## 2 Intelligent Mooring Motion Equation in the Time Domain  
### Motion of a Mooring Ship under Wave Action  
Based on the frequency-domain calculation of the three-dimensional potential flow theory, ship hydrodynamic parameters such as the amplitude operator of motion response, wave drift force, first-order wave force, additional mass and damping coefficient under wave action can be obtained. The frequency domain theory is generally only applicable to the interactions between regular waves and linear constraint systems [23]. However, in practical engineering, mooring ships are jointly constrained by mooring cables and fenders at the same time, and their forces are nonlinear. In the movement process of wharf mooring ships under the combined action of winds, waves and currents, the cables will be strained and relaxed, and the fender will be compressed and rebound. Therefore, the ship's motion is also nonlinear. Due to the difference in the stiffness of the cable and the fender, even if the two are assumed to be linear, the ship's mooring force and its motion response are also complex nonlinear problems, and the ship's motion response is no longer simply a harmonic vibration.  
The six-degree-of-freedom dynamic equation of mooring ships in unidirectional irregular waves proposed by [PERSON] in 1962 [24]--namely, the nonlinear motion equation of mooring ships--is solved by Fourier transformation through the use of hydrodynamic coefficients such as the ocean wave excitation force, the ship's additional mass and the radiation damping calculated in the frequency domain. The wave force, added mass and hysteresis functions of floating bodies such as ships are obtained in the time domain. Finally, the numerical results of the six degrees of freedom of motion (heave, roll, surge, sway, pitch and yaw) and mooring elements such as the mooring force, fender impact force and fender compression deformation are solved according to the coupled motion equation in the time domain--see Figure 2. The expression of the time domain analysis is as follows:  
\[(M+m)X^{\prime\prime}(t)+\int\limits_{0}^{\infty}X^{\prime}(t)R(t-\tau)d\tau+ KX(t)=F^{(1)}+F^{(2)} \tag{1}\]  
where \(X\)_(t)_ is the motion vector of the mooring ship--namely, the motion with six degrees of freedom: pitch, roll, heave, roll, pitch and yaw; \(M\) is the hull mass inertia matrix; \(m\) is the fluid added mass matrix; \(K\) is the still water restoring force coefficient matrix of the hull; \(F^{(1)}\) is the first-order wave excited force vector on the hull; \(F^{(2)}\) is the second order wave excited force vector of the hull; \(R(t\)-\(\tau)\) is the hull velocity pulse function matrix and the convolution integral term represents the fluid memory effect.  
In the above equation, for the added mass, m can be obtained as follows:  
\[m=A(\infty) \tag{2}\]  
where \(A\) is the frequency-dependent additional mass matrix composed of \(A_{jk}\) and \(A_{jk}\) is the KTH additional mass coefficient generated by the rigid body motion of the JTH unit amplitude body, where \(j\) = 1,2... \(N\); \(N\) is the degree of freedom.  
The function \(R(t)\) can be obtained by the inverse Fourier transform of the frequency-dependent radiation damping coefficient, expressed as follows:  
\[R(t)=\frac{2}{\pi}\int\limits_{0}^{\infty}B(\omega)\cos(\omega t)dt \tag{3}\]  
where \(B\) is a frequency-dependent potential damping matrix composed of \(B_{jk}\) and \(B_{jk}\) is the KTH potential damping coefficient generated by the JTH unit amplitude rigid body motion.