# Analysis of Dynamic Characteristics of Attached High Rise Risers
## 2 Natural Vibration Characteristics Analysis
### Constraint Arrangement Analysis  
In the actual project, the elastic constraint simulates the vertical guide support of the VNA system. Multiple guide supports, to some extent, limit the horizontal vibration of the system. Therefore, the layout of the guide supports has an important impact on the VNA system. The qualitative impact of different layout schemes on the natural vibration characteristics of the system is discussed below. In order to highlight the impact of layout, the constraint elastic stiffness is selected at 10\({}^{8}\) N/m. The layout schemes are as follows:  
\(\rightarrow\) Layout 1: Low constraint elevation is 30 m and high constraint elevation is 60 m, \(\rightarrow\) Layout 2: Low constraint elevation is 15 m and high constraint elevation is 75 m, \(\rightarrow\) Layout 3: Low constraint elevation is 40 m and high constraint elevation is 50 m, \(\rightarrow\) Layout 4: Low constraint elevation is 15 m and high constraint elevation is 30 m, \(\rightarrow\) Layout 5: Low constraint elevation is 60 m and high constraint elevation is 75 m.  
Layout 1, Layout 2, and Layout 3 are symmetrical arrangements about the midpoint of the pipeline, while Layout 4 and Layout 5 are respectively low at the root and high at the top. Refer to Table 1 for other physical parameters. The first four-order natural frequencies and vibration mode function curves under different layout schemes can be obtained through the DTM method, as shown in Table 3 and Figure 4.  
Figure 3: Modal shapes of VNA system with different constraint stiffness. (**a**) First mode shapes; (**b**) Second order mode shapes; (**c**) Third order mode shapes; (**d**) Fourth order mode shapes. When the symmetrical layout (Layout 1 to 3) is further away from the midpoint, the higher the first-order frequency of the structure. But in the higher order modes, the frequencies of those symmetric arrangements farther away from and closer to the midpoint are approximate, and those relatively moderate symmetrical arrangements show significant increases or decreases in higher order frequencies. The analysis of structural natural frequency in Layouts 1, 4, and 5 indicates that the closer the two constraints are arranged at the top, the higher the first-order natural frequency of the structure. When the two constraints are simultaneously located in the middle (Layout 1), the natural frequencies are larger compared with the other two arrangements at the second and third order frequencies, but smaller at the fourth order. Figure 4: Modal shapes of VNA system with different restricted location. (**a**) First mode shapes; (**b**) Second order mode shapes; (**c**) Third order mode shapes; (**d**) Fourth order mode shapes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(f_{1}\) (Hz) & \(f_{2}\) (Hz) & \(f_{3}\) (Hz) & \(f_{4}\) (Hz) \\ \hline Layout 1 & 0.9249 & 5.8070 & 7.4536 & 8.7769 \\ Layout 2 & 1.4401 & 3.0100 & 6.0971 & 11.303 \\ Layout 3 & 0.6700 & 4.3996 & 5.0971 & 11.987 \\ Layout 4 & 0.3239 & 2.1522 & 6.0335 & 11.529 \\ Layout 5 & 2.4147 & 2.4887 & 6.5410 & 12.259 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The first four natural frequencies of VNA system with different bracket arrangement. In Figure 4a, the first mode of vibration exhibits an arcing phenomenon between constraints due to the limiting effect of constraints. At the same time, the way of layout 3 is arranged symmetrically near the midpoint causes the high elastic constraint point to move downward; thus, the first, second, and third mode shapes of the structure all exhibit a vibration pattern characteristic of half being relatively stationary and half being substantially oscillating. For the restriction of large stiffness and the different arrangement schemes, the original system vibration modes of each order are changed due to the constraint force, and there are vibration mode curves similar to simple supported beams between the constraints.