## 4 Research on Vortex-Induced Vibration Response of Riser under Different Oscillation Frequencies
### Vibration Displacement Analysis  
By observing the time-history variation of riser dimensionless displacements at different oscillation frequencies in Figure 19, it can be found that with the increase of the oscillation frequency of the superimposed flow, the amplitude of the non-dimensional displacement of the transverse-flow and the forward-flow gradually decreases. The dimensionless displacement of the riser in the transverse-flow direction and the forward-flow direction presents a relatively regular periodic change, and the period is approximately equal to the period of the superposed flow change. That is, the amplitude modulation phenomenon occurs. At the same time, it can be obviously observed that there is a delay in the peak value of the vertical tube's dimensionless displacement in relation to the maximum velocity. The reason is that the leakage vortex and fluid resistance do not change immediately when the velocity changes, so there is a \"delay effect\" between the vibration amplitude of the vertical tube and the velocity change.  
From the three working conditions of f = 0.10f1, f = 0.25f1 and f = 0.35f1, the transverse-flow to dimensionless vibration displacement of the riser under one amplitude modulation period has been taken, respectively, as Figure 20 shows.  
We can see clearly from the figure above that the dimensionless displacement of the riser goes through the \"build up\" phase, the \"lock up\" phase and the \"decay\" phase in one amplitude modulation period. When f = 0.10f1, the risers \"establish\", \"lock\" and \"decline\" phases account for 40%, 30% and 40%, respectively. When f = 0.25f1, the proportion of the \"locked\" stage is the largest, reaching 50%. When f = 0.35f1, with the period of the oscillation decreasing, the amplitude modulation period of the transverse-flow to the dimensionless displacement of the riser also decreases, and the conversion process of the three stages is compressed, with the proportion of the \"locked\" stage of the riser reducing to 25%. The root mean square values of the transverse-flow displacement of the riser to dimensionless under different oscillation frequencies are shown in Figure 21.  
Figure 18: Overlay flow velocity time-history change plot (different oscillation frequencies).  
Figure 19: Time \(-\)history change plot of dimensionless displacement of riser.  
Figure 20: Displacement development process diagram in riser amplitude modulation period.  
Figure 21: Root mean square value of dimensionless displacement in the cross-flow direction of the riser.  
It can be seen from the figure above that under the action of superimposed flows of different oscillation frequencies, the RMS value of the transverse-flow to dimensionless displacement of the riser in the deceleration stage is greater than that in the acceleration stage. To put it in other way, there exists hysteresis. This is because the riser speed accelerates and decelerates periodically under the action of superimposed flow. As the velocity changes, the drain vortex and the fluid lift force do not change immediately, so a hysteresis follows between the vibration amplitude and the velocity change of the riser.