# Vibration Control of Multi-Modular VLFS in Random Sea Based on Stiffness-Adjustable Connectors
## 3 Vibration Control Method Based on the Stiffness-Adjustable Connector
### Mathematical Model of the Floating System
#### 3.1.1 Hydrodynamic Model  
Linear wave theory [36] is used to determine the wave excitation force based on the assumption that the water is irrotational and inviscid. For a wave frequency \(\omega\), the total wave potential can be expressed as  
\[\Phi=\mathrm{Re}\Big{(}\psi e^{-j\omega t}\Big{)} \tag{3}\]where \(j=\sqrt{-1}\). Symbol \(\psi\) is the space velocity potential. For a \(N\)-modular VLFS such as the one shown in Figure 1, the expression of \(\psi\) is  
\[\psi=\psi_{I}+\psi_{D}+j\omega\sum_{i=1}^{N}\left(\psi_{i}\right)^{T}\widetilde {\mathbf{x}}_{i},\ \ i=1,2,\cdots N \tag{4}\]  
where \(\psi_{I}\) is incident potential, and \(\psi_{D}\) is diffraction potential. Vector \(\psi_{i}=\left[\psi_{i}\right.^{1},\cdots,\psi_{i}{}^{6}\right]^{T}\) is the potential produced by the unit amplitude of the \(i\)th module in each degree of freedom. Vector \(\widetilde{\mathbf{x}}_{i}\) is the amplitude vector of the responses of the \(i\)th module. The incident potential \(\psi_{I}\) is associated with wave condition, which can be written as  
\[\psi_{I}=\frac{jga}{\omega}\frac{\cosh k(z+h)}{\cosh(kh)}e^{jk(x\cos\theta+y \sin\theta)} \tag{5}\]  
where \(a\) is the wave amplitude, and \(h\) is the water depth. Symbol \(k\) is the number of waves that can be determined by the wave frequency \(\omega\) and water depth \(h\). The diffraction potential \(\psi_{D}\) and the potential \(\psi_{i}\) can be solved by the Laplace equation along with the conditions of the free water surface, including the surface boundary of the modules, sea bottom, and infinity radiation. Generally, the partial differential equations and boundary conditions for \(\psi_{D}\) and \(\psi_{i}\) are expressed as  
\[\left\{ \begin{array}{l}\
abla^{2}\psi_{D}=0\\ \frac{\partial\psi_{D}}{\partial z}-\frac{\omega^{2}}{g}=0\\ \frac{\partial\psi_{D}}{\partial t}=-\frac{\partial\psi_{D}}{\partial t}\\ \frac{\partial\psi_{D}}{\partial z}|_{z=-h}=0\\ \frac{\partial\psi_{D}}{\partial z}|_{z=-h}=0\end{array} \right.\left\{ \begin{array} []{l}\
abla^{2}\psi_{i}^{q}=0\\ \frac{\partial\psi_{i}}{\partial z}-\frac{\omega^{2}}{g}\psi_{i}=0\\ \frac{\partial\psi_{i}}{\partial z}|_{z}=\mathbf{n}_{i}\\ \frac{\partial\psi_{i}}{\partial z}|_{S_{n}}=0\quad i\
eq m\\ \frac{\partial\psi_{i}}{\partial z}|_{z=-h}=0\\ \lim\limits_{r\to\infty}\sqrt{r}\Big{(}\frac{\partial\psi_{D}}{\partial t}-j \frac{\omega^{2}}{g}\psi_{i}\Big{)}=0\end{array} \right. \tag{6}\]  
where vector \(\mathbf{n}_{i}=\left[n_{i}\right.^{1},\cdots,n_{i}{}^{6}\Big{)}^{T}\) is the projection component in each normal direction. Symbol \(r\) is the displacement between the field point and source point. By using HydroD, the velocity potential is determined through the Laplace equation and the boundary conditions. Then, using [PERSON]'s equation, the wave excitation amplitude acting on the \(i\)th module can be obtained by integrating the wave potential along the module's wet surface, which is expressed as  
\[\mathbf{\tilde{t}}_{i,w}=\iint_{S_{i}}\left[-j\omega\rho\left(\psi_{I}+\psi_{ D}+j\omega\sum_{q=1}^{N}\left(\psi_{q}\right)^{T}\widetilde{\mathbf{x}}_{q} \right)\right]\mathbf{n}_{i}\,\mathrm{d}s \tag{7}\]  
As such, the wave excitation can be written as  
\[ \begin{array}{ll}\mathbf{f}_{i,w}=\mathbf{\tilde{f}}_{i,w}e^{-j\omega t}&= \left(\iint_{S_{i}}\left[-j\omega\rho\left(\psi_{I}+\psi_{D}+j\omega\sum_{q=1} ^{N}\left(\psi_{q}\right)^{T}\widetilde{\mathbf{x}}_{q}\right)\right]\mathbf{ n}_{i}\mathrm{d}s\right)e^{-j\omega t}\\ &=\left(-j\omega\rho\iint_{S_{i}}(\psi_{I}+\psi_{D})\mathbf{n}_{i}\mathrm{d}s \right)e^{-j\omega t}\\ &-\sum_{q=1}^{N}\left(\rho\iint_{S_{i}}\mathrm{Re}\left(\psi_{q}\right) \mathbf{n}_{i}^{T}\mathrm{d}s\right)\left(-\omega^{2}\widetilde{\mathbf{x}}_{ q}e^{-j\omega t}\right)\\ &-\sum_{q=1}^{N}\left(\rho\omega\iint_{S_{i}}\mathrm{Im}\left(\psi_{q}\right) \mathbf{n}_{i}^{T}\mathrm{d}s\right)\left(-j\omega\widetilde{\mathbf{x}}_{q}e^ {-j\omega t}\right)\end{array}  \tag{8}\]  
The harmonic response of the \(i\)th module is expressed by  
\[