Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019
If the time history of the span mid-point which considers the gravity effect (i.e. the black curve) is translated upwards by 9.7930, Figure 5 is plotted. Observation indicates that, for horizontal spans, the gravity effect on the vibration amplitude is very subtle. When gravity is ignored, the maximum displacement-to-diameter ratio is 0.6029; and when gravity is considered, the maximum displacement-to-diameter ratio is 0.6043. The results are very close. The spectral analysis shown in Figure 6 (where PSD refers to power spectral density) also proves that gravity does not affect vibration frequency of the horizontal free-spanning pipeline.
g = 9.8 m/s2 g = 0
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
15 20 25
Dimensionless time t U = 0.5, V = 0.03
Figure 5: Comparison of the vibration amplitude between horizontal free spans considering gravity and ignoring gravity (
t[15, 30]).
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0 2 γ = 0 1 30 -2 -4 -6 -8 -10 -12 15.0 15.5 16.0 16.5
Dimensionless time t U = 0.5, V = 0.03
17.0 (b)
(a) Dimensionless t[0, 30] (b) Dimensionless t[15, 17]
4 6 8
Dimensionless frequency f U = 0.5, V = 0.03
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0 2 γ = 0 1 10 (a)
Figure 7: Cross-flow time history of the mid-point of a free span with a 45° slope.
4 6 8
Dimensionless frequency f U = 0.5, V = 0.03
(a) g = 9.8 m/s2 (b) g = 0
Figure 6: Frequency analysis of the horizontal free span. A-328 10 (b)
If the time history of the span mid-point which considers the gravity effect (i.e. the black curve) is translated upwards by 6.8506, Figure 8 is plotted. It can be observed that, for inclined free spans, the gravity effect will change the vibration amplitude of the system. When gravity is ignored, the maximum displacement-to-diameter ratio is 0.6029; and when gravity is considered, the maximum displacement-to-diameter ratio is 0.5823, and the difference is very distinct. The FFT analysis shown in Figure 9 displays that gravity does not affect vibration frequency of the inclined free-spanning pipeline. However, it has to be admitted that the difference in the vibration amplitude cannot be ignored, thus if the free-spanning pipeline system is over seabed slope, the inclination of the free-spanning pipeline has to be considered when predicting the dynamic behaviors of the system.
©2019: The Royal Institution of Naval Architects
The same comparison is also done for a free span over a 45° slope. The time history results of the mid-point of an inclined free-spanning pipeline is calculated with dimensionless U = 0.5, V = 0.03, as shown in Figure 7. It is found that when the gravity terms are ignored, i.e. g = 0, the vibration center of the span mid-point is z = 0; and when the gravity terms are considered, i.e. g = 9.8 m/s2, the vibration center of the span mid-point deviates to z = -6.8506.
-12 -10 -8 -6 -4 -2 0 2
0 5 10 15 20
Dimensionless time t U = 0.5, V = 0.03
2 0
g = 9.8 m/s2 g = 0
g = 9.8 m/s2 g = 0
25
30 (a)
Displacement-to-diameter ratio z PSD PSD
Displacement-to-diameter ratio z
Displacement-to-diameter ratio z
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