Case

A case with a drill ship which is to perform a marine operation at Haltenbanken was chosen in order to discuss the subject in more detail. The theory presented in the previ-ous section has been used to determine the response of the vessel. The critical responses and their corresponding sea states have been compared to relevant sea states represented by contour plots. Total sea has been used as input.

Wave Spectrum A wave spectrum was needed in order to carry out the analysis, and Section 2.3.1 spoke briefly about wave spectrums. There exists several standardized spectrums today, and for this case the JONSWAP spectrum was chosen: ”Numerous models for the wave frequency spectrum have been proposed over the years. At present the most common model is theJONSWAPspectrum” (p. 117, Haver, 2013). The JON-SWAP spectrum is more peaked than the Pierson Moskowitz spectrum, and this is based on research which was carried out in the North Sea. TheJONSWAPspectrum for

grow-ing wind sea is given as (Haver [10, p. 150])

Verification of theMATLABscript and wave spectrum was done by the following con-siderations:

1. Verifying that the input value ofT_pcorresponded with theT_pof the wave spectrum which was produced: The spectral peak frequency, w_p, is found fromw_p= ^2π_T

p. The input value ofw_pwas then compared to the peak value of the graph in Figure 49, by using a function inMATLABwhich calculates the frequency corresponding to the peak of the graph. An almost perfect match of w_p =1.27 seconds was found.

2. Verifying that the input value ofH_s corresponded with theH_s of the wave

spec-! [rad/s]

trum which was produced: This was done my calcuatingm₀of the wave spectrum:

m_n=

As we can seem₀is the area under the graph, and this area was calculated by us-ing the trapezoidal integration functiontrapzinMATLAB.H_swas then calculated byH_s=4√

m₀. AH_svalue equal to 9.855 metres was found (the input was 1

me-tres). This is considered acceptable, since the trapezoidal method is a numerical method.

Response Amplitude Operator As mentioned in the previous section, an RAO for the structure is needed in order to determine the response spectrumS_η3(ω)and its cor-responding standard deviationσ_η3(ω). AnRAOis a function which represents a certain motion characteristic of some structure. It is the ratio of the response amplitude to the wave amplitude,η3_A/ζ_A. In this problem the heave motion is under consideration:

Table 17: Definition of Response Amplitude Operator Wave process: ζ(t) =ζ_Acos(ωt)

Heave process: η3(t) =η3Acos(ωt+ε) RAO: ^η3A

ζA

For this task we chose to use anRAOfor a drill ship, illustrated in Figure 50. The curve for head seas is used, meaning that we are considering waves which are headed straight towards the bow of ship. Head seas and beam seas (beam seas are waves which are headed towards the side of the ship, 90^◦) have a different impact on the heave response η₃, which can also be seen in the figure. According to Faltinsen [6, p. 83], the larger response from beam seas can be understood by considering

|η₃|_{head sea}=|η₃|_{beam sea}· 2 The relation above shows that the length of the vesselLand the wave numberk deter-mines the size of |η₃|_{head sea}. L is relatively large, meaning that _kL² is small, giving a small value of|η₃|_{head sea}. As the wave periodT increases, kdecreases, increasing the fraction _kL². Therefore,|η₃|_{head sea}and|η₃|_{beam sea}approach each other as the wave pe-riod increases. For relevant wave pepe-riods,|η₃|_{beam sea} is always larger than|η₃|_{head sea} (see Figure 50). A larger version of this figure is found in appendix B.

Points on the curve in Figure 50 were visually interpreted and stored in an array in the MATLAB script, and the results are shown in Figure 51. Next, the points were transformed to be dependant of frequencyω[rad/s]instead of wave period, byω= ^2π_T . This is illustrated in Figure 52. In order to make the data more complete and the curve

smoother, it was interpolated using an interpolation function in MATLAB. The results are shown in Figure 53.

Figure 50: RAOfor Drill Ship

T [s]

0 5 10 15 20 25 30 35 40

RAO [-]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 51: RAOfor Drill Ship

! [rad/s]

Figure 52:RAO(ω)versusω

! [rad/s]

Figure 53: RAO(ω)versusω, interpolated Determining acceptable pairs ofH_sandT_p As stated in Section 3.1.2 we can deter-mine a limiting pair ofH_s andT_pby

An iteration process was performed, producing unique wave spectrumsS(ω)from dif-ferent pairs of (H_s,T_p) and calculatingση3(H_s,T_p)for every iteration. The iteration loop which was used inMATLABlooks like this:

1. H_s andT_pare first set to an arbitrary value, e.g.H_s=1 metre andT_p=5 seconds.

We then have a wave spectrum,S(ω)

2. S_η3(ω)is determined byS_η3(ω) =|RAO(ω)|²·S(ω) 3. Nis determined bym₀,m₂

4. σ_η3 is determined bypR_∞

0 S_η3(ω)dω. If this value is smaller than the right side of Equation 62, the set ofH_s andT_pis acceptable and stored in an array.

This is an iteration process which calculatesσ_η3(H_s,T_p)for many different wave spec-trums. This implies that each iteration starts with a value of H_s and T_p. In the total picture, each value ofH_sis combined with a series ofT_pvalues, and it looks like this:

1. A value ofH_s,H_s1, is combined with a series of differentT_p-values (T_p1,T_p2, ...,T_pn) 2. Taking the next fixed value ofH_s,H_s2, and combining this with the same series of

differentT_p-values

3. Continuing doing this until our last value ofH_sm

In this fashion a large number of different sets of (H_s,T_p) are tested.