# 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
#### 3.1.2 Model of the Novel Stiffness-Adjustable Connector
\begin{array}{c}+\big{[}x_{p},y_{p},z_{p}\big{]}^{T}\\ \mathbf{p}_{p+1}=\mathbf{\overline{p}}_{p+1}\begin{array}{c}+\big{[}x_{p+1},y_ {p+1},z_{p+1}\big{]}^{T}\\ \mathbf{d}_{p}^{q}=\mathbf{\overline{p}}_{p}+\mathit{t}\mathit{n}\big{(}a_{p}, \mathbf{\beta}_{p},\mathbf{\gamma}_{p}\big{)}\cdot\mathbf{\overline{d}}_{p}^{q} \\ \mathbf{d}_{p+1}^{q}=\mathbf{\overline{p}}_{p+1}+\mathit{t}\mathit{n}\big{(}a_{p+1},\mathbf{\beta}_{p+1},\mathbf{\gamma}_{p+1}\big{)}\cdot\mathbf{\overline{d}}_{p+1}^{q} \end{array} \tag{18}\]where \(tran(\text{g})\) is the rotation matrix [37], which can be written as  
\[\begin{array}{l}tran(\alpha,\beta,\gamma)\\ =\left[\begin{array}{ccc}\cos\gamma\cos\beta&-\sin\gamma\cos\alpha+\cos\gamma \sin\beta\sin\alpha&\sin\gamma\sin\alpha+\cos\gamma\sin\beta\cos\alpha\\ \sin\gamma\cos\beta&\cos\beta\cos\alpha+\sin\gamma\sin\beta\sin\alpha&-\cos \gamma\sin\alpha+\sin\gamma\sin\beta\cos\alpha\\ -\sin\beta&\cos\beta\sin\alpha&\cos\beta\cos\alpha\end{array}\right]\end{array} \tag{19}\]  
Thus, the actuator force of the \(q\)th actuator in the \(p\)th group in Equation (17) can be projected onto each degree of freedom. The control force of the \(q\)th actuator vector acting on the mass center of the \(p\)th module can be expressed as [38]  
\[\begin{array}{l}\mathbf{\tilde{I}}_{a}{}^{q,p}=f_{a}{}^{q,p}\cdot\left[ \begin{array}{c}\mathbf{d}_{p+1}^{q}-\mathbf{d}_{p}^{q}\\ \frac{\mathbf{d}_{p+1}^{q}-\mathbf{d}_{p}^{q}}{\|\mathbf{d}_{p+1}^{q}-\mathbf{ d}_{p}^{q}\|}\end{array}\right]\\ \left[\begin{array}{c}\mathbf{d}_{p+1}^{q}-\mathbf{d}_{p}^{q}\\ \frac{\mathbf{d}_{p+1}^{q}-\mathbf{d}_{p}^{q}}{\|\mathbf{d}_{p+1}^{q}-\mathbf{ d}_{p}^{q}\|}\end{array}\right]\end{array}q=1,\cdots Nc;\ p=1,\cdots N-1 \tag{20}\]  
where the first three components of the control force vector are the translational forces, and last three components are the moments.  
It should be noted that the control force vector in Equation (20) contains geometric nonlinearity due to module rotation. If the rotation angles are small, then the control force can be treated approximately by linearization at the equilibrium point. Let \(\mathbf{f}_{a,1}^{q,p}\) denote the linearized control force vector acting at the mass center of the \(p\)th module, and similarly \(\mathbf{f}_{a,2}^{q,p}\) for the \((p+1)\)th module. The linearized control force vectors are respectively expressed as  
\[\begin{array}{l}\mathbf{f}_{a,1}^{q,p}=\overline{\mathbf{H}}^{q,p}\left[ \begin{array}{cccc}\mathbf{x}_{1}{}^{T}&\mathbf{x}_{2}{}^{T}&\cdots&\mathbf{ x}_{N}{}^{T}\end{array}\right]^{T}\\ \mathbf{f}_{a,2}^{q,p}=-\overline{\mathbf{H}}^{q,p}\left[\begin{array}{cccc} \mathbf{x}_{1}{}^{T}&\mathbf{x}_{2}{}^{T}&\cdots&\mathbf{x}_{N}{}^{T}\end{array} \right]^{T}\end{array} \tag{21}\]  
The symbol of \(\overline{\mathbf{H}}^{q,p}\in\mathbb{R}^{6\times 6N}\) is the linearized matrix, which is given by  
\[\overline{\mathbf{H}}^{q,p}=\frac{1}{\delta}\left[\begin{array}{c}\left( \delta_{1,x}^{q,p}-\delta_{2,x}^{q,p}-\triangle x\right)\\ \left(\delta_{1,y}^{q,p}-\delta_{2,y}^{q,p}\right)\\ \left(\delta_{1,z}^{q,p}-\delta_{2,z}^{q,p}\right)\\ \left(\delta_{2,y}^{q,p}\delta_{1,z}^{q,p}-\delta_{1,y}^{q,p}\cdot\delta_{2,z }^{q,p}\right)\\ \left(\delta_{1,x}^{q,p}\delta_{2,z}^{q,p}-\delta_{2,x}^{q,p}\cdot\delta_{1,z }^{q,p}-\triangle x\cdot\delta_{1,z}^{q,p}\right)\\ \left(\triangle x\cdot\delta_{1,y}^{q,p}-\delta_{1,x}^{q,p}\cdot\delta_{2,y}^ {q,p}+\delta_{2,x}^{q,p}\cdot\delta_{1,y}^{q,p}\right)\end{array}\right] \end{array} \right]\cdot\left[\begin{array}{l}\mathbf{\lambda}{}^{q,p}, \lambda_{2}{}^{q,p},\cdots,\lambda_{N}{}^{q,p}\right] \tag{22}\]  
where \(\delta=\sqrt{\left(\triangle x-\delta_{1,x}^{q,p}+\delta_{2,x}^{q,p}\right)^ {2}+\left(\delta_{1,y}^{q,p}-\delta_{2,y}^{q,p}\right)^{2}+\left(\delta_{1,z}^ {q,p}-\delta_{2,z}^{q,p}\right)^{2}}\), and the symbol \(\triangle x\) is the initial horizontal distance between the mass centers of the \(p\)th module and the \((p+1)\)th module. Assume that the deployment of the actuators in all of the connection groups is the same.