*2.4. Equation of Motion*

In this study, the wind and wave energy combined system consists of two bodies having in total 12 degrees of freedom (six degrees for each body), which requires comprehensive consideration of multiple degree-of-freedom systems [29]. The multi-body equation of motion is given as follows:

$$
\begin{array}{c}
\left(\begin{array}{c}
(M+m)\_{11} \\
m\_{21}
\end{array}\right)
\left(\begin{array}{c}
\bar{\mathbf{x}}\_{1}(t) \\
\bar{\mathbf{x}}\_{2}(t)
\end{array}\right) + \left[\begin{array}{c}
\bar{\mathbf{x}}\_{1}(t) \\
\bar{\mathbf{x}}\_{2}(t)
\end{array}\right] + \left[\begin{array}{c}
\bar{\mathbf{x}}\_{1}(t) \\
\bar{\mathbf{x}}\_{2}(t)
\end{array}\right] + \left[\begin{array}{c}
\bar{\mathbf{x}}\_{1}(t) \\
\bar{\mathbf{x}}\_{2}(t)
\end{array}\right] + \left[\begin{array}{c}
\mathbf{x}\_{1}(t) \\
\mathbf{x}\_{2}(t)
\end{array}\right] = \left[\begin{array}{c}
\mathbf{f}^{\text{wind}}(t) \\
0
\end{array}\right] + \left[\begin{array}{c}
\mathbf{f}^{\text{wind}}(t) \\
\mathbf{f}^{\text{wind}}(t)
\end{array}\right] + \left[\begin{array}{c}
\mathbf{f}^{\text{wind}}(t) \\
\mathbf{f}^{\text{wind}}(t)
\end{array}\right] \\
\left[\begin{array}{c}
\mathbf{f}^{\text{wind}}\_{1}(t) \\
\mathbf{f}^{\text{wind}}\_{2}
\end{array}\right] + \left[\begin{array}{c}
\mathbf{f}^{\text{wind}}\_{1}(t) \\
\mathbf{f}^{\text{wind}}\_{2}
\end{array}\right] \\
\end{array}\tag{18}
$$

where *<sup>m</sup>* is the added mass matrix, *<sup>x</sup>*, . *<sup>x</sup>*, and .. *x* are the displacement, velocity, and acceleration matrix in the time domain, respectively, *κ*(*τ*) is the retardation function, which is based on the added mass and potential damping matrix, and *f* is the summation of the external forces in time domain. The subscripts 1 and 11 refer to the variables of body 1 (braceless); subscripts 2 and 22 refer to the variables of body 2 (WEC); and subscripts 12 and 21 present the coupling terms between the braceless and WEC. The vertical (heave) quadratic damping of the braceless and WEC terms is modeled by the quadratic damping matrix on the left side of Equation (18). The term *f wind* denotes wind load on the turbine rotor, while *f wave* is the wave forces applied on the braceless platform and WEC. The

interface forces *f*<sup>1</sup> *inter f ace* between the two bodies include horizontal contact forces and vertical friction forces. *FPTO* is the PTO forces. Each term of the interface and PTO forces is applied on the two bodies with the same value but in different directions.
