## 2 Analysis Methods
### Equation of Motion of a Constrained Rigid Body System
#### 2.1.2 External Forces Acting on a Floating Body  
The external forces acting on a floating body can be expressed as Equation (8). \[\mathbf{F}_{Float}=\mathbf{F}_{Weight}+\mathbf{F}_{Hydrostatic}+\mathbf{F}_{ Froude-Key}+\mathbf{F}_{Diffraction}+\mathbf{F}_{Radiation}+\mathbf{F}_{Constraint} \tag{8}\]  
Figure 1: Description of a hinge joint. where \(\mathbf{F}_{Weight}\) and \(\mathbf{F}_{Hydrostatic}\) are the self-weight and hydrostatic force on the body, respectively. \(\mathbf{F}_{Froude-Krylov}\) is the Froude-Krylov force, \(\mathbf{F}_{Diffraction}\) is the diffraction force, \(\mathbf{F}_{Radiation}\) is the wave radiation force, and \(\mathbf{F}_{Constraint}\) is the constraint force. The self-weight acts in a vertically downward direction from the center of gravity because the float is assumed to be a rigid body. In this study, a body-nonlinear hydrostatic force considering the submerged surface of the float was calculated at every time step. The self-weight and body-nonlinear hydrostatic force of the float are presented in Equation (9). \[\mathbf{F}_{Weight}+\mathbf{F}_{Hydrostatic}= \begin{bmatrix}0&0&-mg&0&0&0\end{bmatrix}  ^{T}-\iint\limits_{S_{B}(t)}\rho_{w}gz\,\mathbf{n}\,dS \tag{9}\]  
where \(m\) is the mass of the float; \(g\) is the gravitational acceleration; \(\rho_{w}\) is the water density; \(S_{B}\) is the submerged surface of the float; \(z\) is the vertical displacement from the still water level; \(\mathbf{n}\) is the normal vector of the body surface. The Froude-Krylov and diffraction forces constituting the external wave force can be regarded as linear or body-nonlinear forces. The magnitude and phase angle of the linear force can be obtained through a frequency domain analysis using the boundary element method. Body-nonlinear Froude-Krylov forces must be considered to perform a precise dynamic analysis of HPA-type WECs [33]. When a body-nonlinear Froude-Krylov force is applied to a floating body, the tendency and characteristics of the dynamic response are different from when the linear Froude-Krylov force is considered [19]. The body-nonlinear Froude-Krylov force was calculated by integrating the dynamic pressure acting on the submerged surface of the body at each time step. In this study, the dynamic pressure was calculated based on the linear wave theory (or Airy wave theory). \[\mathbf{F}_{Froude-Krylov}(t)=-\iint\limits_{S_{B}(t)}\rho_{w}p_{FK}(t)\mathbf{ n}\,dS \tag{10}\]  
\[p_{FK}(t)=\frac{\rho_{w}gH}{2}\frac{\cosh(k(h+z))}{\cosh(kh)}\cos(k(x\cos \alpha+y\sin\alpha)-\omega t+\delta(\omega)) \tag{11}\]  
where \(p_{FK}\) is the dynamic pressure; \(H\) is the wave height; \(h\) is the water depth; \(k\) is the wavenumber; \(x\) and \(y\) are the body surface points; \(\alpha\) is the incident wave direction; \(\omega\) is the wave frequency, and \(\delta\) is the phase angle of the incident wave. In the analysis of HPA-type WECs, the nonlinear wave diffraction effect can be neglected because the diameter of the floating body is very small compared to the wavelength [33]. In this study, the linear diffraction force was considered, and the magnitude and phase angle of the linear diffraction force were obtained using WAMIT, a frequency domain analysis program [26]. The time history of the linear diffraction force can be calculated through the product of the magnitude of the force obtained by frequency domain analysis and the trigonometric function as follows:  
\[\mathbf{F}_{Diff}(t)=\frac{\rho_{w}\mathbf{g}H}{2}\cdot\left|\mathbf{F}_{Diff} (\omega)\right|\cdot\cos(k\cdot(x\cos\alpha+y\sin\alpha)-\omega t+\varphi( \omega)) \tag{12}\]  
where \(\left|\mathbf{F}_{Diff}\right|\) and \(\varphi\) denote the magnitude and the phase angle of linear diffraction force, respectively. The linear wave radiation force caused by the motion of a floating body was obtained using the Cummins' equation. This equation is expressed as the convolutional integral of the velocity and retardation function of the float to account for the time memory effect [34; 35].