# Vibration Control of Multi-Modular VLFS in Random Sea Based on Stiffness-Adjustable Connectors
## 3 Vibration Control Method Based on the Stiffness-Adjustable Connector
### Feedback Control Gains and the Parameters of Hydraulic-Actuators
For regular wave excitation at \(\omega\) with the unit of wave height, the module response can be expressed as  
\[\mathbf{A}(\omega)=\left(-\omega^{2}\mathbf{M}+j\omega\mathbf{C}+\mathbf{K} \right)^{-1}\cdot\mathbf{\bar{f}}_{i,\omega} \tag{26}\]  
For a wave spectrum \(\mathrm{S}(\omega)\) of a random sea condition, the 1-in-1000 amplitude of the extreme responses for all modules in all directions can be expressed as [31]  
\[\mathbf{R}_{e}=3.72\sqrt{\int_{0}^{\infty}\mathbf{A}(\omega)\ \circ\ \mathbf{A}( \omega)\mathrm{S}(\omega)d\omega} \tag{27}\]  
where the mathematical symbol \(\circ\) is the Hadamard product, which represents the multiplication of the corresponding elements in two vectors. The optimal function is set as  
\[ \begin{array}{l}\min\
olimits/=\sum_{i=1}^{N}\sum_{k=1}^{6}W_{k}\mathbf{R}_ {e}^{(i-1)\times 6+k}\\ s.t.\quad\|f_{d}^{d,p}\|\leq f_{max}\quad\quad q=1,\cdots,Nc;\quad\quad p=1, \cdots N-1\\ \quad\quad\mathbf{M}\mathbf{x}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x} =\mathbf{\bar{f}}_{i,\omega}e^{-j\omega t}\end{array}  \tag{28}\]  
to determine the feedback gains, where \(\mathbf{R}_{e}^{q}\) indicates the \(q\)th element of the vector \(\mathbf{R}_{e}\). Symbol \(\kappa_{k}\) is the weight of the motion in the \(k\)th degree of freedom. If the motion in the \(k\)th direction cannot be controlled, then the weight \(\kappa_{k}\) is set to be zero. The values of the weights for the concerned motions will be set based on designer preference. In Equation (28), the design variables for optimization are the control gains, which are the 1-in-1000 amplitude variables of the extreme responses of the floating system. Since the control force of the actuators cannot be infinite, the optimizing process is constrained by the maximal control output of each actuator \(f_{max}\). Of course, the force-response relationship should also satisfy [PERSON]'s law in the expression of Equation (1), which is a constraint as well. For a floating system with \(N\) modules, there are \(N-1\) groups of connectors. Each group has \(Nc\) connectors. For each flexible connector, there are \(6N\) control gains. There are \((N-1)\times Nc\times 6N\) optimal variables to be determined in total. The SQP is chosen because of its efficiency in dealing with the optimization of multiple variables. The SQP method solves a quadratic programming (QP) sub-problem to determine the search direction and uses a proper merit function with the promise of convergence to determine the step size. In this paper, the SQP software package in MATLAB is employed to obtain the feedback gains and further the control forces. To implement the control force, a hydraulic cylinder with a single-piston rod was used as the actuator, as shown in Figure 5. The parameters of the hydraulic cylinder, such as the pressure, displacement, and velocity of the piston movement, have to be associated with the required control forces. Assume that the velocity of the hydraulic fluid entering the left chamber is \(v_{l}\) and that the pressure of the left chamber is \(p_{l}\). The velocity of the hydraulic fluid as it flows out of  
Figure 5: The schematic diagram of a hydraulic actuator with a single-piston rod. the right chamber is \(v_{r}\), and the pressure of the right chamber is \(p_{r}\). The diameters of the left oil orifice, the right oil orifice, the piston head, and the piston rod are \(d_{l}\), \(d_{r}\), \(D\), and \(d\), respectively. As such, the actuator force defined in Equation (17) can be expressed as  
\[f_{a}\lx@note{footnote}{{$=$}}\eta_{m}\bigg{[}p_{l}\frac{\pi D^{2}}{4}-p_{r} \bigg{(}\frac{\pi D^{2}}{4}-\frac{\pi d^{2}}{4}\bigg{)}\bigg{]} \tag{29}\]  
where \(\eta_{m}\) is the efficiency coefficient of the hydraulic cylinder. Suppose that the hydraulic fluid is incompressible. According to the theorem of momentum, we have  
\[