## 2 Theoretical Models
### Solving for the Conductance Function of a Thin Underwater Cylindrical Shell
 \tag{17}\]  
Figure 3 shows an example of the calculation results for the vertical conductance of the thin cylindrical shell of Figure 1. Due to the complexity of the calculations, Figure 4 compares the results of the hydroacoustic pressure calculations obtained by the above analytical algorithm with the results of the finite element method obtained from the LMS Virtual Lab software environment for the same structural parameters, in order to check the reliability of the analytical model. It is clear from the comparison that the results of the two methods are in general agreement. The individual peaks in the graph are resonance peaks, which are the result of the resonance of the coupled system at this excitation force. However, the resonant peak frequency of the thin cylindrical shell obtained from the virtual.lab simulation is slightly lower than the resonant peak frequency of the analytical solution. This is due to the fact that vitural.lab uses an AML layer approach for the infinite waters wrapped around the outside of the thin cylindrical shell, which ignores a large amount of water mass and thus leads to the advancement of the resonant peak of the thin cylindrical shell. Figure 3: Example of calculation results for the vertical conductor. Level Drop, Transmitted Power Flow, Sound Pressure at the Hydroacoustic Field and Total Radiated Sound Power of the Coupled System  
For elastic foundation vibration isolation systems, the transferred power flow (the time-averaged power of the excitation forces transmitted to the foundation structure through the isolation bearings) is a theoretical indicator for evaluating the effectiveness of vibration isolation design. However, as the transferred power flow is difficult to measure in practice, the effect of the actual vibration isolation system is generally assessed by measuring the vibration level drop. For the system in Figure 1, the transmitted power flow and vibration level drop can be combined with the sound pressure and radiated sound power of the water acoustic field to make a comprehensive analysis and evaluation of the acoustic vibration characteristics of the coupled system. After substituting the results of Equation (17) into Equation (7), the excitation force \(\mathbf{F}_{\text{C}}\) transmitted to the cylindrical shell through the vibration isolation support can be obtained from Equation (3), and then the velocity response \(\mathbf{V}_{\text{C}}\) of the cylindrical shell at the point of coupling with the vibration isolation support can be derived from Equation (2) of Equation (3), so the transferred power flow of the system can be known as  
\[P_{\text{T}}=\frac{1}{2}\text{Re}\Big{\{}\mathbf{F}_{\text{C}}^{\text{H}}\cdot\mathbf{ C}\cdot\mathbf{F}_{\text{C}}\Big{\}} \tag{18}\]  
where the superscript \"H\" indicates the Hermitian transpose. The vibration level drop of the floating raft isolator is the ratio of the mean square value of the acceleration response (in dB) between the isolator and the base structure (here the base structure is a thin cylindrical shell) and the junction point of the vibration source equipment, so it is  
\[D_{\text{a}}=10\log\frac{\mathbf{V}_{\text{C}}^{\text{H}}\mathbf{V}_{\text{C}}/4}\mathbf{{V}_{\text{A2}}^{\text{H}}\mathbf{V}_{\text{A2}}/8}=10\log\left(2\frac{\mathbf{F}_{ \text{C}}^{\text{H}}\mathbf{C}^{\text{H}}\mathbf{C}\mathbf{F}_{\text{C}}}{\mathbf{V}_{\text{A2 }}^{\text{H}}\mathbf{V}_{\text{A2}}}\right) \tag{19}\]  
where \(V_{\text{A2}}\) is calculated from Equation (4). In order to calculate the radiated hydroacoustic field formed by the excitation of \(\mathbf{F}_{\text{C}}\) = [\(F_{\text{C1}}\), \(F_{\text{C2}}\), \(F_{\text{C3}}\), \(F_{\text{C4}}\)]T in the cylindrical thin shell in the coupled system of Figure 1, take out the elements \(F_{\text{C1}}\) (\(i\) = 1, 2, 3, 4) in \(\mathbf{F}_{\text{C}}\) in turn.