## 3 Contribution of Froude-Krylov Forces to the Free Response  
Before exploring the control system design of Section 4, it is helpful to examine the relative effects of the terms in Equation (11). Instead of a nondimensional assessment, we will use the example buoy used for the control system design and rewrite Equation (11) as  
\[M\zeta+b\zeta-F_{b}-F_{fk,d}=F_{c} \tag{13}\]  
The buoy characteristics are given in Table 1, where \(m_{a}\) and \(b\) were obtained using WAMIT for the buoy geometry of Figure 1. The incident regular, linear wave has a period of 18 s and an amplitude of \(A=0.35\) m. The amplitude was selected to yield maximum buoy displacement such that it nearly leaves the water. While this is not physically possible for a free-floating buoy, it provides a limiting scenario. Assuming the water depth, \(h_{w}\), is large such that \(\chi h_{w}\gg 1\), the dispersion equation.  
\[\omega^{2}=g\chi\tanh\left(\chi h_{w}\right)\approx g\chi \tag{14}\]  
gives \(\chi\approx\frac{\omega^{2}}{g}=0.0125\) with wavelength \(\lambda=\frac{2\pi}{\chi}=502.65\)m.  
It should be noted that under some conditions, the dynamic FK force is small compared to \(F_{b}\) and could be neglected. For example, for large wavelength, small wave amplitudes, small buoy displacements, and small buoy size such that \(\chi\eta\ll 1\), \(\chi\xi\ll 1\) and, \(\chi h\ll 1\), the Taylor series expansion of Equation (8) gives \(F_{fk,d}\approx 0\).  
A Simulink simulation using a fixed-step solver, ODE3, with a 0.01 s time step was created using Equation (13) where the dynamic FK force could be selectively included. The model parameters are provided in Table 1 where the hydrodynamic coefficients \(m_{a}\) and \(b\) were calculated using WAMIT's boundary-element-method solver [40]. The hourglass buoy geometry was discretized by a 221-panel mesh, and the solution was calculated over the angular frequency range of 0.1-12 rad/s with 0.01 rad/s increments. The equilibrium configuration was with the buoy submerged to its halfway point. To check convergence, the mesh was increased to 349 panels yielding nearly identical results as the 221-panel model. Figure 2 compares the three noninertial forces of Equation (11), as well as their sum, for the regular wave described above with \(F_{c}=0\). The buoyancy and dynamic FK forces, \(F_{b}\) and \(F_{fk,d}\), are of similar amplitude but phase shifted nearly \(180^{\circ}\). The radiation damping force, \(b_{b}^{\zeta}\), is small in comparison. The force sum is also shown and illustrates an interesting result arising from the combination of \(F_{b}\) and \(F_{fk,d}\). Namely, the force sum changes sign often, which is not the case if \(F_{fk,d}\) is omitted. This has a dramatic effect on the buoy motion.  
To further investigate the effect of the combination of \(F_{b}\) and \(F_{fk,d}\) a simulation was performed with \(F_{fk,d}=0\) with the results shown in Figure 3. For these conditions, the buoy behaves as a wave-follower with very little vertical motion and thus very low \(F_{b}\), about 15 kN compared to the 600 kN of Figure 2. When \(F_{fk,d}\) is omitted, the buoy response cannot fully excite its buoyancy force's nonlinearity. In contrast, when \(F_{fk,d}\) is included, the buoy motion is excited, effectively turning on the nonlinearity that leads to even greater motion.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Feature** & **Symbol** & **Value** & **Units** \\ \hline Height & \(2h\) & 5 & m \\ Cone Angle & \(\alpha\) & 60 & degree \\ Draft & \(h_{0}\) & 2.5 & m \\ Mass & \(m\) & 50,376 & kg \\ Density & \(\rho_{b}\) & 1000 & kg/m\({}^{3}\) \\ Added Mass & \(m_{a}\) & 59,250 & kg \\ Damping & \(b\) & 20,000 & N/(m/s) \\ \hline \end{tabular}
\end{table}
Table 1: Hourglass buoy parameters.