## 3 Results and Discussion
### Natural Vibration Frequency in Water  
The water depth (0.6 m), top tension force (5 N/65 N), end condition (S-S/S-F) and riser material (Al/PMMA/UPVC) would affect the natural vibration frequency of the riser. Beating the riser with equal force and equal time intervals, the time-domain strain response can be obtained (Figure 3).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Materials** & **Top Tension Force (N)** & **End Conditions** & **Velocities (m/s)** \\ \hline Al/PMMA/UPVC & 5/65 & S-S/S-F & 0.2/0.6/1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Values for each parameter used in the experiment.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **Parameters** & **No.** & **Parameters** & **No.** & **Parameters** \\ \hline
1 & Al, 5 N, S-S, 0.2 m/s & 2 & Al, 5 N, S-S, 0.6 m/s & 3 & Al, 5 N, S-S, 1.0 m/s \\
4 & Al, 5 N, S-F, 0.2 m/s & 5 & Al, 5 N, S-F, 0.6 m/s & 6 & Al, 5 N, S-F, 1.0 m/s \\
7 & Al, 65 N, S-S, 0.2 m/s & 8 & Al, 65 N, S-S, 0.6 m/s & 9 & Al, 65 N, S-S, 1.0 m/s \\
10 & Al, 65 N, S-F, 0.2 m/s & 11 & Al, 65 N, S-F, 0.6 m/s & 12 & Al, 65 N, S-F, 1.0 m/s \\
13 & PMMA, 5 N, S-S, 0.2 m/s & 14 & PMMA, S, S-S, 0.6 m/s & 15 & PMMA, 5 N, S-S, 1.0 m/s \\
16 & PMMA, 5 N, S-F, 0.2 m/s & 17 & PMMA, S, N-S, 0.6 m/s & 18 & PMMA, 5 N, S-F, 1.0 m/s \\
19 & PMMA, 65 N, S-S, 0.2 m/s & 20 & PMMA, 65 N, S-S, 0.6 m/s & 21 & PMMA, 65 N, S-S, 1.0 m/s \\
22 & PMMA, 65 N, S-F, 0.2 m/s & 23 & PMMA, 65 N, S-F, 0.6 m/s & 24 & PMMA, 65 N, S-F, 1.0 m/s \\
25 & UPVC, 5 N, S-S, 0.2 m/s & 26 & UPVC, 5 N, S-S, 0.6 m/s & 27 & UPVC, 5 N, S-S, 1.0 m/s \\
28 & UPVC, 5 N, S-F, 0.2 m/s & 29 & UPVC, 5 N, S-F, 0.6 m/s & 30 & UPVC, 5 N, S-F, 1.0 m/s \\
31 & UPVC, 65 N, S-S, 0.2 m/s & 32 & UPVC, 65 N, S-S, 0.6 m/s & 33 & UPVC, 65 N, S-S, 1.0 m/s \\
34 & UPVC, 65 N, S-F, 0.2 m/s & 35 & UPVC, 65 N, S-F, 0.6 m/s & 36 & UPVC, 65 N, S-F, 1.0 m/s \\ \hline \hline \end{tabular}
\end{table}
Table 3: Experiment cases.  
After that, these signals are treated using Fourier transform to obtain the riser's natural vibration frequency (Table 4). As mentioned in Section 2.2, there are 12 strain gauges for each riser model, and in this step, the strain responses under beating from all these strain gauges are similar, except for the amplitude, since no flow velocity is involved. In this paper, the results from the strain gauges at z = 0.5 m are selected because the vibration in this location is the most obvious.  
From Table 4 it can be seen that the S-F end condition would lead to a larger natural frequency compared with the S-S end condition when the material and top tension force are same; when the material and end condition are set, the Al riser has the biggest natural vibration frequency, followed by the PMMA riser and the UPVC riser, and the natural vibration frequency rises with the increase in the top tension force when the material and end condition are fixed. In sum, there is a positive correlation between the riser's properties (modulus of the riser, end condition, and top tension force) and the natural vibration frequency; i.e., larger modulus, more constraints and bigger top tension force will lead to higher natural vibration frequency.