## 2 Numerical Simulation Method of Riser Vortex Shock Vibration
### Vibration Frequency Analysis  
Figure 15 reveals the power spectral density diagram of the transverse-flow and forward-flow vibration response at the midpoint of the riser under four different flow ratios. The transverse-flow and forward-flow main control frequencies in the figure are summarized in Table 4. The results show that: (1) there is a two-fold relationship between the upstream and cross-flow main control frequencies; (2) with the velocity ratio increasing, the instantaneous velocity of superimposed flow increases, and the main control frequency of vertical cross-flow direction and forward-flow direction increases; (3) under the influence of the cyclic change of the superimposed flow rate, the vibration response of the riser has a wide frequency band, which stimulates the multi-mode vibration of the riser. Under the condition \(\alpha\) = 0.57, the transverse-flow vibration response frequency band is the widest and has multiple peaks. Figure 16 depicts the transverse-flow spectrum diagram of the riser under different flow ratios: when \(\alpha\) = 0.50, the vibration spectrum diagram at the axial sixth equipartition point of the riser has a frequency band of 1.5-3 Hz, in which the main control frequency is 1.98 Hz between the first-order natural frequency (0.99 Hz) and second-order natural frequency (2.34 Hz) of the riser. The reason for the formation of the frequency band may be that the vibration mode of the cross-flow direction of the riser changes as the flow rate changes with time, of which the second-order mode occupies most of the time, and the first-order mode appears. When \(\alpha\) = 0.57, the vibration spectrum diagram at the axial sixth equipartition of the riser has a frequency band of 1-3.5 Hz, in which the main control frequency is 2.52 Hz, slightly larger than the second-order natural frequency 9 (2.34 Hz) of  
Figure 14: Riser cross–flow direction vibration envelope diagram. the riser, which is consistent with the larger proportion of the second-order formation in the cross-flow vibration envelope of the riser. As \(\alpha\) = 0.62, the vibration spectrum diagram at the axial sixth equipartition of the riser has a frequency band of 1.8-3.8 Hz, where the main control frequency is 2.68 Hz, exceeding the second-order natural frequency of the riser (2.34 Hz) and failing to reach the third-order natural frequency of the riser (4.64 Hz), which is consistent with the three-order formation beginning to appear in the cross-flow vibration envelope of the riser. When \(\alpha\) = 0.67, the vibration spectrum diagram at the axial sixth equal equinox of the riser has a frequency band of 1.5-4 Hz, in which the main control frequency is 2.87 Hz, which is still between the second-order natural frequency (2.34 Hz) and the third-order natural frequency (4.64 Hz) of the riser. Figure 17 shows the vibration spectrum diagram of the riser with different flow rate ratios: as \(\alpha\) = 0.50, there are 3-5 Hz frequency bands in the sixth division position of the riser. In addition, the main control frequency is 4.05 Hz, which is close to the third-order natural frequency of the riser (4.64 Hz). When \(\alpha\) = 0.57, there is a 3-6 Hz frequency band in the risers along the flow to the sextile position, whose main control frequency is 4.90 Hz, slightly greater than the third-order natural frequency of the risers (4.64 Hz). As \(\alpha\) = 0.62, there is a 4-6 Hz frequency band in the risers along the flow to the sextant position, and its main control frequency is 5.37 Hz, between the third-order natural frequency (4.64 Hz) and the fourth-order natural frequency (7.67 Hz) of the risers. When \(\alpha\) = 0.67, there is a 4.5-6.5 Hz frequency band in the risers along the flow to the sextile position, with a main control frequency of 5.72 Hz, which is between the third- and fourth-order natural frequency of the risers. Figure 15: Riser vibration response power spectral density diagram.