Vibration Control of Multi-Modular VLFS in Random Sea Based on Stiffness-Adjustable Connectors

Abstract

The response control of a very large floating structure (VLFS) is a crucial issue that affects the stability and safety of the structure. This paper presents a vibration control method for stabilizing the motion of a multi-modular VLFS by using a set of stiffness-adjustable connectors. The proposed connector consists of a cylindrical spring with an embedded actuator, making its stiffness adjustable. In a case study, a layout for such connectors is suggested to reduce the surge, heave, pitch, and yaw motions of the VLFS in random seas. To control the vibration responses of the VLFS, a mathematical model of the floating structure with the proposed connectors is established. A state feedback control scheme is developed using Sequential Quadratic Programming, which is able to adapt to varying wave conditions. Numerical studies indicate that the control method based on the stiffness-adjustable connectors was able to greatly reduce the responses of the modules when compared to flexible connectors and was also able to reduce the connector loads when compared to hinged connectors. Most importantly, this control method enables the elimination of resonant responses by changing the system stiffness.

1. Introduction

A very large floating structure (VLFS) is a unique concept for an oceanic structure that embraces a range of unprecedented parameters, being thousands of meters long, weighing several tons, and costing between USD 5000~15,000 million, etc., [1]. VLFS could be used as floating fuel storage facilities, floating airports, floating cities, and so on. These prospective applications have aroused enormous enthusiasm in researchers to explore the VLFS-related theories and technologies that have been around since the 1980s [2]. One of the most famous VLFS is the mega-float model, which was built and tested in Tokyo Bay [3,4]. The mega-float model is a flat structure that is usually modeled as a beam or plate, the dynamic characteristics of which were investigated through the use of hydroelastic theory [5,6]. Mega-float models have the merits of easy manufacturing, deployment, and maintenance, but huge bending stress occurs in the structure, making it vulnerable when deployed to rough seas. To overcome this problem, a multi-modular type of VLFS has been considered [7], which consists of a number of floating modules connected by connectors. These VLFS modules could be broadly classified into two types: pontoon types and semi-submersible types [2,8]. The pontoon type is the only type that is suitable for deployment in calm waters [9], and the semi-submersible type can withstand large waves in open seas [10]. For multi-modular VLFS, connectors are the key components to hold the integrity of the VLFS and greatly influence the responses and the stability of multi-modular VLFS.

Since the operation environment of VLFS is prone to changes and is rough, the vibration response of VLFS has become a key concern for its operational stability and security. To depress the response of VLFS, vibration control for VLFS has been considered. There are two fundamental approaches for vibration reduction: passive methods and active methods. Passive methods include employing breakwaters and anti-motion devices. Breakwaters, including bottom-founded breakwaters and floating breakwaters, are usually effective in reducing wave excitation on protected structures [11,12,13] in particular sea conditions but are poor at adapting to changing sea conditions. Some researchers have studied anti-motion devices that are attached to VLFS, such as plates [14], boxes [15] or oscillating water columns (OWC) [16], among others [17,18,19,20,21,22]. All kinds of anti-motion devices can dissipate the wave energy and reduce the responses of a VLFS to a certain extent, but not as efficiently as active ones. In recent years, a semi-active control method [23] was proposed to reduce the vibration of a floating airport by intermittently tuning the stiffness of the connectors based on the amplitude death mechanism. This method can greatly reduce the vibration responses of a VLFS, but it may fail to perform in certain sea conditions. Active control methods are relatively complicated but can efficiently reduce the vibration of a VLFS in varying sea conditions. Pneumatic actuators combined with the PID algorithm have been used to decrease the heave motion of a VLFS [24]. Dynamic position (DP) [25,26,27] is a popular active method that can manipulate and alienate floating bodies into their designated positions by using a group of thrusts. The DP system is in fact a location positioning approach that does not take care of the vibration reduction of an individual floating body. In recent years, researchers attempted to decrease the responses of a VLFS using thrusts [28,29], and encountered tremendous control output requirements, which make it difficult to implement this control method in real engineering practice.

In this paper, a vibration control method is that uses a stiffness-adjustable connectors (briefly called novel connectors when at rest) is proposed for VLFS, which has the benefit of having relatively small output requirements. The novel connector consists of a cylindrical spring with an embedded actuator inside. The cylindrical spring is made of a high-strength polymer material with stiffness in three-translational directions. The actuator is hydraulically driven and is able to provide adaptive control forces to stabilize the floating system according to a control algorithm. Thus, the cylindrical spring is the passive component, and the actuator is the active component. During a control process, the cylindrical spring shares a part of the connection forces with the actuator. The combination of both of the components actually makes the connector stiffness adjustable. A feedback control scheme that is based on the responses of floating bodies is proposed, and the feedback gains are determined by Sequential Quadratic Programming (SQP) [30] with the minimization of spectral VLFS responses in random sea conditions. By using the novel connectors, the resonances of VLFS systems can be eliminated in all wave conditions, and the vibration response can be effectively reduced in random sea conditions with reasonable small control forces.

This paper is organized as follows: In the next section, the model of the stiffness-adjustable novel connector is proposed for a VLFS. Then, a vibration control method is developed based on the mathematical VLFS model and is connected by the novel connectors and Sequential Quadratic Programming to determine feedback gains of the different states and the hydraulic parameters of the actuators. Numerical examples are provided to verify the effectiveness of the vibration control method in different wave conditions. To highlight the performance of the control scheme, a comparison study for a VLFS that is connected using the flexible and hinged connectors [31,32] is conducted. Finally, the results are discussed.

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 $x_i{y}_i{z}_i{\alpha }_i{\beta }_i{\gamma }_i$ 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 $0°$ wave angle is parallel to the $x$ axis, and the 9$0°$ 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 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. The embedded actuator provides an additional axial force to minimize the relative displacement between the two modules by a state-feedback control scheme that is able to adapt to wave conditions. The actuator actually changes the force–displacement relationship of the connector and equivalently alters the flexibility of the connector in an adaptive way to cope with the changes in the wave condition.

The stiffness-adjustable connector is an essential element that can be deployed between two adjacent modules to create a connection in a desirable position and direction. Configuring a number of connectors in different positions and directions can achieve multi-directional vibration control of a VLFS. As the excitation frequency and angle of the waves change, the connection flexibility can be adjusted accordingly using the actuators, which can at least avoid the system resonances resulting from random sea conditions.

The major benefit of introducing these stiffness-adjustable connectors is that it can greatly reduce the required control outputs, which will be illustrated in numerical studies. Although the use of a hydraulic actuator is unusual for VLFS connection, the use of a hydraulic actuator is very popular in different industries. The use of a hydraulic actuator is a sophisticated and common technique that is widely used in heavy-duty machines, where actuation loads can reach thousands of tons [33,34]. The mechanism of combining hydraulic actuators with coil springs has also been broadly utilized in passenger vehicles and for vibration attenuation in heavy machines [35]. We attempt to shift these mature techniques to marine engineering in order to develop a new vibration control technique for VLFS that can significantly enhance the stability of floating systems in seas and that can avoid the occurrence of resonant disasters.

