## 4 Results and Discussion
### Free Vibration of Finite Pipe  
In this part, the free vibration properties, including the natural frequency and mode shape, of the fluid-conveying pipes with periodically varying cross-section will be examined by using Equations (28) and (29). The substructure length is taken as \(l_{\text{A}}=l_{\text{B}}\), and despite the number of cells, the total length of the pipe is fixed to 3 m in order to examine the effect of number of cells on the natural properties of the present periodic pipe conveying fluid under the same pipe length. Figure 9 presents the first six natural frequencies of the pipe for different numbers of cells. Corresponding to the foregoing study, the periodic models with 4, 7, and 10 cells are also employed for analysis. It can be seen that under the same total length, with increasing the number of cells, the natural frequencies of the pipe are decreased, meaning the rigidity is reduced, and this is obvious especially for higher frequencies. Furthermore, this performance is also obvious for smaller numbers of cells. For the pipes with larger numbers of cells, the natural frequencies almost remain constant. Corresponding to the natural frequencies in Figures 9 and 10 exhibits the first six mode shapes of free vibration for the pipe. To be comparable, the curves in each single subgraph have been plotted in the same dimension of coordinate system, although the dimensions of some subgraphs have been modified for observation. Moreover, it should be noted that the present mode shapes are basically asymptotic since they are quite sensitive to the corresponding natural frequencies, and due to lack of adequate nodes, only the first three modes of the pipe with 4 cells can be plotted. The modes without fluid loading are also displayed. It can be seen that the effect of FSI is evident. Figure 8: Vibration attenuation shapes of a representative fluid-conveying periodic pipe with 7 cells in different BG regions: (**a**) BG1, (**b**) BG2. (the thick and thin curves denote the two segments of a cell.) Blue—\(U=20\) m/s, Red—without fluid loading. Figure 10: Mode shapes of the fluid-conveying periodic pipe with different numbers of cells: (**a**) Mode 1, (**b**) Mode 2, (**c**) Mode 3, (**d**) Mode 4, (**e**) Mode 5, (**f**) Mode 6. Blue—_U_ = 20 m/s, Red—without fluid loading. Figure 9: Natural frequencies of the fluid-conveying periodic pipe with different numbers of cells. To further discuss the effects of FSI and the number of cells on the free vibration of the periodic pipe, Figure 11 displays the evolution of natural frequencies with the flow velocity for different numbers of cells. It is clear that due to the influence of FSI, the first four natural frequencies of the pipes with 4 and 10 cells all demonstrate a decreasing trend with the increase of flow velocity. When the critical flow velocity is reached, at which the first natural frequency vanishes, the static instability occurs and the pipe is subject to buckling. Lower natural frequencies and critical flow velocity of the periodic pipe with 4 and 10 cells are very close, while higher ones will present difference. The third and fourth natural frequencies of the pipe with 10 cells are first lower than those of the pipe with 4 cells, and will become higher than the pipe with 4 cells when the flow velocity is increased to higher values, especially for the fourth natural frequency. This implies that the FSI has a greater effect on the pipes with smaller numbers of cells. In addition, it is worth noting in Figure 11 that when the flow velocity exceeds the critical value, it is necessary to consider other natural frequencies since if the flow velocity continues increasing up to a certain value, the first natural frequency will be present again, and gradually coincide with the second natural frequency, leading to a'mode-coalescence flutter' of the pipe. With increasing the flow velocity, such'mode-coalescence flutter' will occur involving the third and higher frequencies.