# Vortex-Induced Vibration of Deep-Sea Mining Pipes: Analysis Using the Slicing Method
## 4 Numerical Simulation and Results
### Impact of Oscillatory Flow on Vortex-Induced Vibration of Deep-Sea Mining Pipes
#### 4.2.3 Vibration Frequency Analysis of Deep-Sea Mining Pipe Vibration Frequency Characteristics  
Figure 12 vividly illustrates that regardless of the variations in oscillation frequencies applied, the frequency characteristics of the cross-flow vibrations of the deep-sea mining pipe remain consistent. This observation underscores that the alterations in the frequency of superimposed oscillatory flows do not significantly impact the inherent vibration frequencies of the mining pipe. Notably, across five specified axial positions along the pipe, the amplitude frequencies are remarkably uniform, lying within a band from 0.5 Hz to 4.0 Hz, where the predominant frequency consistently hovers around 2.7 Hz. The stability of this frequency band can be attributed to the adjustments in the vibration frequency within the cross-flow direction, which correlate directly with changes in the velocity of the oscillating flow. Furthermore, an examination of the amplitude reveals that the axial vibration amplitude of the mining pipe demonstrates a gradual enhancement from the top towards the bottom, illustrating a clear gradient in vibrational intensity along the pipe's length.  
Influence of Relay Bin Weight Variation on Vortex-Induced Vibration Characteristics of Deep-Sea Mining Tubes  
In this section, the vortex-induced vibration characteristics of deep-sea mining tubes under different bottom-end forces W\({}_{\text{b}}\) will be investigated. In this section, the properties of the mining tubes are selected to be the same as those of the mining tubes under the uniform flow in the previous section, and the properties of the intrinsic frequencies of the mining tubes are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**T** & **f1 (Hz)** & **f2 (Hz)** & **f3 (Hz)** & **f4 (Hz)** \\ \hline Wb = 2000 N & 0.47 & 1.42 & 2.06 & 3.04 \\ Wb = 1500 N & 0.39 & 0.01 & 1.52 & 2.15 \\ Wb = 1000 N & 0.30 & 0.72 & 0.07 & 1.50 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Intrinsic frequency of deep-sea mining tubes in Intermediate warehouses of different weights.  
The instantaneous dimensionless displacement envelopes for u = 0.2 m/s, Wb = 1000 N, Wb = 1500 N, and Wb = 2000 N in the transverse flow direction of the deep-sea mining tube are given in Figure 13, respectively.  
At a top tension of Wb = 1000 N, the envelope of vibration displacement in the cross-flow direction of the deep-sea mining pipe predominantly exhibits the second-order mode, with the first and third-order modes appearing less distinctly. When the top tension in  
Figure 12: Frequency amplitude plots of the transverse flow direction of deep-sea mining tubes at different stacking frequencies.  
Figure 13: Envelope plots of instantaneous dimensionless displacements in the transverse flow direction of deep-sea mining tubes with different weight intermediate warehouses.  
creases to WB = 1500 N, the displacement envelope still primarily shows the second-order mode, but it now includes a more prominent third-order mode, resulting in a more complex overall displacement pattern. At Wb = 2000 N, the primary mode of vibration shifts to the first order, with a less pronounced second-order mode also observable. With incremental increases in top tension, T, the dominant vibration mode excited by the cross-flow diminishes in complexity at a constant flow velocity, transitioning from a second-order mode to a more simplified first-order mode as the top tension ranges from WB = 1000 N to Wb = 2000 N at a flow velocity of u = 0.2 m/s. Concurrently, the maximum dis-placement observed at the bottom end of the pipe decreases progressively as the intermediate warehouse weight increases.