3. Vibration Control Method Based on the Stiffness-Adjustable Connector

A vibration control method is proposed to suppress VLFS motion through adjusting the stiffness of the connectors using actuators. According to different wave conditions, the output forces of the actuators can be determined by the state feedback law. The coefficients of the feedback law, which are as the adjustable stiffness of the system, are derived through an optimizing process by means of the minimization of the spectral response of the VFLS. As such, the mathematical model of the VLFS needs to be formulated first.

3.1. Mathematical Model of the Floating System

Considering a three-dimensional model of a floating system with $N$ modules in a column, the governing equation of each module based on Newton’s law can be expressed as

$$M_i{\ddot{x}}_i+\left(K_{M,i}+R_i\right)x_i=f_{i,w}+f_{i,c}, i=1,2,\cdots ,N$$

(1)

where symbol $x_i={\left[x_i,y_i,z_i,{\alpha }_i,{\beta }_i,{\gamma }_i\right]}^T$ is the displacement vector of the ith module, representing the displacements of surge, sway, heave, roll, pitch, and yaw, respectively. It also indicates the displacement of the local frame in the global frame. On the left-hand side of Equation (1), symbol $K_{M,i}$ is the stiffness matrix of the mooring system, which is simplified as a constant matrix. Symbols $M_i\in {ℝ}^{6\times 6}$ and $R_i\in {ℝ}^{6\times 6}$ are the mass matrix and the buoyant restoring stiffness matrix of water, respectively, which is expressed by

$$\begin{array}{c}{\mathbf{M}}_i=\left[\begin{array}{cccccc}m& 0& 0& 0& 0& 0\\ 0& m& 0& 0& 0& 0\\ 0& 0& m& 0& 0& 0\\ 0& 0& 0& I_x& 0& 0\\ 0& 0& 0& 0& I_y& 0\\ 0& 0& 0& 0& 0& I_z\end{array}\right]\hfill \\ {\mathbf{R}}_i=-\rho g\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& -A_w& 0& {\int \int }_{\mathit{Aw}}xdA& 0\\ 0& 0& 0& -\left(V_w\left(z_b-z_g\right)+{\int \int }_{\mathit{Aw}}{y}^{2}dA\right)& 0& 0\\ 0& 0& {\int \int }_{\mathit{Aw}}xdA& 0& -\left(V_w\left(z_b-z_g\right)+{\int \int }_{\mathit{Aw}}{x}^{2}dA\right)& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]\hfill \end{array}$$

(2)

where symbol $m$ is the mass of the ith module. The symbols $I_x$, $I_y,$ and $I_z$ are the rotary inertias for the $x_i$, $y_i,$ and $z_i$ axes, respectively. Symbol $\rho$ is the density of water, and $g$ is the acceleration of gravity. Symbol $A_w$ is the water plane area, and $V_w$ is the volume of the water. Symbol $z_b$ is the design draft, and $z_g$ is the height of the center of gravity. On the right-hand side of Equation (1), symbol $f_{i,w}$ expresses the wave excitation vector determined by the linear wave theory [36]. Symbol $f_{i,c}$ indicates the connector force vector acting on the ith module. Vectors $f_{i,w}$ and $f_{i,c}$ are derived as discussed in the paragraphs below.

3.1.1. Hydrodynamic Model

Linear wave theory [36] is used to determine the wave excitation force based on the assumption that the water is irrotational and inviscid. For a wave frequency $\omega$, the total wave potential can be expressed as

$$\mathrm{\Phi }=\mathrm{Re}\left(\psi e^{-j\omega t}\right)$$

(3)

where $j=\sqrt{-1}$. Symbol $\psi$ is the space velocity potential. For a N-modular VLFS such as the one shown in Figure 1, the expression of $\psi$ is

$$\psi ={\psi }_I+{\psi }_D+j\omega {\displaystyle \sum _{i=1}^N{\left({\psi }_i\right)}^T{\tilde{X}}_i}, i=1,2,\cdots N$$

(4)

where ${\psi }_I$ is incident potential, and ${\psi }_D$ is diffraction potential. Vector ${\psi }_i={⎡{\psi }_i{}^1,\cdots ,{\psi }_i{}^6⎤}^T$ is the potential produced by the unit amplitude of the ith module in each degree of freedom. Vector ${\tilde{\mathit{x}}}_i$ is the amplitude vector of the responses of the ith module.

The incident potential ${\psi }_I$ is associated with wave condition, which can be written as

$${\psi }_I=\frac{jga}{\omega }\frac{\cosh k\left(z+h\right)}{\cosh \left(kh\right)}{e}^{jk\left(x\cos \theta +y\sin \theta \right)}$$

(5)

where $a$ is the wave amplitude, and $h$ is the water depth. Symbol $k$ is the number of waves that can be determined by the wave frequency $\omega$ and water depth $h$.

The diffraction potential ${\psi }_D$ and the potential ${\psi }_i$ can be solved by the Laplace equation along with the conditions of the free water surface, including the surface boundary of the modules, sea bottom, and infinity radiation. Generally, the partial differential equations and boundary conditions for ${\psi }_D$ and ${\psi }_i$ are expressed as

$$\left\{\begin{array}{l} {\nabla }^2{\psi }_D=0\\ \frac{\partial {\psi }_D}{\partial z}-\frac{{\omega }^2}{g}=0\\ \frac{\partial {\psi }_D}{\partial n}=-\frac{\partial {\psi }_I}{\partial n}\\ {\left.\frac{\partial {\psi }_D}{\partial z}\right|}_{z=-h}=0\\ \underset{r\to \infty }{\lim }\sqrt{r}\left(\frac{\partial {\psi }_D}{\partial r}-j\frac{{\omega }^2}{g}{\psi }_D\right)=0\end{array}\right.\left\{\begin{array}{l}{\nabla }^2{\psi }_i{}^q=0\\ \frac{\partial {\psi }_i}{\partial z}-\frac{{\omega }^2}{g}{\psi }_i=0\\ {\left.\frac{\partial {\psi }_i}{\partial n}\right|}_{S_i}=n_i\\ {\left.\frac{\partial {\psi }_i}{\partial n}\right|}_{S_m}=0\left(i\ne m\right)\\ {\left.\frac{\partial {\psi }_i}{\partial z}\right|}_{z=-h}=0\\ \underset{r\to \infty }{\lim }\sqrt{r}\left(\frac{\partial {\psi }_i}{\partial r}-j\frac{{\omega }^2}{g}{\psi }_i\right)=0\end{array}\right.$$

(6)

where vector $n_i={\left[n_i{}^1,\cdots ,n_i{}^6\right]}^T$ is the projection component in each normal direction. Symbol $r$ is the displacement between the field point and source point.

