# Analysis of Dynamic Characteristics of Attached High Rise Risers
## 2 Natural Vibration Characteristics Analysis
### Outer Diameter Function on Vibration Mode Analysis  
The slender and high-rise structure of the VNA system is made up of multi-segment fiberglass pipes, which are assembled together by flanges. The inevitable installation deviations would cause the outer diameter of the VNA system to not have a constant value. Therefore, it is necessary to discuss the impact of different outer diameter functions on system vibration modes and natural frequencies.  
When the outer diameter is constant, the calculation parameters refer to the values in Table 1. In order to ensure the premise of similar costs, it should be noted that different outer diameter functions are designed to make the total weight of the structure equal. The first and second outer diameter functions can be obtained as follows:  
\[D_{out}(x)=-0.000445x+3.06 \tag{12}\]  
\[D_{out}(x)=3.02+0.00133x-0.0000148x^{2} \tag{13}\]  
where \(x\) is the coordinate along the length of the pipeline. Because the outer diameter function is no longer a constant value, the VNA system becomes a variable cross-section riser, and the bending stiffness and mass per unit length also change with \(x\). That makes it difficult to solve the fundamental frequency and mode of vibration of the structure theoretically. Using piecewise linear processing will make the boundary conditions multiply and the dimensions of the solution matrix increase sharply, consuming too much time and energy. Using the finite element method can solve the natural vibration characteristics of a VNA system with a higher-order polynomial outer diameter, but it is not conducive to carrying out parametric analysis. Therefore, the DTM method introduced in this chapter would be a beneficial and efficient way to solve this kind of problem. Through calculation, the first four natural frequencies of the outer diameter under different conditions can be obtained, as shown in Table 4.  
From Table 4, it can be seen that changes in the outer diameter function have a certain impact on the system structure's fundamental frequency and a significant impact on its first-order frequency. Taking the constant value diameter as a benchmark, the first-order natural frequency of the VNA system increases by 18.7% when the outer diameter is linear and increases by 5.2% when the outer diameter is quadratic. The frequencies of other orders change slightly, all below 5%.  
Figure 5a indicates the difference in the first-order mode shapes under the conditions of each outer diameter function is not significant, but in the second, third, and fourth-order  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**External Diameter** & \(f_{1}\) **(Hz)** & \(f_{2}\) **(Hz)** & \(f_{3}\) **(Hz)** & \(f_{4}\) **(Hz)** \\ \hline Constant value & 0.3200 & 1.2140 & 3.2245 & 6.1556 \\ Linear function & 0.3798 & 1.2645 & 3.2312 & 6.1269 \\ Quadratic function & 0.3367 & 1.1608 & 3.0977 & 5.9339 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The first four natural frequencies of VNA system with different outer diameter functions.  
mode shapes, the outer diameter of the quadratic function shows significant differences in mode shape, with wave peaks and valleys moving upward and the curve near the root changing more smoothly. In addition, the vibration modes of the linear function outer diameter and the constant outer diameter are relatively close. The analysis of the influence of different outer diameter functions on the natural vibration characteristics of structures has important theoretical guiding significance for structural design optimization and seismic response spectrum calculation.