# 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}{cc}\mathbf{x}_{p}^{T}&\mathbf{x}_{p+1}^{T}\end{array} \big{]}^ {T}\]  
Then, the force term \(\mathbf{f}_{i,s}\) in Equation (12) is the summation of all of the individual connector forces in Equation (15), which can be written as  
\[\mathbf{f}_{i,s}=\left\{ \begin{array}{ll}\sum\limits_{q=1}^{Nc}\mathbf{f}_{s,1}^{q,i}=\sum\limits_{q=1}^{Nc}-\Big{(}\mathbf{W}_{1}^{q,i}\Big{)}^{T} \mathbf{K}_{s}\Big{[}\begin{array}{cc}\mathbf{W}_{1}^{q,i}&-\mathbf{W}_{2}^{ q,i}\end{array}\Big{]}\big{[}\begin{array}{cc}\mathbf{x}_{i}^{T}& \mathbf{x}_{i+1}^{T}\end{array}\big{]}^{T},&i=1\\ \sum\limits_{q=1}^{Nc}\mathbf{f}_{s,2}^{q,i-1}=\sum\limits_{q=1}^{Nc}\Bigg{(} \begin{array}{cc}\mathbf{W}_{2}^{q,i-1}\end{array}\Big{)}^{T}\mathbf{K}_{s} \Big{[}\begin{array}{cc}\mathbf{W}_{1}^{q,i-1}&-\mathbf{W}_{2}^{q,i-1}\end{array} \Big{]}\big{[}\begin{array}{cc}\mathbf{x}_{i-1}^{T}&\mathbf{x}_{i}^{T}\end{array} \big{]}^{T}\\ -\Big{(}\mathbf{W}_{1}^{q,i}\Big{)}^{T}\mathbf{K}_{s}\Big{[}\begin{array}{ cc}\mathbf{W}_{1}^{q,i}&-\mathbf{W}_{2}^{q,i}\end{array}\Big{]}\big{[} \begin{array}{cc}\mathbf{x}_{i}^{T}&\mathbf{x}_{i+1}^{T}\end{array}\big{]}^ {T}\end{array} \Bigg{)},&i=2\cdots N-1\\ \sum\limits_{q=1}^{Nc}\mathbf{f}_{s,2}^{q,i-1}=\sum\limits_{q=1}^{Nc}\Big{(} \mathbf{W}_{2}^{q,i-1}\Big{)}^{T}\mathbf{K}_{s}\Big{[} \begin{array}{cc} \mathbf{W}_{1}^{q,i-1}&-\mathbf{W}_{2}^{q,i-1}\end{array} \Big{]}\big{[} \begin{array}{cc}\mathbf{x}_{i-1}^{T}&\mathbf{x}_{i}^{T}\end{array} \big{]}^{T},& i=N\end{array}\right. \tag{16}\]  
For an actuator, the output force of the actuator of the \(q\)th connector is expressed by \(f_{a}^{q,p}\), which is determined by a control gain and module state. In this paper, a state feedback control scheme will be developed to determine a control gain vector for each actuator. Let \(\mathbf{\lambda}_{m}^{q,p}\in\mathbb{R}^{1\times 6}\) denote the control gain vector for the \(m\)th module state \(\mathbf{x}_{m}\). The actuator force of the \(q\)th connector in the \(p\)th group is the summation of all of the control gain vectors multiplied with the module states, which can be expressed as  
\[f_{a}^{q,p}=-\sum\limits_{m=1}^{N}\mathbf{\lambda}_{m}^{q,p}\cdot\mathbf{x}_{m} \tag{17}\]  
This actuator force is applied at the mounting positions \(\mathbf{\overline{d}}_{p}^{q}=\left[\delta_{1,x^{\prime}}^{q,p}\delta_{1,y^{ \prime}}^{q,p}\delta_{1,z}^{q,p}\right]\) and \(\mathbf{\overline{d}}_{p+1}^{q}=\left[\delta_{2,x^{\prime}}^{q,p}\delta_{2,y^ {\prime}}^{q,p}\delta_{2,z}^{q,p}\right]\) along the axial direction of the connector. The actuator force defined in Equation (17) may generate three translational forces and three moments at the mass centers of the \(p\)th module and the \((p+1)\)th module. Therefore, to study the control force at the mass center of a module, the actuator force has to be projected onto each degree of freedom at the mass center to form a control force vector. Let \(\mathbf{\overline{p}}_{p}\) and \(\mathbf{\overline{p}}_{p+1}\) denote the initial positions of the mass centers of the \(p\)th module and the \((p+1)\)th module in the global coordinate system. After the modules move, the mass centers of the \(p\)th and the \((p+1)\)th modules are relocated to \(\mathbf{P}_{p}\) and \(\mathbf{P}_{p+1}\), and the mounting points of the \(q\)th connector are relocated to \(\mathbf{d}_{p}^{q}\) and \(\mathbf{d}_{p+1}^{q}\) in the global coordinate system, which can be expressed as  
\[\mathbf{p}_{p}=\mathbf{\overline{p}}_{p}