By using HydroD, the velocity potential is determined through the Laplace equation and the boundary conditions. Then, using Bernoulli’s equation, the wave excitation amplitude acting on the ith module can be obtained by integrating the wave potential along the module’s wet surface, which is expressed as

$${\overline{f}}_{i,w}^{}={\displaystyle {\iint }_{S_i}\left[-j\omega \rho \left({\psi }_I+{\psi }_D+j\omega {\displaystyle \sum _{q=1}^N{\left({\psi }_q\right)}^T{\tilde{x}}_q}\right)\right]}{n}_i^{} \mathrm{d}s$$

(7)

As such, the wave excitation can be written as

$$\begin{array}{cc}\hfill {\mathbf{f}}_{i,w}^{}={\overline{\mathbf{f}}}_{i,w}^{}{e}^{-j\omega t}& =\left({\int \int }_{\mathrm{Si}}\left[-j\omega \rho \left({\psi }_I+{\psi }_D+j\omega {\displaystyle \sum _{q=1}^N}{\left({\psi }_q\right)}^T{\tilde{\mathbf{x}}}_q\right)\right]{\mathbf{n}}_i^{}ds\right)e^{-j\omega t}\hfill \\ & =\left(-j\omega \rho {\int \int }_{\mathrm{Si}}\left({\psi }_I+{\psi }_D\right){\mathbf{n}}_i^{}ds\right)e^{-j\omega t}\hfill \\ & -{\displaystyle \sum _{q=1}^N}\left(\rho {\int \int }_{\mathrm{Si}}Re\left({\psi }_q\right){\mathbf{n}}_i^{T}ds\right)\left(-{\omega }^2{\tilde{\mathbf{x}}}_q{e}^{-j\omega t}\right)\hfill \\ & -{\displaystyle \sum _{q=1}^N}\left(\rho \omega {\int \int }_{\mathrm{Si}}Im\left({\psi }_q\right){\mathbf{n}}_i^{T}ds\right)\left(-j\omega {\tilde{\mathbf{x}}}_q{e}^{-j\omega t}\right)\hfill \end{array}$$

(8)

The harmonic response of the ith module is expressed by

$$\begin{array}{l}{x}_i^{}={\tilde{x}}^{}{}_i^{}{e}^{-j\omega t}\\ {\dot{x}}^{}{}_i^{}=-j\omega {\tilde{x}}_i^{}{e}^{-j\omega t}\\ {\ddot{x}}^{}{}_i^{}=-{\omega }^2{\tilde{x}}_i^{}{e}^{-j\omega t}\end{array}$$

(9)

If we substitute Equation (9) into Equation (8), we arrive at

$$f_{i,w}={\overline{f}}_{i,w}{e}^{-j\omega t}-{\displaystyle {\displaystyle \sum }_{q=1}^N}\left(\Delta M_{i,q}{\ddot{x}}_q+\Delta C_{i,q}{\dot{x}}_q\right)$$

(10)

where

$$\begin{array}{l}{\overline{f}}_{i,w}^{}=-j\omega \rho {\displaystyle {\iint }_{S_i}\left({\psi }_I+{\psi }_D\right)n_i^{} \mathrm{d}s}\\ \Delta M_{i,q}=\rho {\displaystyle {\iint }_{S_i}\mathrm{Re}\left({\psi }_q\right)n_i^T \mathrm{d}s}\\ \Delta C_{i,q}=\rho \omega {\displaystyle {\iint }_{S_i}\mathrm{Im}\left({\psi }_q\right)n_i^T \mathrm{d}s}\end{array}$$

(11)

In Equation (11), the first term is the incident wave force, the second term is the added mass, and the third one is the added damping.

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 $f_{i,c}$ denotes the connector force imposed on the ith module, which can be expressed as

$$f_{i,c}=f_{i,s}+f_{i,a}$$

(12)

where vector $f_{i,s}$ is the spring force vector, and $f_{i,a}$ 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 qth novel connector in the pth groups, the position of the mounting point at the pth module is ${\overline{\mathbf{d}}}_p^q=\left[{\delta }_{1,x}^{q,p},{\delta }_{1,y}^{q,p},{\delta }_{1,z}^{q,p}\right]$, and the position of the mounting point at the (p+1)th module is ${\overline{\mathbf{d}}}_{p+1}^q=\left[{\delta }_{2,x}^{q,p},{\delta }_{2,y}^{q,p},{\delta }_{2,z}^{q,p}\right]$ in their local frames. Due to the movement of the two connected modules, the displacements of the two mounting points in the global frame ${\mathsf{\mu }}_1^{q,p}$ and ${\mathsf{\mu }}_2^{q,p}$ can be expressed as [25]

$$\begin{array}{c}{\mathsf{\mu }}_1^{q,p}={\mathbf{W}}_1^{q,p}·{\mathbf{x}}_p=\left[\begin{array}{cccccc}1& 0& 0& 0& {\delta }_{1,z}^{q,p}& -{\delta }_{1,y}^{q,p}\\ 0& 1& 0& -{\delta }_{1,z}^{q,p}& 0& {\delta }_{1,x}^{q,p}\\ 0& 0& 1& {\delta }_{1,y}^{q,p}& -{\delta }_{1,x}^{q,p}& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]·{\mathbf{x}}_p\\ {\mathsf{\mu }}_2^{q,p}={\mathbf{W}}_2^{q,p}·{\mathbf{x}}_{p+1}=\left[\begin{array}{cccccc}1& 0& 0& 0& {\delta }_{2,z}^{q,p}& -{\delta }_{2,y}^{q,p}\\ 0& 1& 0& -{\delta }_{2,z}^{q,p}& 0& {\delta }_{2,x}^{q,p}\\ 0& 0& 1& {\delta }_{2,y}^{q,p}& -{\delta }_{2,x}^{q,p}& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]·{\mathbf{x}}_{p+1}\end{array}\hspace{1em}q=1,\cdots Nc;\hspace{1em}p=1,\cdots N-1$$

(13)

where $W_1^{q,p}$ and $W_2^{q,p}$ 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

$$f_s{}^{q,p}=K_s\left({\mu }_2^{q,p}-{\mu }_1^{q,p}\right)=-\left[\begin{array}{cccccc}{k}_a& 0& 0& 0& 0& 0\\ 0& k_d& 0& 0& 0& 0\\ 0& 0& k_d& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{cc}{W}_1^{q,p}& -W_2^{q,p}\end{array}\right]{\left[\begin{array}{cc}{x}_p{}^T& x_{p+1}{}^T\end{array}\right]}^T$$

