# Vibration Control of Multi-Modular VLFS in Random Sea Based on Stiffness-Adjustable Connectors
## 2 The Model of Stiffness-Adjustable Connector  
The vibration control method to be developed is based on stiffness-adjustable connectors. Therefore, the novel connector type and the related VLFS system are introduced first. The VLFS consists of a column comprising \(N\) number of semi-submersible modules,as shown in Figure 1. Each module is composed of an upper hull, several columns, and lower hulls. A global coordinate system \(XYZ\) is set on the surface of still water, and the \(Z\) axis faces upward, perpendicular to the water surface. For the convenience of deriving the mathematical model of the connector, a local coordinate system \(x_{i}y_{i}z_{i}a_{i}\beta_{i}\gamma_{i}\) is set at the mass center of each module, as shown in Figure 1. The responses of a module include the three translational displacements of surge, sway, and heave along the \(x,y,\) and \(z\) axes and the three rotational displacements of roll, pitch, and yaw, respectively, around \(x,y,\) and \(z\) axes. Ajacent modules are connected by connectors (to be discussed later). For \(N\) modules, there are \(N-1\) groups of connectors. The wave propagation direction is defined where the \(0^{\circ}\) wave angle is parallel to the \(x\) axis, and the \(90^{\circ}\) wave angle is parallel to the \(y\) axis. The connector is a key component for the connection of a multi-modular VLFS, which could significantly affect the vibration responses of floating modules. Here, we propose a sketch model of a stiffness-adjustable novel connector, as shown in Figure 2. The connector consists of an annular spring and a hydraulic actuator. The spring has a hollow cylindrical structure comprising an interlayer of steel plates and polymer materials, which provides stiffness both in the axial and radial directions (here, the bending, rotation, and torsion stiffness are ignored). The spring is considered to be a passive component to link the two adjacent modules, which provides a basic constant stiffness in three-translational directions, regardless of whether the actuator performs. The spring can withstand the compound loads of compression, tensile, and shear forces. The stiffness of the spring can be determined by the number of interlayers, steel plates, polymer materials, and geometrical dimensions. The hydraulic actuator is considered to be an active component that is mounted inside the inner hollow space of the spring and is assembled with the two-end covers by two spherical joints. Oil orifices allow hydraulic fluid to flow in and out to manipulate the inner pressure of the hydraulic cylinder, which generates desirable force in the axial direction. The output force of the actuator is time dependent and is determined by a control scheme to reduce the motion of the floating system according to wave conditions, which can reduce the response of the modules in random sea conditions. At each end of the connector, there is a convex cone and a groove, which is used to fasten the connector onto the end faces of the adjacent modules. When the convex cone slides into the conical concave hole in a floating module, a gate fastens inside the module, which is then pushed into the connector groove and fastens the connector firmly to the module. Between the two modules, there could be several connectors mounted according to the requirements of the designers. For example, Figure 3 shows the layout of the novel stiffness-adjustable connectors. There are four connectors that have been horizontally installed in parallel at the four corners of the two adjacent modules, and another one  
Figure 1: The schematic diagram of a multi-modular VLFS. is vertically installed in the middle of the modules. The combinatorial action of these connectors can simultaneously suppress the surge, heave, pitch, and yaw motions between the modules. The mechanism of the novel stiffness-adjustable connector can be stated as follows: The annular spring provides a restoring constraint force that is induced by the relative displacement between the two adjacent connected modules.