the waveguide. Traditionally, the investigation of the waveguide of the testing objects is carried out mainly empirically by measuring the Hsu-Nielsen lead break response at various points of the structure. In this study an analytical approach is implemented due to limited access to the dragline constructive elements. Load-bearing metal structures of dragline such as back legs and booms represent hollow tubes with a diameter of 20 inches which could be considered as a single-layer cylindrical waveguide. Ordinary form of the waveguide allows to calculate AE signal propagation analytically using modal analysis [20]. \begin{table}
\begin{tabular}{c c c} \hline \hline _D, 1/mm_ & \(\delta\)_D, \%_ & \(R^{2}\) \\ \hline
137.5 & 0.93 & 0.97 \\
149.6 & 0.87 & 0.96 \\
144.9 & 1.4 & 0.92 \\
152.3 & 0.82 & 0.98 \\
151.4 & 0.93 & 0.96 \\
155.3 & 1.4 & 0.99 \\
165.1 & 3.1 & 0.95 \\
160.5 & 1.5 & 0.94 \\
153.2 & 1.3 & 0.98 \\
141.3 & 2.1 & 0.97 \\
144.9 & 1.4 & 0.92 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: AE emissivity estimation—the Palmer–Heald model parameters. Figure 5: Amplitudes of AE impulses: (**a**) AE impulse amplitudes \(A\) and loading force \(P\) vs. time \(t\); (**b**) AE impulse amplitudes distribution. Modal analysis is a common method for the waveguide calculating based on the properties of orthogonality and completeness of normal waves. It allows to calculate in Fourier space the displacement of the testing structure at the point \(z\) caused by the action of a point source with the waveform \(u(0,t)\). The result signal is determined using the expression  
\[u(z,f)=u(0,f)\cdot a_{p}(z,f) \tag{2}\]  
where \(u(0,t)\) is the signal emitted by the AE source located in the point \(z=0\), and \(u(0,f)=F[u(0,t)]\) is its Fourier transform. It was shown in [21] that when the external force is a point source, the coefficients \(a_{p}(z)\) can be determined using the simplified formula:  
\[a_{p}(z)=\mathrm{e}^{\mathrm{i}\mathrm{j}\eta z}/4 \tag{3}\]  
where \(k_{p}=(2\pi f)/c_{ph}(f)\) is the wave number; \(c_{ph}(f)\) is the phase velocity of the normal wave, \(f\)--frequency. In the case of a cylindrical waveguide with a large radius of curvature, Lamb waves can be considered as the main type of normal waves. One of the main aspects related to the influence of the waveguide is attenuation. The attenuation effect is a difficult problem from the point of view of interpreting the AE data, since different wave modes and different frequency components are characterized by different attenuation coefficients, which has a significant effect on the waveform of AE impulse. The attenuation coefficients of the Lamb waves modes \(\gamma_{S0}\) and \(\gamma_{A0}\) are a linear combination of the attenuation coefficients of the longitudinal wave \(\alpha\) and transverse wave \(\beta\):  
\[\left\{ \begin{array}{rcl}\gamma_{S0}&=&A_{S0}\alpha\ +\ B_{S0}\beta\\ \gamma_{A0}&=&A_{A0}\alpha\ +\ B_{A0}\beta\end{array} \right. \tag{4}\]  
where \(A_{S0}\), \(B_{S0}\), \(A_{A0}\), \(B_{A0}\)--coefficients that are determined by the type of Lamb wave, relative plate thickness, and Poisson's ratio. Analytical expressions of the coefficients are given in [22], their values depend on frequency \(f\) and thickness of the testing structure \(h\). The analytical method based on the modal analysis, supplemented by analytical consideration of the attenuation coefficient and the impulse response of AE sensor, allows simulating AE signals of a realistic form [23]. Figure 6 represents the calculated signal (Figure 6a) and experimental signal obtained using the Hsu-Nielsen simulator (Figure 6b) for wall thickness 20 mm at 4 m from the AE sensor.