(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 pth module and the (p + 1)th module are

$$\begin{array}{l}{f}_{s,1}^{q,p}={\left(W_1^{q,p}\right)}^T{f}_s{}^{q,p}=-{\left(W_1^{q,p}\right)}^T{K}_s\left[\begin{array}{cc}{W}_1^{q,p}& -W_2^{q,p}\end{array}\right]{\left[\begin{array}{cc}{x}_p{}^T& x_{p+1}{}^T\end{array}\right]}^T\\ f_{s,2}^{q,p}=-{\left(W_2^{q,p}\right)}^T{f}_s{}^{q,p}={\left(W_2^{q,p}\right)}^T{K}_s\left[\begin{array}{cc}{W}_1^{q,p}& -W_2^{q,p}\end{array}\right]{\left[\begin{array}{cc}{x}_p{}^T& x_{p+1}{}^T\end{array}\right]}^T\end{array}$$

(15)

Then, the force term $f_{i,s}$ in Equation (12) is the summation of all of the individual connector forces in Equation (15), which can be written as

$${\mathbf{f}}_{i,s}=\left\{\begin{array}{cc}{\displaystyle \sum _{q=1}^{Nc}}{\mathbf{f}}_{s,1}^{q,i}={\displaystyle \sum _{q=1}^{Nc}}-{\left({\mathbf{W}}_1^{q,i}\right)}^T{\mathbf{K}}_s\left[\begin{array}{cc}{\mathbf{W}}_1^{q,i}& -{\mathbf{W}}_2^{q,i}\end{array}\right]{\left[\begin{array}{cc}{{\mathbf{x}}_i}^T& {{\mathbf{x}}_{i+1}}^T\end{array}\right]}^T,\hfill & i=1\hfill \\ {\displaystyle \sum _{q=1}^{Nc}}{\mathbf{f}}_{s,2}^{q,i-1}+{\mathbf{f}}_{s,1}^{q,i}={\displaystyle \sum _{q=1}^{Nc}}\left(\begin{array}{c}{\left({\mathbf{W}}_2^{q,i-1}\right)}^T{\mathbf{K}}_s\left[\begin{array}{cc}{\mathbf{W}}_1^{q,i-1}& -{\mathbf{W}}_2^{q,i-1}\end{array}\right]{\left[\begin{array}{cc}{{\mathbf{x}}_{i-1}}^T& {{\mathbf{x}}_i}^T\end{array}\right]}^T\hfill \\ -{\left({\mathbf{W}}_1^{q,i}\right)}^T{\mathbf{K}}_s\left[\begin{array}{cc}{\mathbf{W}}_1^{q,i}& -{\mathbf{W}}_2^{q,i}\end{array}\right]{\left[\begin{array}{cc}{{\mathbf{x}}_i}^T& {{\mathbf{x}}_{i+1}}^T\end{array}\right]}^T\hfill \end{array}\right),\hfill & i=2\cdots N-1\hfill \\ {\displaystyle \sum _{q=1}^{Nc}}{\mathbf{f}}_{s,2}^{q,i-1}={\displaystyle \sum _{q=1}^{Nc}}{\left({\mathbf{W}}_2^{q,i-1}\right)}^T{\mathbf{K}}_s\left[\begin{array}{cc}{\mathbf{W}}_1^{q,i-1}& -{\mathbf{W}}_2^{q,i-1}\end{array}\right]{\left[\begin{array}{cc}{{\mathbf{x}}_{i-1}}^T& {{\mathbf{x}}_i}^T\end{array}\right]}^T,\hfill & i=N\hfill \end{array}\right.$$

(16)

For an actuator, the output force of the actuator of the qth 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 ${\lambda }_m^{q,p}\in {ℝ}^{1\times 6}$ denote the control gain vector for the mth module state $x_m$. The actuator force of the qth connector in the pth group is the summation of all of the control gain vectors multiplied with the module states, which can be expressed as

$$f_a{}^{q,p}=-{\displaystyle \sum _{m=1}^N{\lambda }_m{}^{q,p}\cdot x_m}$$

(17)

This actuator force is applied at the mounting positions ${\overline{\mathbf{d}}}_p^q=\left[{\delta }_{1,x}^{q,p},{\delta }_{1,y}^{q,p},{\delta }_{1,z}^{q,p}\right]$ and ${\overline{\mathbf{d}}}_{p+1}^q=\left[{\delta }_{2,x}^{q,p},{\delta }_{2,y}^{q,p},{\delta }_{2,z}^{q,p}\right]$ 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 pth 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 ${\overline{p}}_p$ and ${\overline{p}}_{p+1}$ denote the initial positions of the mass centers of the pth module and the (p + 1)th module in the global coordinate system. After the modules move, the mass centers of the pth and the (p + 1)th modules are relocated to $P_p$ and $P_{p+1}$, and the mounting points of the qth connector are relocated to $d_p^q$ and $d_{p+1}^q$ in the global coordinate system, which can be expressed as

$$\begin{array}{l}{p}_p={\overline{p}}_p +{\left[x_p,y_p,z_p\right]}^T\\ p_{p+1}={\overline{p}}_{p+1} +{\left[x_{p+1},y_{p+1},z_{p+1}\right]}^T\\ d_p^q={\overline{p}}_p+tran\left({\alpha }_p,{\beta }_p,{\gamma }_p\right)\cdot {\overline{d}}_p^q\\ d_{p+1}^q={\overline{p}}_{p+1}+tran\left({\alpha }_{p+1},{\beta }_{p+1},{\gamma }_{p+1}\right)\cdot {\overline{d}}_{p+1}^q\end{array}$$

(18)

where tran(g) is the rotation matrix [37], which can be written as

$$tran\left(\alpha ,\beta ,\gamma \right)=\left[\begin{array}{ccc}\cos \gamma \cos \beta & -\sin \gamma \cos \alpha +\cos \gamma \sin \beta \sin \alpha & \sin \gamma \sin \alpha +\cos \gamma \sin \beta \cos \alpha \\ \sin \gamma \cos \beta & \cos \beta \cos \alpha +\sin \gamma \sin \beta \sin \alpha & -\cos \gamma \sin \alpha +\sin \gamma \sin \beta \cos \alpha \\ -\sin \beta & \cos \beta \sin \alpha & \cos \beta \cos \alpha \end{array}\right]$$

(19)

Thus, the actuator force of the qth actuator in the pth group in Equation (17) can be projected onto each degree of freedom. The control force of the qth actuator vector acting on the mass center of the pth module can be expressed as [38]

$${\overline{f}}_a{}^{q,p}=f_a{}^{q,p}\cdot \left[\begin{array}{l}\frac{d_{p+1}^q-d_p^q}{‖d_{p+1}^q-d_p^q‖}\\ \frac{d_{p+1}^q-d_p^q}{‖d_{p+1}^q-d_p^q‖}\times \left(d_p^q-p_p\right)\end{array}\right] q=1,\cdots Nc; p=1,\cdots N-1$$

(20)

where the first three components of the control force vector are the translational forces, and last three components are the moments.

