# Incorporating Epistemic Uncertainties in Ship Operability Study
## 2 Methods
### Seakeeping Analysis
#### 2.1.1 Transfer Functions  
Transfer functions for heave and pitch are calculated using semi-analytical closed-form expressions (CFE) [22]. These simple formulas are determined for the motion of a prismatic barge in the vertical plane using standard linear strip theory assumptions, assuming a constant sectional added mass equal to the displaced water volume and ignoring the coupling components between pitch and heave. The main ship attributes--length \(L\), breadth \(B\), draft \(T\), block coefficient \(Cb\), and speed \(V\)--are the only necessary input data for the procedure.  
In this section, the procedure is briefly outlined, while details may be found in the original reference [22].  
The frequency response functions for have (\(\Phi_{w}\)) and pitch (\(\Phi_{\theta}\)) are given as:  
\[\Phi_{w}=\eta F, \tag{1}\]  
\[\Phi_{\theta}=\eta G, \tag{2}\]  
where \(F\) and \(G\) are heave and pitch forcing function, and:  
\[\eta=\left(\sqrt{\left(1-2 kT\alpha^{2}\right)^{2}+\left(\frac{A^{2}}{kB\alpha^ {2}}\right)^{2}}\right)^{-1}, \tag{3}\]  
where \(k\) is the wave number, \(T\) is draft, \(A\) is approximation of the sectional hydrodynamic damping, \(B\) is breadth of ship, and \(\alpha\) is parameter:  
\[\alpha=1-F_{n}\sqrt{kL}\text{cos}\beta, \tag{4}\]  
where \(F_{n}\) is Froude number, \(L\) is the ship length, and \(\beta\) is heading of the ship.  
The transfer functions of vertical motion \(\Phi_{u}\) and vertical acceleration \(\Phi_{v}\) at the forward perpendicular are given as:  
\[\Phi_{u}=\sqrt{\Phi_{w}^{2}+x_{\text{PP}}^{2}\Phi_{\theta}^{2}}, \tag{5}\]  
\[\Phi_{v}=\overline{\omega}^{2}\Phi_{u}, \tag{6}\]  
where \(x_{\text{PP}}\) is the longitudinal distance of the forward perpendicular from the center of gravity, and \(\overline{\omega}\) is the encounter frequency.  
The transfer function for roll motion is obtained by solving the equation of the motion for roll in regular waves, and it can be written as:  
\[\Phi_{\varphi}=\frac{|M|}{\left(\left[-\overline{\omega}^{2}(\text{T}_{N}/2 \pi)^{2}+1\right]^{2}C_{44}^{2}+\overline{\omega}^{2}B_{44}^{2}\right)^{1/2}}, \tag{7}\]  
where \(|M|\) is the amplitude of the excitation moment; \(T_{N}\) is the natural roll period; \(C_{44}\) is the restoring moment coefficient; and \(B_{44}\) is the hydrodynamic roll damping for the ship which is considered 20% in this study.  
The CFEs are semi-analytical formulae, so they can be easily calculated using a usual spreadsheet application. CFEs have been used for seakeeping predictions in conceptual design, enabling an efficient sensitivity analysis regarding operational profiles and different wave environments [25]. CFEs have also been used in hydro-elastic studies [26], for the computation of extreme wave loads using the hindcast wave database, and for tuning the transfer function for the prediction of ship responses [27].  
In the present paper, CFEs are considered for the development of the risk-based ship DSS for heavy weather maneuvering. The numerical efficiency of DSS could be crucial if DSS is used for decision making onboard the ship on heavy seas [19]. CFEs have already been considered for DSS application in [28].