## 2 Theoretical Models  
### Conductivity Analysis of Coupled System Substructures  
In view of the increasing use of floating raft vibration isolators in various ship systems, this paper presents a dynamic modelling and dynamic response solution for the coupled system of the vibration source machine--floating raft device--thin cylindrical shell--hydroacoustic field in Figure 1, then carries out a numerical simulation of the acoustic vibration characteristics of the coupled system. In Figure 1, vibration source machines A\({}_{1}\) and A\({}_{2}\) (simulating multiple units) are installed on the floating raft R. Each machine is installed on the raft by four elastic supports, and the raft is connected with the thin cylindrical shell by four elastic supports. The rectangular coordinate system \(Oxyz\) is established, where \(Oz\) axis is the center line of the cylinder, and the \(Oy\) axis is the vertical line. In order to simplify the analysis, only the vertical translational vibration and rotational vibration around \(Oz\) or \(Ox\) axis of the system is considered. Therefore, the vertical exciting force \(F_{1}\), \(F_{2}\) and excitation torque \(T_{1}\), \(T_{2}\) in the direction of Oz axis to the center of mass of each machine is applied. In addition, set machine A\({}_{1}\), A\({}_{2}\) and R are symmetrical about the \(yOz\) plane, and the figures of \(b_{1}\), \(b_{2}\), \(h_{12}\), \(h_{12}\), \(h_{12x}\), \(h_{2x}\), \(H_{x}\) and \(L_{\text{R}}\) is the relevant installation dimensional parameter of the system, while \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\), \(\sigma_{4}\) indicates the connection point between the vibration isolation support below the floating raft and the thin cylindrical shell. Figure 2 decomposes the system in Figure 1 into three subsystems according to the power transfer relationship: the source machine (A), the floating raft device (B) and the submerged cylindrical shell (C), where the submerged cylindrical shell contains the acoustic coupling between the thin cylindrical shell and the peripheral hydroacoustic field. In Figure 2, \(F_{\text{e}}=[F_{1}\), \(T_{1}\), \(F_{2}\), \(T_{2}]^{\text{T}}\) is the excitation vector acting on the source device, \(F_{\text{A}}\), \(F_{\text{B1}}\), \(F_{\text{B2}}\) and \(F_{\text{C}}\) are the dynamic forces transmitted by the floating raft device at its coupling points with the source device and the shell (vectors) and \(V_{\text{A1}}\), \(V_{\text{A2}}\), \(V_{\text{B1}}\), \(V_{\text{B2}}\) and \(V_{\text{C}}\) are the velocity responses at the corresponding coupling points (vectors). The dynamic transfer characteristics of the vibration source machine A, the floating raft device B and the submerged cylindrical shell C are described in terms of the conductance matrices \(A\), \(B\) and \(C\), respectively. Figure 1: Coupling system diagram: (**a**) 3D view of the system (**b**) \(x\) direction; (**c**) \(z\) direction. Figure 2: Power transfer relationship between mechanical equipment and ship body coupling system. \[\left\{ \begin{array}{c}\left[\begin{array}{c}\mathbf{V}_{\rm A1}\\ \mathbf{V}_{\rm A2}\end{array}\right]=\mathbf{A}\left[\begin{array}{c}\mathbf{F}_{\rm e} \\ \mathbf{F}_{\rm A}\end{array}\right]=\left[\begin{array}{cc}\mathbf{A}_{11}&\mathbf{A}_{12} \\ \mathbf{A}_{21}&\mathbf{A}_{22}\end{array}\right]\left[\begin{array}{c}\mathbf{F}_{\rm e} \\ \mathbf{F}_{\rm A}\end{array}\right]\\ \left[\begin{array}{c}\mathbf{V}_{\rm B1}\\ \mathbf{V}_{\rm B2}\end{array}\right]=\mathbf{B}\left[\begin{array}{cc}\mathbf{F}_{\rm B 1}\\ \mathbf{F}_{\rm B2}\end{array}\right]=\left[\begin{array}{cc}\mathbf{B}_{11}&\mathbf{B}_{ 12}\\ \mathbf{B}_{21}&\mathbf{B}_{22}\end{array}\right]\left[\begin{array}{c}\mathbf{F}_{\rm B1 }\\ \mathbf{F}_{\rm B2}\end{array}\right]\\ \mathbf{V}_{\rm C}=\mathbf{C}\cdot\mathbf{F}_{\rm C}\end{array} \right.