## 4 Pipeline System Modeling
### Identification Using Modal Macro Strain Method  
Modal macro strain is calculated using the rotational displacements which are perpendicular to the radial direction measured by installed distributed angular displacement sensors, and then power spectral density is acquired via Fast Fourier Transform (FFT). By extracting the peak values, the modal macro strain relative values (normalized modal macro strain) are achieved. Grange length is chosen to be 0.1 m, and eight groups of macro strain dynamic responses are captured from eight angle direction (0\({}^{\circ}\), 45\({}^{\circ}\), 90\({}^{\circ}\), 135\({}^{\circ}\), 180\({}^{\circ}\), 225\({}^{\circ}\), 270\({}^{\circ}\), 315\({}^{\circ}\)). Figure 8 shows the power spectral density of the dynamic strain response of the intact structure under the excitation of random force 1, and the power spectral density of macro strain dynamic responses is extracted from four groups of signals (angle 0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), 270\({}^{\circ}\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Frequency Order** & **D0 (Hz)** & **D1 (Hz)** & **D2 (Hz)** & **D3 (Hz)** \\ \hline
1 st order (Lateral Bending) & 233.55 & 232.85 & 232.13 & 230.79 \\
2 nd order (Vertical Bending) & 315.62 & 312.67 & 312.64 & 310.11 \\
3 rd order (Torsion) & 824.38 & 822.90 & 822.48 & 820.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Structural frequency. Figure 8: Power spectrum density of intact structure (Rand1). (**a**) Angle 0\({}^{\circ}\). (**b**) Angle 90\({}^{\circ}\). (**c**) Angle 180. (**d**) Angle 270. As shown in Figures 9-12, modal macro strain vectors are extracted from eight-angles of each section, and 10 long gauges in the longitudinal direction of the pipeline are acquired. The reference macro strain location is \(Z=0\) m.  
From Figure 9, we know that for the case of D0, two supports' locations are well identified, and for angle = 90\({}^{\circ}\), the MMS shows a hump compared with other cases, and the curves at 0.42-0.48 m and 0.52-0.58 m keep steady. From Figure 10, we can see for the case of D1 (Location: 0.42-0.48 m), two supports' locations are well identified, and for angle = 90\({}^{\circ}\), the MMS shows a hump compared with other cases, and for angle = 0\({}^{\circ}\) and angle = 180\({}^{\circ}\) (the edges of damage areas), the MMS of the location 0.42-0.48 m shows a hump compared with other cases, but for the angle = 270\({}^{\circ}\) (the center of damage area), the MMS of the location 0.52-0.58 m shows a hump, this might be the result of an abrupt change of relative stiffness. Figure 10: Modal macro strain (Rand1, D1). Figure 9: Modal macro strain (Rand1, D0). From Figure 11, we can see for the case of D2 (Location: 0.52-0.58 m), two supports' locations are well identified, and for angle = 90\({}^{\circ}\), the MMS shows a hump compared with other cases, and for angle = 0\({}^{\circ}\) and angle = 180\({}^{\circ}\) (the edges of damage areas), the MMS of the location 0.52-0.58 m shows a hump compared with other cases, but for the angle = 2700\({}^{\circ}\) (the center of damage area), the MMS of the location 0.42-0.48 m shows a hump, and this also might be as a result of abrupt change of relative stiffness. From Figure 12, we can see for the case of D3 (Location: 0.42-0.48 m + 0.52-0.58 m), two supports' locations are well identified, and for angle = 90\({}^{\circ}\), the MMS shows a hump compared with other cases, and almost for all the angles, the MMSs of the location 0.42-0.48 m and 0.52-0.58 m are evident, especially for the case when angle = 2700\({}^{\circ}\). Figure 13 indicates the normalized modal macro strain distribution of 10 long gauges under rand1 excitation. Figure 11: Modal macro strain (Rand1, D2). Figure 12: Modal macro strain (Rand1, D3).