# Vibration Control of Multi-Modular VLFS in Random Sea Based on Stiffness-Adjustable Connectors
## 4 Numerical Assessment on the Performance
### Performance of the Vibration Control Method  
The performance of the novel stiffness-adjustable connector (named \"novel\") was assessed by comparing it with its two counterparts. One of these was a flexible connector (named \"flexible\"), which has the same stiffness as the actuator-based flexible connector but \begin{table}
\begin{tabular}{c c c c} \hline \hline  & Length & Breadth & Depth \\ upper hull & 300 m & 100 m & 6 m \\ lower hull & 30 m & 96 m & 5 m \\ \hline \multirow{2}{*}{column} & diameter & length & distance in \(x\)-axis & distance in \(y\)-axis \\  & 18 m & 16 m & 58 m & 60 m \\ \hline \multirow{6}{*}{others} & \(z_{g}\) & \(l_{x}\) & \(l_{y}\) & \(l_{z}\) \\  & 12 m & \(1.09\times 10^{11}\) kg\(\cdot\)m\({}^{2}\) & \(7.28\times 10^{11}\) kg\(\cdot\)m\({}^{2}\) & \(7.28\times 10^{11}\) kg\(\cdot\)m\({}^{2}\) \\ \cline{1-1} \cline{2-4}  & design draft & displacement & \(\mathbf{K}_{M,1}\) and \(\mathbf{K}_{M,5}\) \\ \cline{1-1}  & 12 m & \(9.36\times 10^{7}\) kg & \(10^{5}\times\text{diag}\big{(}[5,5,1.5,3.5\times 10^{3},3.5\times 10^{4},1\times 10^{5}] \big{)}\) \\ \cline{1-1} \cline{2-4}  & & The horizontal gap between two modules is 5 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the VLFS. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Group \(i\)** & **Module \(i\) (\(\overline{d}_{i}^{q}\))** & **Module \(i+1\) (\(\overline{d}_{i+1}^{q}\))** \\ \hline Connector 1 & \(\{\)150 m, 40 m, 12 m\(\}\) & \(\{\)\(-150\) m, 40 m, 12 m\(\}\) \\ Connector 2 & \(\{\)150 m, \(-40\) m, 12 m\(\}\) & \(\{\)\(-150\) m, \(-40\) m, 12 m\(\}\) \\ Connector 3 & \(\{\)150 m, 40 m, 6 m\(\}\) & \(\{\)\(-150\) m, 40 m, 6 m\(\}\) \\ Connector 4 & \(\{\)150 m, \(-40\) m, 6 m\(\}\) & \(\{\)\(-150\) m, \(-40\) m, 6 m\(\}\) \\ Connector 5 & \(\{\)152.5 m, 0 m, 12 m\(\}\) & \(\{\)\(-152.5\) m, 0 m, 6 m\(\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Local coordinates of the novel connectors. without the actuator. The other was a hinged connector (named \"hinged\") that is commonly used by scholars [32]. The stiffness of the novel connector can be adjusted through the control of the actuators. The suppression of the surge, heave, pitch, and yaw motions is intended by using the layout of the novel stiffness-adjustable connectors (in Figure 3). Figure 6 shows the results of the responses of all five VLFS modules are illustrated for the three types of connectors under wave excitation (the wave incident angle is 0\({}^{\circ}\)). The vertical axis denotes the 1-in-1000 amplitude of the extreme responses calculated by Equation (27), and the horizontal axis denotes the number of the module. We first looked at the results of the intended surge, heave, pitch, and yaw motions to be controlled. Figure 6a,f show the surge and yaw motions of the five modules. Apparently, the motions with control of the novel connectors are much smaller than those of the flexible connector without control. The surge motion of the novel connector is almost the same as that of the hinged connectors, and the yaw motion of novel the connector is larger than that of the hinged connectors. Because hinged connectors are almost completely rigid in the \(x,y,\,\alpha\), and \(\beta\) directions, the surge, sway, roll, and yaw motions of the VLFS with hinged connectors are usually small. Figure 6c,e show the heave and pitch motions, respectively. We can see that these two motions are significantly reduced by the control strategy in comparison to the flexible and hinged connectors. It seems a bit peculiar that the heave and pitch motion of the first hinged module is larger. One reason for this is that in heading waves, the first few modules usually have larger responses than others especially, for the pitch motions that are unconstrained by hinged connectors.