It should be noted that the control force vector in Equation (20) contains geometric nonlinearity due to module rotation. If the rotation angles are small, then the control force can be treated approximately by linearization at the equilibrium point. Let $f_{a,1}^{q,p}$ denote the linearized control force vector acting at the mass center of the pth module, and similarly $f_{a,2}^{q,p}$ for the (p + 1)th module. The linearized control force vectors are respectively expressed as

$$\begin{array}{l}{f}_{a,1}^{q,p}={\stackrel{-}{H}}_{}^{q,p}{\left[\begin{array}{cccc}{x}_1{}^T& x_2{}^T& \cdots & x_N{}^T\end{array}\right]}^T\\ f_{a,2}^{q,p}=-{\stackrel{-}{H}}_{}^{q,p}{\left[\begin{array}{cccc}{x}_1{}^T& x_2{}^T& \cdots & x_N{}^T\end{array}\right]}^T\end{array}$$

(21)

The symbol of ${\stackrel{-}{H}}^{q,p}\in {ℝ}^{6\times 6N}$ is the linearized matrix, which is given by

$${\stackrel{-}{H}}_{}^{q,p}=\frac{1}{\delta }\left[\begin{array}{c}\left({\delta }_{1,x}^{q,p}-{\delta }_{2,x}^{q,p}-△x\right)\\ \left({\delta }_{1,y}^{q,p}-{\delta }_{2,y}^{q,p}\right)\\ \left({\delta }_{1,z}^{q,p}-{\delta }_{2,z}^{q,p}\right)\\ \left({\delta }_{2,y}^{q,p}\cdot {\delta }_{1,z}^{q,p}-{\delta }_{1,y}^{q,p}\cdot {\delta }_{2,z}^{q,p}\right)\\ \left({\delta }_{1,x}^{q,p}\cdot {\delta }_{2,z}^{q,p}-{\delta }_{2,x}^{q,p}\cdot {\delta }_{1,z}^{q,p}-△x\cdot {\delta }_{1,z}^{q,p}\right)\\ \left(△x\cdot {\delta }_{1,y}^{q,p}-{\delta }_{1,x}^{q,p}\cdot {\delta }_{2,y}^{q,p}+{\delta }_{2,x}^{q,p}\cdot {\delta }_{1,y}^{q,p}\right)\end{array}\right]\cdot \left[{\mathsf{\lambda }}_1{}^{q,p},{\mathsf{\lambda }}_2{}^{q,p},\cdots ,{\mathsf{\lambda }}_N{}^{q,p}\right]$$

(22)

where $\delta =\sqrt{{\left(△x-{\delta }_{1,x}^{q,p}+{\delta }_{2,x}^{q,p}\right)}^2+{\left({\delta }_{1,y}^{q,p}-{\delta }_{2,y}^{q,p}\right)}^2+{\left({\delta }_{1,z}^{q,p}-{\delta }_{2,z}^{q,p}\right)}^2}$, and the symbol $△x$ is the initial horizontal distance between the mass centers of the pth module and the (p + 1)th module.

Assume that the deployment of the actuators in all of the connection groups is the same. Then, the total control force vector $f_{i,a}$ in Equation (12) from all of the related actuators imposed on the ith module can be written as

$${\mathbf{f}}_{i,a}=\left\{\begin{array}{c}{\displaystyle \sum _{q=1}^{Nc}}{\mathbf{f}}_{a,1}^{q,i}={\displaystyle \sum _{q=1}^{Nc}}{\overline{\mathbf{H}}}_{}^{q,i}{\left[{{\mathbf{x}}_1}^T,{{\mathbf{x}}_2}^T,\cdots ,{{\mathbf{x}}_N}^T\right]}^T, i=1\hfill \\ {\displaystyle \sum _{q=1}^{Nc}}\left({\mathbf{f}}_{a,2}^{q,i-1}+{\mathbf{f}}_{a,1}^{q,i}\right)={\displaystyle \sum _{q=1}^{Nc}}\left(\begin{array}{c}-{\overline{\mathbf{H}}}_{}^{q,i-1}{\left[{{\mathbf{x}}_1}^T,{{\mathbf{x}}_2}^T,\cdots ,{{\mathbf{x}}_N}^T\right]}^T+\hfill \\ {\overline{\mathbf{H}}}_{}^{q,i}{\left[{{\mathbf{x}}_1}^T, ,{{\mathbf{x}}_2}^T,\cdots ,{{\mathbf{x}}_N}^T\right]}^T\end{array}\right), i=2\cdots N-1\hfill \\ {\displaystyle \sum _{q=1}^{Nc}}{\mathbf{f}}_{a,2}^{q,i-1}={\displaystyle \sum _{q=1}^{Nc}}-{\overline{\mathbf{H}}}_{}^{q,i-1}{\left[{{\mathbf{x}}_1}^T,{{\mathbf{x}}_2}^T,\cdots ,{{\mathbf{x}}_N}^T\right]}^T, i=N\hfill \end{array}\right.$$

(23)

3.2. Feedback Control Gains and the Parameters of Hydraulic-Actuators

The control flow chart is shown in Figure 4. With the wave parameters, the feedback gains $\left[{\lambda }_1^{q,p},{\lambda }_2^{q,p},\cdots ,{\lambda }_N^{q,p}\right]$ are obtained by using Sequential Quadratic Programming (SQP), where the optimization goal is the minimization of the 1–in–1000 amplitude of the extreme responses of all modules in the intended directions. Using the feedback gains and the VLFS states, the outputs of the actuators can be obtained, which are the forces imposed on the VLFS to suppress the vibration of the system.

Substituting Equation (23), Equation (16), Equation (12), and Equation (11) into Equation (1), the mathematical model of the VLFS can be rewritten in the simple form

$$M\ddot{x}+C\dot{x}+Kx={\overline{f}}_{i,w}^{}{e}^{-j\omega t}$$

(24)

