# 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  
Two adjacent modules are connected together by a group of connectors. For \(N\) modules VLFS, there are \(N-1\) groups of connectors. In each group, there are \(Nc\) novel connectors, and each connector consists of a spring and a hydraulic actuator. The vector \(\mathbf{f}_{i,c}\) denotes the connector force imposed on the \(i\)th module, which can be expressed as  
\[\mathbf{f}_{i,c}=\mathbf{f}_{i,s}+\mathbf{f}_{i,a} \tag{12}\]  
where vector \(\mathbf{f}_{i,s}\) is the spring force vector, and \(\mathbf{f}_{i,a}\) indicates the actuator force vector. Both the spring force and the actuator force act on the modules through the mounting points. The bending moment of the spring due to the relative pitch motion between two adjacent modules is trivial and is thus ignored. In addition, the twist moment of the spring is ignored since the relative roll motion between two adjacent modules is insignificant and neglectable. Therefore, the spring only provides connection forces in the \(x_{i},y_{i},\) and \(z_{i}\) directions. The spring force is based on the stiffness of the spring and relative displacements at the two ends of the mounting points. The actuator only provides an active control force in the axial direction. For the \(q\)th novel connector in the \(p\)th groups, the position of the mounting point at the \(p\)th module is \(\overline{\mathbf{d}}_{p}^{q}=\left[\delta_{1,x^{\prime}}^{q,p},\delta_{1,y^{ \prime}}^{q,p},\delta_{1,z}^{q,p}\right]\), and the position of the mounting point at the \((p+1)\)th module is \(\overline{\mathbf{d}}_{p+1}^{q}=\left[\delta_{2,x}^{q,p},\delta_{2,y^{\prime }}^{q,p},\delta_{2,z}^{q,p}\right]\) in their local frames. Due to the movement of the two connected modules, the displacements of the two mounting points in the global frame \(\mathbf{\mu}_{1}^{q,p}\) and \(\mathbf{\mu}_{2}^{q,p}\) can be expressed as [25]  
\[\mathbf{\mu}_{1}^{q,p}=\mathbf{W}_{1}^{q,p}\cdot\mathbf{x}_{p}=\left[ \begin{array} []{cccccc}1&0&0&0&\delta_{1,z}^{q,p}&-\delta_{1,y}^{q,p}\\ 0&1&0&-\delta_{1,z}^{q,p}&0&\delta_{1,x}^{q,p}\\ 0&0&1&\delta_{1,y}^{q,p}&-\delta_{1,x}^{q,p}&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array} \right]\cdot\mathbf{x}_{p}\] \[\mathbf{\mu}_{2}^{q,p}=\mathbf{W}_{2}^{q,p}\cdot\mathbf{x}_{p+1}= \left[ \begin{array}{cccccc}1&0&0&0&\delta_{2,z}^{q,p}&-\delta_{2,y}^{q,p} \\ 0&1&0&-\delta_{2,y}^{q,p}&0&\delta_{2,x}^{q,p}\\ 0&0&1&\delta_{2,y}^{q,p}&-\delta_{2,x}^{q,p}&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array} \right]\cdot\mathbf{x}_{p+1} \tag{13}\]  
where \(\mathbf{W}_{1}^{q,p}\) and \(\mathbf{W}_{2}^{q,p}\) are the transformation matrices to translate the displacements of the mass centers of the modules to the displacements of the mounting points. Assuming that the stiffness of the spring is linear, the force of the spring is defined by \[\mathbf{f}_{s}^{q,p}=\mathbf{K}_{s}\Big{(}\mathbf{\mu}_{2}^{q,p}-\mathbf{\mu}_{1}^{ q,p}\Big{)}=-\left[ \begin{array}{ccccc}k_{a}&0&0&0&0&0\\ 0&k_{d}&0&0&0&0\\ 0&0&k_{d}&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{array} \right]\big{[} \begin{array}{cc}\mathbf{W}_{1}^{q,p}&- \mathbf{W}_{2}^{q,p}\end{array} \big{]}\big{[} \begin{array}{cc}\mathbf{x}_{p} ^{T}&\mathbf{x}_{p+1}^{T}\end{array} \big{]}^{T} \tag{14}\]  
where \(k_{a}\) is the axial stiffness, and the \(k_{d}\) is the radial stiffness of the connector. Then, the forces acting on the mass centers of the \(p\)th module and the \((p+1)\)th module are  
\[\mathbf{f}_{s,1}^{q,p} =\Big{(}\mathbf{W}_{1}^{q,p}\Big{)}^{T}\mathbf{f}_{s}^{q,p}=- \Big{(}\mathbf{W}_{1}^{q,p}\Big{)}^{T}\mathbf{K}_{s}\big{[} \begin{array}{ cc}\mathbf{W}_{1}^{q,p}&-\mathbf{W}_{2}^{q,p}\end{array} \big{]}\big{[} \begin{array}{cc}\mathbf{x}_{p}^{T}&\mathbf{x}_{p+1}^{T}\end{array} \big{]}^ {T} \tag{15}\] \[\mathbf{f}_{s,2}^{q,p} =-\Big{(}\mathbf{W}_{2}^{q,p}\Big{)}^{T}\mathbf{f}_{s}^{q,p}= \Big{(}\mathbf{W}_{2}^{q,p}\Big{)}^{T}\mathbf{K}_{s}\big{[} \begin{array}{ cc}\mathbf{W}_{1}^{q,p}&-\mathbf{W}_{2}^{q,p}\end{array} \big{]}\big{[}