## 4 Results and Discussion
### BG of an Infinite Pipe
In addition, it can be seen in Figures 4 and 5 that under the considered pipe period, the representative BG evolutions have been fully displayed, and the designed stepped pipe can also be readily fulfilled in fabrication. Therefore, the model proposed is practicable. Figure 3: Imaginary part of the present fluid-conveying pipe with periodically varying cross-section. The FSI effect on the BG is further discussed in Figure 6. It is noted that the impact of flow velocity is analyzed herein since it is the chief parameter reflecting the FSI effect of fluid-conveying structures. It is observed that the location and width of the BG regions are hardly changed at lower flow velocities, and only higher flow velocities can lead to evident movement of the BG regions, revealing that the FSI has significant effect on the BG only at higher flow velocities. In addition, the region of BG1 has a special evolution. When the flow velocity is ranged over 113 m/s-122 m/s, BG1 exhibits divergence. If the flow velocity continues increasing from 122 m/s, BG1 will cover the frequency range from 0, and the pipe presents a complete ultra-low frequency BG. Thanks to the presence of such complete ultra-low frequency BG in the PC pipe conveying fluid, the incident waves with frequency of 0-50 Hz cannot propagate along the slender pipe, meaning the  
Figure 4: BGs as function of the ratio of substructure outer radius. Figure 5: BGs as function of the ratio of substructure length. vibration within this frequency range will be effectively suppressed. This performance of the system will facilitate to expand the application of PC pipes conveying fluid in ultra-low frequency conditions. However, such characteristic will vanish when the flow velocity exceeds 193 m/s. In order to validate the vibration suppression effect of the periodic pipe, the frequency response and vibration attenuation shape of a finite pipe are further presented in terms of Equation (26). Applying a periodic excitation of \(M_{\text{L}}=1\text{e}^{\text{i}\omega t}\text{N}\cdot\text{m}\) (others = 0) at the left end of the pipe, we can obtain the logarithmic relationship between the amplitude of the right end and the incident frequency, as shown in Figure 7, in which the periodic pipes with 4, 7 and 10 cells, as an example, are calculated for comparison. It is clear that within the specified frequency range, there are four BG regions, which just correspond to those in Figure 3. Under the parameter values used, the vibration is attenuated better in BG2 and BG4. With increasing the number of cells, the frequency ranges of BGs of various pipes are almost constant, indicating that the BGs of the periodic pipes are independent of the number of cells, while the vibration attenuation is getting greater. This is just the contribution of periodicity of the structure. Figure 6: BGs as function of the flow velocity. Figure 7: Frequency responses of the fluid-conveying periodic pipe with different numbers of cells. By exerting the above excitation and selecting different excitation frequencies inside and outside the first two BG regions as examples, the response shapes of a representative pipe with 7 cells are illustrated in Figure 8. Obviously, the vibration of the pipe has been highly suppressed by the periodic structure when the incident frequency falls in the BG regions. There is hardly any vibration occurring for the frequency in BG2. Even in BG1, as shown in Figure 8a, there is still some response present near the left end of the pipe, it is not able to transmit along the pipe. This confirms the vibration suppression effect of the present periodic pipe conveying fluid. Moreover, the attenuation shapes of the pipe without fluid loading are also shown in the figure for comparison. A significant effect of FSI can be observed.