# Analysis of Dynamic Characteristics of Attached High Rise Risers
## 2 Natural Vibration Characteristics Analysis
### Constraint Stiffness on Vibration Mode Analysis  
It is assumed that the section and material properties are consistent along the length, and only gravity and concentrated mass are considered. The physical parameters of the model are shown in Table 1.  
Table 2 shows the first five natural frequencies of the VNA riser under the two calculation methods, and the results are quite consistent. The calculation result of the finite element method will be easily affected by the number of divided elements. The calculation accuracy of the differential transformation method depends on the number of its expansion terms. The result of the differential transformation method is close to the exact solution in theory [38] and is convenient for parameter analysis.  
In the engineering installation, EPDM rubber will be filled between the annular guide support and VNA riser to reduce vibration and local stress. The diversity of rubber elastic stiffness caused by different processes will significantly affect the change in vibration mode. Therefore, the system vibration mode can be obtained by using the DTM method for different constraint stiffnesses.  
Figure 3a shows that stiffness has an obvious impact on the first mode shape. When the stiffness is \(10^{5}\) N/m, the vibration mode of the cantilever riser pipe still shows characteristics similar to those of a cantilever beam. However, with the gradual increase in the elastic constraint stiffness, the displacement-limited effect of the elastic constraint point becomes obvious, and the arch appears between the constraint points. As seen in Figure 3b, the stiffness has a relatively small impact on the second-order vibration mode. When the stiffness is less than \(10^{7}\) N/m, the vibration mode of the cantilever riser pipe is similar to that of a cantilever beam. Although the vibration mode collapses near the constraint point when the stiffness is equal to \(10^{7}\) N/m, it can still be approximately equal to the vibration mode of the cantilever beam. Especially when the stiffness is equal to \(10^{8}\) N/m, as a consequence of the displacement-limited effect, the second vibration mode of the structure is similar to the fourth vibration mode of low constraint stiffness. That is because the change in constraint increases the system stiffness and then changes the sequence of the system mode. By using this method, the precision instrument at the key position of the high-rise structure can be avoided from producing a large range of vibration due to resonance under second-order frequency excitation. It can be seen from Figure 3c,d that when the stiffness is less than \(10^{6}\) N/m, the vibration mode of the cantilever riser pipe is very close to that of the cantilever beam. When the stiffness is equal to \(10^{8}\) N/m, the vibration mode changes significantly.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Elastic modulus & \(1.4\times 10^{10}\) MPa & Low position constraint stiffness & \(1\times 10^{5}\) N/m \\ Outer diameter & 1.52 m & High position constraint stiffness & \(1\times 10^{5}\) N/m \\ Inner diameter & 1.5 m & Low position constraint elevation & 30 m \\ Riser density & 1856 kg/m\({}^{3}\) & High position constraint elevation & 60 m \\ Additional mass & 2000 kg & riser length & 90 m \\ Additional rotational inertia & 5000 kg-m\({}^{2}\) & Gas density & 1.225 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical parameters of VNA.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\omega_{1}\) **(rad/s)** & \(\omega_{2}\) **(rad/s)** & \(\omega_{3}\) **(rad/s)** & \(\omega_{4}\) **(rad/s)** & \(\omega_{5}\) **(rad/s)** \\ \hline DTM & 2.0108 & 7.6238 & 20.2602 & 38.6768 & 62.5487 \\ FEM & 2.0086 & 7.5970 & 20.1703 & 38.4927 & 62.2990 \\ \hline \hline \end{tabular}
\end{table}
Table 2: First five natural frequencies of VNA.