where the matrices are

$$\begin{array}{cc}\hfill & \mathbf{M}=\left[\begin{array}{cccc}{\mathbf{M}}_1+\Delta {\mathbf{M}}_{1,1}& \Delta {\mathbf{M}}_{1,2}& \cdots & \Delta {\mathbf{M}}_{1,N}\\ \Delta {\mathbf{M}}_{2,1}& {\mathbf{M}}_2+\Delta {\mathbf{M}}_{2,2}& \cdots & \Delta {\mathbf{M}}_{2,N}\\ ⋮& ⋮& \ddots & ⋮\\ \Delta {\mathbf{M}}_{N,1}& \Delta {\mathbf{M}}_{N,2}& \cdots & {\mathbf{M}}_N+\Delta {\mathbf{M}}_{N,N}\end{array}\right]\hfill \\ \hfill & \mathbf{C}=\left[\begin{array}{cccc}\Delta {\mathbf{C}}_{1,1}& \Delta {\mathbf{C}}_{1,2}& \cdots & \Delta {\mathbf{C}}_{1,N}\\ \Delta {\mathbf{C}}_{2,1}& \Delta {\mathbf{C}}_{2,2}& \cdots & \Delta {\mathbf{C}}_{2,N}\\ ⋮& ⋮& \ddots & ⋮\\ \Delta {\mathbf{C}}_{N,1}& \Delta {\mathbf{C}}_{N,2}& \cdots & \Delta {\mathbf{C}}_{N,N}\end{array}\right]\hfill \\ \hfill & \mathbf{K}=\left[\begin{array}{cccc}{\mathbf{R}}_1& & & \\ & {\mathbf{R}}_2& & \\ & & \ddots & \\ & & & {\mathbf{R}}_N\end{array}\right]+\left[\begin{array}{cccc}{{\mathbf{K}}_{M,}}_1& & & \\ & {{\mathbf{K}}_{M,}}_2& & \\ & & \ddots & \\ & & & {{\mathbf{K}}_{M,}}_N\end{array}\right]+{\displaystyle \sum _{q=1}^{Nc}}\left[\begin{array}{c}{\overline{\mathbf{H}}}_{}^{q,1}\\ -{\overline{\mathbf{H}}}_{}^{q,1}+{\overline{\mathbf{H}}}_{}^{q,2}\\ ⋮\\ -{\overline{\mathbf{H}}}_{}^{q,N-2}+{\overline{\mathbf{H}}}_{}^{q,N-1}\\ -{\overline{\mathbf{H}}}_{}^{q,N-1}\end{array}\right]\hfill \\ \hfill & +{\displaystyle \sum _{q=1}^{Nc}}\left[\begin{array}{ccccc}{\left({\mathbf{W}}_1^{q,1}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_1^{q,1}& -{\left({\mathbf{W}}_1^{q,1}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_2^{q,1}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ -{\left({\mathbf{W}}_2^{q,1}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_1^{q,1}& \left(\begin{array}{cc}\hfill & {\left({\mathbf{W}}_1^{q,2}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_1^{q,2}\hfill \\ \hfill & +{\left({\mathbf{W}}_2^{q,1}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_2^{q,1}\hfill \end{array}\right)& -{\left({\mathbf{W}}_1^{q,2}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_2^{q,2}& \mathbf{0}& \mathbf{0}\\ & \ddots & \ddots & \ddots & \\ \mathbf{0}& \mathbf{0}& -{\left({\mathbf{W}}_2^{q,N-2}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_1^{q,N-2}& \left(\begin{array}{cc}\hfill & {\left({\mathbf{W}}_1^{q,N-1}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_1^{q,N-1}\hfill \\ \hfill & +{\left({\mathbf{W}}_2^{q,N-2}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_2^{q,N-2}\hfill \end{array}\right)& -{\left({\mathbf{W}}_1^{q,N-1}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_2^{q,N-1}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& -{\left({\mathbf{W}}_2^{q,N-1}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_2^{q,N-1}& {\left({\mathbf{W}}_2^{q,N-1}\right)}^T{\mathbf{K}}_s{\mathbf{W}}_2^{q,N-1}\end{array}\right]\hfill \end{array}$$

(25)

From the matrix $K$ in Equation (25), we can see that the control effects are actually converted to a stiffness matrix expression. In view of the motion equation of the VLFS in Equation (24), the actuators in the connectors actually change the system stiffness and are adjustable according to sea conditions. This is why we call the novel connectors stiffness-adjustable connectors.

For regular wave excitation at $\omega$ with the unit of wave height, the module response can be expressed as

$$A\left(\omega \right)={\left(-{\omega }^{2}M+j\omega C+K\right)}^{-1}·{\overline{f}}_{i,w}$$

(26)

For a wave spectrum $\mathrm{S}\left(\omega \right)$ 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]

$$R_e=3.72\sqrt{{{\displaystyle \int }}_0^{\infty }A\left(\omega \right) \circ A\left(\omega \right)\mathrm{S}\left(\omega \right)d\omega }$$

(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

$$\begin{array}{l}\min J={{\displaystyle \sum }}_{i=1}^N{{\displaystyle \sum }}_{k=1}^6{W}_k{R}_e^{\left(i-1\right)\times 6+k}\\ s.t. \Vert f_a^{q,p}\Vert \le f_{max} q=1,\cdots ,Nc; p=1,\cdots N-1\\ M\ddot{x}+C\dot{x}+Kx={\overline{f}}_{i,w}{e}^{-j\omega t}\end{array}$$

(28)

to determine the feedback gains, where $R_e{}^q$ indicates the qth element of the vector $R_e$. Symbol ${\kappa }_k$ is the weight of the motion in the kth degree of freedom. If the motion in the kth 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 Newton’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 $\left(N-1\right)\times Nc\times 6N$ 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 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

$$f_a{}^{q,p}={\eta }_m\left[p_l\frac{\pi D^2}{4}-p_r\left(\frac{\pi D^2}{4}-\frac{\pi d^2}{4}\right)\right]$$

(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

$$\begin{array}{l}{p}_l\frac{\pi D^2}{4}-f_a{}^{q,p}-p_r\left(\frac{\pi D^2}{4}-\frac{\pi d^2}{4}\right)=\rho p_l\left(v_0-\frac{d_l{}^2}{D^2}{v}_l\right)\\ p_r\left(\frac{\pi D^2}{4}-\frac{\pi d^2}{4}\right)+f_a{}^{q,p}-p_l\frac{\pi D^2}{4}=-\rho p_r\left(v_0-\frac{d_r{}^2}{D^2-d^2}{v}_r\right)\end{array}$$

(30)

where $v_0$ is the velocity of the piston rod, and $\rho$ is the density of the hydraulic fluid. From Equation (30), we can derive the relationship between the velocities of the hydraulic fluid flowing through the oil orifices and the pressures in the two chambers, which is

$$\frac{d_l{}^2}{D^2}{p}_l{v}_l-\frac{d_r{}^2}{D^2-d^2}{p}_r{v}_r=0$$

(31)

From Equations (29) and (31), the proper chamber pressures for the required control force of the actuator can be determined.

As for the vibration control method using the proposed connector, the actual stiffness is in fact adjustable because of the control force of the hydraulic actuator. As indicated in Equation (25), the last term of $K$ is the stiffness contribution from the actuators, which is a state-dependent matrix that is associated with the feedback gains adapted to the wave conditions. This term changes the flexibility of the connectors and thus makes the whole floating system avoid the resonances in critical wave frequencies.

The control process can be summarized as follows: When the wave conditions change, such as the wave excitation angle and wave frequency, the corresponding control gains can be obtained using the SQP optimization process based on the feedback states of the floating structure. With the required control forces, the control parameters of the hydraulic actuators can be determined by Equations (29) and (31) to exercise the control. Since the feedback gains are adaptive to the wave conditions, the control scheme can keep the floating system away from resonant vibrations. Since the present work does not involve the prototype design, we cannot provide specific dimensions and parameters for the actuator in the numerical analysis.

