## 3 Methodology
### Scaling of Excitation Coefficient Database  
As can be seen in Figure 3, the excitation coefficient curve at a given non-dimensional frequency consists of two parabolic curves. It is defined by at least four parameters, i.e., \((A/D)_{C_{GCF}=0}\), \((A/D)_{C_{GCF}=max}\), \(Ce_{CF=max}\), and \(Ce_{CF,(A/D)=0}\), refer to Table 1. It is difficult and not efficient to derive a reasonable new excitation coefficient curve by randomly varying these parameters. Instead, two scaling parameters (\(\gamma_{A/D},\gamma_{C}\)) are introduced, which scales the amplitude parameters \((A/D)_{C_{GCF}=0}\), \((A/D)_{C_{GCF}=max}\), and excitation coefficient parameters, \(Ce_{CF=max}\) and \(Ce_{CF,(A/D)=0}\), respectively. Representative excitation coefficient curves can be generated by systematically varying these two scaling parameters.  
The scaling is carried out in two steps as explained in the following:  
Step 1: Amplitude modification process: The \(A/D\) modification factor is termed as  
\[\gamma_{A/D}=\frac{(A/D)_{C_{GCF}=0}}{\left((A/D)_{C_{GCF}=0}\right)_{def}} \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \(\overset{\wedge}{f}_{CF}\) & \((A/D)_{C_{GC}=0}\) & \((A/D)_{C_{GC}=max}\) & \(Ce_{CF}\)=\(max\) & \(Ce_{CF,(A/D)=0}\) \\ \hline
0.120 & 0.149 & 0.100 & 0.100 & 0.000 \\
0.172 & 0.900 & 0.430 & 0.800 & 0.400 \\
0.310 & 0.160 & 0.100 & 0.100 & 0.000 \\ \hline \end{tabular}
\end{table}
Table 1: Default set of parameters at three non-dimensional frequencies.  
Figure 3: Illustration of the parameter scaling procedure by three curves: Curve 1: default CF excitation coefficient curve at a particular non-dimensional frequency in VIVANA, Curve 2: after amplitude modification, Curve 3: after the excitation coefficient modification.  
where \(\left(A/D\right)_{C_{CCF}=0}\) and \(\left(\left(A/D\right)_{C_{CCF}=0}\right)_{def}\) are defined as the modified maximum response amplitude ratio and default maximum response amplitude ratio, respectively. \(\gamma_{A/D}\) is the amplitude modification factor.  
From VIVANA, \(\left(\left(A/D\right)_{C_{CCF}=0}\right)_{def}\) is 0.9 at \(\hat{f}_{CF}\) of 0.172; and in the present analysis, the amplitude ratio modification factor is ranged between 0.5-1.3. For example, when \(\gamma_{A/D}=0.8\),  
\[\left(A/D\right)_{C_{CCF}=0}=\left(\left(A/D\right)_{C_{CCF}=0}\right)_{def} \times\gamma_{A/D}=0.9\times 0.8=0.72 \tag{8}\]  
Figure 3 presents the modification process of hydrodynamic coefficients. The curve 2 in Figure 3 shows the modified excitation coefficient curve after applying a modification factor less than 1. It is obvious that \(A/D\) values at points B and C become smaller than the original curve, but the \(Cc\) values at points A and B remain the same.  
Step 2: Excitation Coefficient Modification: The \(Cc\) modification factor is defined as:  
\[\gamma_{Cc}=\frac{Ce_{A/D=max}}{\left(Ce_{A/D=max}\right)_{def}} \tag{9}\]  
where \(Ce_{A/D=max}\) and \(\left(Ce_{A/D=max}\right)_{def}\) are defined as the modified excitation coefficient and default excitation coefficient, respectively. \(\gamma_{Cc}\) is the excitation coefficient modification factor.  
The default peak value from VIVANA for \(\left(Ce_{A/D=max}\right)_{def}\) is 0.8 at \(\hat{f}_{CF}=0.172\). In the present analysis, the excitation coefficient modification factor (\(\gamma_{Cc}\)) ranges between 0.5-2.25. For example, if \(\gamma_{Cc}=0.8\) then  
\[Ce_{A/D=max}=\left(Ce_{A/D=max}\right)_{def}\times\gamma_{Cc}=0.8\times 0.8=0.64 \tag{10}\]  
As shown in Figure 3, by applying the amplitude ratio modification factor (step 1) at each non-dimensional frequency, the curve 2 coefficient curve is obtained. Based on step 1, the final excitation coefficient curve (curve 3) is obtained after applying the excitation coefficient modification factor (step 2).  
These two scaling parameters are systematically varied to create sufficient amount of different excitation coefficient databases.