*3.2. Equations of Motion*

The total hydrodynamic loads (forces and moment) on the twin-hull structure consist of the Froude-Krylov loads, which are solely due to the undisturbed incident field *ϕI*(**x**), the diffraction loads caused by the pressure field generated by the presence of the fixed floating body and the radiation loads due to the pressure fields of the wavefields "radiated" by the oscillating body. Based on the calculated propagating potential *ϕp*(**x**)(consisting of the incident and diffraction potentials), the summation of the Froude-Krylov and the diffraction-induced hydrodynamic forces, as well as the corresponding moment (*Fk*, *k* = 2, 3, 4), are calculated using surface integration, as follows:

$$F\_k = i\omega \rho \int\_{\partial D\_1 \cup \partial D\_3} \rho\_P(\mathbf{x}) \cdot \mathbf{N}\_k(\mathbf{x}) d\ell(\mathbf{x}), \\ k = 2, 3, 4, \ \mathbf{x} \in \left(\partial D\_1 \cup \partial D\_3\right) \tag{10}$$

where *ρ* denotes the fluid (water) density and **N***k*, *k* = 2, 3, 4, is the generalized normal vector on the wetted surface (also defined in Section 2). Moreover, from the radiation potentials *ϕk*(**x**), *k* = 2, 3, 4, the hydrodynamic coefficients are calculated using:

$$
\omega^2 A\_{kl} + i\omega B\_{kl} = i\omega \rho \Pi\_{kl}, \; l, k = 2, 3, 4, \; \text{where} \tag{11a}
$$

$$\Pi\_{kl} = \int\_{\partial D\_1 \cup \partial D\_3} \varphi\_l(\mathbf{x}) \mathbf{N}\_k(\mathbf{x}) d\ell(\mathbf{x}), \ l, k = 2, 3, 4, \ \mathbf{x} \in \left(\partial D\_1 \cup \partial D\_3\right) \tag{11b}$$

In the above expressions, **A**(3×3) is the (symmetric) matrix of the added inertial coefficients, which, for each frequency, corresponded to the proportion of the radiation loads in phase with the structure's acceleration (in the frequency domain). **B**(3×3) is the corresponding matrix of hydrodynamic damping coefficients, which consists of the part of the radiation loads in phase with the structure's velocity. Details about the definitions of the hydrodynamic forces and coefficients, as well as the system of equations of motion, can be found in [15] or in ship hydrodynamics textbooks (see, e.g., [16,24]). The latter quantities allow us to formulate and solve the equations of motion of the floating body in the inhomogeneous domain. The general form of the equations of motion in the frequency domain for the 2D twin-hull structure considered is:

$$\left\{-\omega^{2}[\mathbf{M} + \mathbf{A}(\omega)] - i\omega\,\mathbf{B}(\omega) + \mathbf{C}\right\}\tilde{\boldsymbol{\zeta}} = \boldsymbol{F} \tag{12}$$

where **C** is the hydrostatic restoring forces and moments acting on the structure.

Due to the symmetry of the body with respect to the vertical axis *x*3, the component **N**<sup>3</sup> of the generalized normal vector is symmetric, while the components **N***k*, *k* = 2, 4, are antisymmetric. Assuming that the seabed profile variations do not significantly alter the radiation potentials *ϕk*(**x**), *k* = 2, 3, 4, near the floating structure, the potential function *ϕ*3(**x**) is also symmetric and the functions *ϕk*(**x**), *k* = 2, 4 are antisymmetric. This fact implies that Π<sup>32</sup> = Π<sup>34</sup> = 0 and Π<sup>23</sup> = Π<sup>42</sup> = 0. Therefore, the dynamic equations of motion relating to the oscillations of the body are simplified in the following form, where the heaving motion (*ξ*3) is decoupled from the sway and roll motions (*ξ*2, *ξ*4) of the twin hull:

$$\left[-\omega^2(M+A\_{22})-i\omega B\_{22}\right]\xi\_2-\left(\omega^2A\_{24}+i\omega B\_{24}\right)\xi\_4=F\_{24}\tag{13a}$$

$$\left[-\omega^2(M+A\mathfrak{z}\mathfrak{z})-i\omega B\mathfrak{z}\mathfrak{z}+2\rho\mathfrak{g}B\_{(H)}\right]\mathfrak{z}\mathfrak{z}=F\mathfrak{z},\tag{13b}$$

$$\left[-\omega^2 A\_{42} - i\omega B\_{42}\right]\xi\_2 + \left[-\omega^2 (I\_{44} + A\_{44}) - i\omega B\_{44} + Mg \cdot GM\right]\xi\_4 = F\_{4\prime} \tag{13c}$$

where *B*(*H*) is the breadth of each individual hull and *GM* denotes the metacentric height. The total mass equals *M* = *ρ* · ∇, referring to unit length in the transverse direction (*kg*/*m*), where *ρ* denotes the fluid's density and ∇ is the displacement volume of the structure. Moreover, due to the symmetry of the floating structure, its center of buoyancy (*B*) is

located on the vertical line *x*<sup>2</sup> = 0 and its *x*<sup>3</sup> coordinate is calculated as the center of area of the submerged volume's cross-section. The center of gravity (*G*) is also located at *x*<sup>2</sup> = 0 due to symmetry of the configuration and its *x*<sup>3</sup> coordinate is considered to be located at the waterplane (*x*<sup>3</sup> = 0). A total radius of gyration per unit length in the transverse direction of *RG* = - *B*(*T*) − *B*(*H*) /2 is considered about the longitudinal axis (*x*1), where *B*(*T*) is the total breadth of the twin-hull structure and, therefore, *I*<sup>44</sup> = *M*(*RG*) 2 . The metacentric radius was evaluated as *BM* = *I*/∇, where *I* is the second moment of area of the waterplane, calculated using applying Steiner's theorem as:

$$I = 2\left[\left(\frac{B\_{(H)}}{12}\right) + B\_{(H)} \cdot \left(\frac{B\_{(T)} - B\_{(H)}}{2}\right)^2\right],\tag{14}$$

which also refers to the unit length in the transverse direction (*x*1). Finally, the metacentric height was calculated as *GM* = *KB* + *BM* − *KG*, where *K* is a reference point with coordinates (0, *x*3). The above equations can also be modified to include other external forces, as e.g., mooring forces or spring terms (see, e.g., Section 3.5 of [25]). The solution of the above system (13) provides us with the complex amplitudes of the corresponding motions of the twin hull: *ξk*, *k* = 2, 3, 4. Then, the total wave potential is obtained using Equation (4), from which the hydrodynamic pressure is obtained using Bernoulli's theorem. The wave loads on the floating structure are calculated using pressure integration on the wetted surface *∂D*<sup>1</sup> ∪ *∂D*3.