4. Numerical Assessment on the Performance

In this section, numerical simulations are carried out to assess the performance of the proposed vibration control method. For comparison, connectors with/without a control effect are studied in terms of the frequency responses of a floating system five modules. The frequency response represents the characteristic of the 1–in–1000 amplitude of the extreme responses of the modules in a frequency domain. It describes the transmissibility from the wave excitation to the module response with the unit wave height. The stiffness-adjustable connector is marked as the novel model (named “novel”) in order to compare it to two other types of connectors. One of these types is the flexible connector without the actuator involvement (named “flexible”). The other is the hinged connector (named “hinged”) [32], which have been used by many scholars. The flexible connectors are mounted in the same positions, and the VLFS modules are also the same for the three types of connectors.

4.1. Parameters of the VLFS and the Random Sea Condition

The wave spectrum represents the energy distribution of the random waves in the frequency domain, which usually represents a semi-empirical relationship formularized from observed wave datum such as wave height, excitation angle, etc. In random sea conditions, individual waves may be variant and irregular, but the statistic characteristics of random waves can be described by a wave spectrum. In this paper, the Jonswap spectrum [39] is used for the numerical studies, which can be expressed as

$$\mathrm{S}\left(\omega \right)={\alpha }_s{g}^2{\omega }^{-5}\exp \left(-1.25{\omega }_s{}^4{\omega }^{-4}\right){\gamma }_s{}^{\exp \left(-\frac{{\left(\omega -{\omega }_s\right)}^2}{2{\sigma }^2{\omega }_s{}^2}\right)}$$

(32)

where

$$\begin{gathered} {\alpha }_s=\frac{5}{16}\left(\frac{A_s{}^2{\omega }_s{}^4}{g^2}\right)\left[1-0.278\ln \left({\gamma }_s\right)\right] \\ \sigma =\left\{\begin{array}{c}0.07 \omega \le {\omega }_s\\ 0.09 \omega >{\omega }_s\end{array}\right. \end{gathered}$$

(33)

In Equations (32) and (33), symbols $A_s$, ${\omega }_s,$ and ${\gamma }_s$ are the significant wave height, the peak wave frequency, and the peak enhancement factor, respectively. In numerical simulations, we take $A_s=5\mathrm{m}$, ${\omega }_s=2\pi /10 \mathrm{rad}/\mathrm{s}$, and ${\gamma }_s=3.3$. The wave depth is 300 m. The number of modules of the VLFS is N = 5, and each module is the same [38], and the parameters can be found in

Table 1. The parameters of the VLFS.

	Length		Breadth		Depth
upper hull	300 m		100 m		6 m
lower hull	30 m		96 m		5 m
column	diameter	length	distance in x-axis		distance in y-axis
	18 m	16 m	58 m		60 m
others	$z_g$		$I_x$	$I_y$	$I_z$
	12 m		$1.09\times {10}^{11}$$\mathrm{kg}·{\mathrm{m}}^2$	$7.28\times {10}^{11}$$\mathrm{kg}·{\mathrm{m}}^2$	$7.28\times {10}^{11}$$\mathrm{kg}·{\mathrm{m}}^2$
	design draft		displacement	$K_{M,1}$ and $K_{M,5}$
	12 m		$9.36\times {10}^7$ kg	${10}^5\times \mathrm{diag}\left(\left[5,5,1.5,3.5\times {10}^3,3.5\times {10}^4,1\times {10}^5\right]\right)$
	The horizontal gap between two modules is 5 m

. It is worth noting that only the first module and the last module are connected with mooring chains. The units of the first three moor stiffness elements, $K_{M,1}$ and $K_{M,5}$, are N/m, and the units of the last three elements are $\mathrm{N}·\mathrm{m}/\mathrm{rad}$.

Between two adjacent modules, there are four novel connectors that are mounted horizontally in parallel at the four corners of the modules, and another one is vertically installed in the middle, as illustrated in Figure 3. The four connectors at the corners mainly suppress the surge, pitch, and yaw motions, and the middle connector can suppress the heave motion of the modules. This connector configuration was specifically designed to suppress the four unwanted motions in this paper. It should be remarked there are many possible layouts for the novel stiffness-adjustable novel connectors depending on designer preference and on motion control. In numerical simulations, the coordinates of the mounting points of the five connectors in the local frames are shown in

Table 2. Local coordinates of the novel connectors.

Group i	Module i $\left({\overline{d}}_{\mathit{i}}^{\mathit{q}}\right)$	Module i + 1 $\left({\overline{d}}_{\mathit{i}+1}^{\mathit{q}}\right)$
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}

. All of the novel connectors are identical. Each individual connector has the same radial and axial stiffness of ${10}^7 \mathrm{N}/\mathrm{m}$, as used in [38].

4.2. 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 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 $0°$). 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. It is interesting to see that the controlled motions of all of the modules in a specific direction are at almost the same level, which implies that the control scheme realizes the equal weights assigned to the motions of different modules. For the unintended suppression directions of the sway and roll motions shown in Figure 6b,d, the novel connectors with control significantly outperform the flexible connectors because of the couple between the different degrees of freedom.

The performance of the vibration control method with the stiffness-adjustable connectors can be evaluated based on individual wave frequency. The results in Figure 6 come from the compound results calculated by Equation (26) and by the wave spectrum. Figure 7 only shows the intended motions of surge, heave, pitch, and yaw for the first module over the span of the wave frequency, based on Equation (26). For the surge motion, as shown in Figure 7a, the response of the novel connector is close to that of the hinged connector and is far smaller than that of the flexible connector. For the heave and pitch motions shown in Figure 7b,c, the novel connector (the blue curve) outperforms the other two connectors since the actuator of the novel connector can eliminate or reduce resonance near wave frequency of 0.628 rad/s and 0.9 rad/s by changing the connector stiffness. In the yaw response shown in Figure 7d, the rigid connector has lowest response because of its rigidity. The novel connector greatly reduces the peak of the flexible connector (red line) at 0.9 rad/s, but the novel connector (blue line) has a peak in the very low frequency below 0.25 rad/s. We can state that at low frequency, this peak at low is in fact very small and ignorable, in view of the quantity order of the yaw response. Note that the wave energy is mainly distributed at around 0.628 rad/s. The contribution of the responses from lower frequencies would be insignificant to the 1–in–1000 amplitude.

Next, we present the scenario where the incident angle was $45°$. Figure 8a–d show the responses of the surge, heave, pitch, and yaw motions of the third module versus the wave frequency. In surge motion, the novel connector and rigid connector perform almost equally well and much better than the flexible connector. In the heave and pitch motions, the novel connector outperforms the other two connectors because the actuator of the novel connector takes effect. For the yaw motion, the novel connector performs better than the flexible connector around the main wave frequency of 0.628 rad/s but performs worse at very low frequencies. Since the peak of the yaw response is about 0.027 rad at low frequency, its contribution to the 1–in–1000 extreme responses is insignificant. Figure 8e–h show the 1–in–1000 extreme responses for the third module, which were induced by the wave spectrum in Equation (32). For the surge, heave, and pitch responses, the novel connector performs the best among the three types of connectors. As for the yaw motion, the novel connector greatly outperforms the flexible one. For a slender floating system, the hydrodynamic load on the floating structure is closely related to the wave incident angle. It is necessary to study the effect of the wave incident angle on the performance of the novel connector.

Next, the performance of the vibration control method was studied at different wave incident angles. To present the results neatly, we used the mean response of all of the modules at each degree of freedom to represent the essential nature of the system motion in that direction. For example, the symbol $\overline{x}=\frac{1}{N}{\displaystyle \sum }_{i=1}^N{R}_e\left(x_i\right)$ denotes the mean 1–in–1000 extreme surge responses of the whole floating structure in x direction. The mean responses of the five-module floating structure are presented in Figure 9, where the horizontal axis is the wave incident angle from $0°$ to $90°$.

For the controlled motions of surge, heave, pitch, and yaw shown in Figure 9a,c,e,f, the control method by the novel connector (blue line) performs much better than that of the flexible connector. Especially in the surge, heave, and pitch motions, the result of the novel connector is close to that of the hinged connector due to its rigidity. By comparing the peak responses of the flexible connector, the novel connector can greatly reduce the surge by 91.2%, the heave by 88.8%, the pitch by 74.0%, and the yaw by 84.3%. These results indicate that the control method can effectively stabilize the modular floating structure and that it can greatly improve how stationary the system is. For those uncontrolled sway and roll responses, such as those illustrated in Figure 8b,d, the hinged connector performs better in general because of its rigidity feature. The novel stiffness-adjustable connector performs better than the flexible connector does because the interaction effects of the responses among the different degrees of freedom can be reduced when the actuators are activated.

The benefit of the vibration control method is also reflected in the dramatic reduction of connector loads when compared to the hinged connector. In this numerical simulation model, there are four groups of connectors to connect the five-module floating platform. The loads of the different connectors correspond to the frequency response in the domains of wave frequency and angle. The loads of the novel connector are calculated through Equation (12). For the flexible connector, the connector load is equal to the spring load and can be calculated through Equation (14). The load of the hinged connector is derived in article [32]. Basically, the load level among the different connector groups is the same. In a group, there are five connectors, among which the first four connectors are horizontally deployed in parallel, as indicated in Table 2. The load levels of the four connectors are the same, and the load level of the fifth connector is smaller than that of the first four. Figure 10 only illustrates the loads of the first connector in all of the connection groups. The variation in the load is investigated in a domain that is spanned by wave frequency and the wave incident angle. To make the presentation neat, only the load along x direction is presented here because the loads along other directions are much smaller.

Figure 10a–d depict the loads of the novel connectors, which include the actuator force and the spring load. Although the direction of the actuator force may not always be identical to the direction of the spring load, in the worst case, the peak load of the novel connector is about $3.63\times {10}^4$ KN. The hot spots mainly appear at wave frequencies less than 0.3 rad/s. When comparing the four maps, the distribution and maximum values of the novel connector loads in four connector groups are almost the same. Figure 10e–h show the load distribution of the flexible connector, where the peak load is $3.46\times {10}^4$ KN. The peak load appears at the spot where the wave incident angle falls in the interval from $45°$ to $70°$, and the wave frequency is about 0.8 rad/s. The loads in the first and last groups are larger than those in the second and third groups, but the distribution of the load for the four groups is very similar. In comparison, the peak load of the novel connector is larger than that of the flexible connector by about 4.9%. This means that there is a cost to suppressing the multi-DOF motions by increasing the load of the novel connector. Figure 10i shows the contour map of the load of the hinged connector, and the peak load level is about $8.99\times {10}^4$ KN, which is much greater than that of the novel and flexible connectors. Even worse, the high-level load appears to be in much broader area across the wave frequency and wave incident angle. The actuator output control force is calculated using Equation (17). These hot spots on the cloud map in Figure 10j appear in the domain spanned by the wave frequency and wave incident angle. The peak control force is up to the level of $2.35\times {10}^4$ KN, which is much smaller than the load level of the novel connector and the flexible connector. This means that with a small control force, the novel connector can achieve largely improved VLFS stability. Compared to the maximum control force of $5\times {10}^4~7\times {10}^4$ KN when using a full-control strategy [29] in the same VLFS, the control force of the novel connector is about 1/2 to 2/3 smaller in magnitude. Since the load difference between the novel connector and the flexible connector is smaller than the actuator peak load, the spring force of the flexible connector is larger than that of the novel connector when the actuator is switched on.

Overall, we would like to comment that the control method with the novel stiffness-adjustable connector generally works well to suppress the intended surge, heave, pitch, and yaw motions with the connector layout shown in Figure 6. As for the roll and sway motions, there are no specific novel connectors arranged in the directions to control the motions. If additional novel connectors are deployed in these directions, the motions can be significantly reduced. Based on the stiffness-adjustable connectors, the vibration control method can apparently improve the stability of the floating structure even though the control forces expend little effort. The proposed method can also eliminate the resonances of the VLFS, making it more adaptive to various wave conditions.

5. Conclusions

In this paper, we proposed a novel stiffness-adjustable connector and a related vibration control method for multi-modular VLFS. The novel connector is composed of an annular spring to provide basic flexibility and an embedded hydraulic actuator to provide control force. The benefit of using the novel connector can greatly reduce the control forces, making its application possible. These individual connectors can be combined into a desirable connection layout to suppress all types of movement experienced by a modularized VLFS in random sea conditions. In the study, a connector layout to suppress suppressing the surge, heave, pitch, and yaw motions of a five-module floating structure is suggested. To estimate the performance of the vibration control method, a mathematical model is developed that is associated with a state-feedback control scheme. The feedback gains are obtained by using Sequential Quadratic Programming. In the numerical studies, we verified the vibration control method in the different wave frequencies and wave incident angles and compared the performance of the stiffness-adjustable connector with flexible connectors and hinged connectors. The results show that the control method can greatly reduce the responses of the VLFS in random sea conditions while using small control forces. In the connector load, the peak load of the novel connector is slightly larger than that of the flexible connector but much smaller than that of the hinged connector. More importantly, the vibration control method enables the modular floating system get rid of resonant responses to random sea conditions, adaptive to the changes in wave conditions.

In this study, the modules are treated as rigid bodies so that the case study remains linear to the motion equation. The nonlinearity is not considered, which could be caused by a large deflection in the flexible modules or mooring systems. The loads of the novel connector could be estimated in both extreme responses and non-extreme responses, and the loads of non-extreme states could be used for fatigue-driven design [40]. The resonant response of floating structures is strongly influenced by non-radiative damping, which is usually represented by empirically calibrated linear damping, which fits the approach of this study. The frequency domain approach is unsuitable for dealing with transient responses. For transient and nonlinear responses, the implementation of a time-domain-adaptive control strategy would be required. The connector model was studied in a long-and-slender VLFS with multiple modules with a simple 2D structure. In fact, there are no technical barriers for the potential application of these connectors to in complicated topological VLFS with three dimensions, which could be of great interest for future studies.