## 2 Theoretical Models
### Solving for the Conductance Function of a Thin Underwater Cylindrical Shell
After the modal influence factor is obtained, substitute into Equations (11) and (13) respectively to obtain the vibration displacement responses \(U(z,\,\varphi)\),\(V(z,\,\varphi)\),\(W(z,\,\varphi)\) and the external hydroacoustic field sound pressure \(P(r,\,\varphi,\,z)\) of the cylindrical thin shell, and also the displacement derivative function of the axially concentrated excitation force \(F_{z}\cdot{\rm e}^{\mathrm{j}\omega t}\) for the three coordinate directions (i.e., axial, tangential and radial) of \(z\),\(\varphi\) and \(r\): \(Y_{uu}(\sigma,\,\varphi_{\rm e})=U/F_{z}\), \(Y_{uu}(\sigma,\,\varphi_{\rm e})=V/F_{z}\), \(Y_{uu}(\sigma,\,\varphi_{\rm e})=W/F_{z}\). Similarly, a simple harmonic excitation force \(F_{\varphi}\cdot{\rm e}^{\mathrm{j}\omega t}\) along the tangential direction (in the direction of the \(\varphi\) coordinate axis) or a simple harmonic excitation force \(F_{r}\cdot{\rm e}^{\mathrm{j}\omega t}\) along the radial direction (in the direction of the \(r\) coordinate axis) can be applied to the inside of the thin cylindrical shell \(\varphi_{\rm e}(\varphi_{\rm e},\,z_{\rm e})\), the displacement derivative functions of the tangential or radial concentrated excitation on the three directions \(z\),\(\varphi\) and \(r\) are obtained according to the same method as above and are noted as \(Y_{uv}(\sigma,\,\varphi_{\rm e})\), \(Y_{v\overline{v}}(\sigma,\,\varphi_{\rm e})\), \(Y_{v\overline{v}}(\sigma,\,\varphi_{\rm e})\), \(Y_{v\overline{v}}(\sigma,\,\varphi_{\rm e})\), \(Y_{v\overline{v}}(\sigma,\,\varphi_{\rm e})\), \(Y_{v\overline{v}}(\sigma,\,\varphi_{\rm e})\), \(Y_{v\overline{v}}(\sigma,\,\varphi_{\rm e})\). For the case of applied tangential excitation, it is noted that the symmetric nature of the cylindrical thin shell and sound field modes is different from the case of axial and radial excitation, and Equations (11) and (13) need to be modified as  
\[\left\{ \begin{array}{c}U=\sum_{i,j=0}^{\infty}q_{ij}\cdot U_{i\overline{j}} =\sum_{i,j=0}^{\infty}q_{ij}\cdot A_{ij}(z)a_{j}(z)\cdot\sin[i(\varphi- \varphi_{\rm e})]\\ V=\sum_{i,j=1}^{\infty}q_{ij}\cdot V_{i\overline{j}}=\sum_{i,j=1}^{\infty}q_{ij} \cdot B_{i\overline{j}}(z)\beta_{j}(z)\cdot\cos[i(\varphi-\varphi_{\rm e})] \\ W=\sum_{i,j=0}^{\infty}q_{ij}\cdot W_{i\overline{j}}=W\sum_{i,j=0}^{\infty}q_{ij} \cdot C_{i\overline{j}}(z)\gamma_{j}(z)\cdot\sin[i(\varphi-\varphi_{\rm e})] \end{array} \right\} \tag{15}\]  
\[P(r,\varphi,z)=\frac{\rho_{0}\omega^{2}}{2\pi}\sum_{i=0,j=1}^{\infty}q_{ij} \sin[i(\varphi-\varphi_{\rm e})]\int_{-\infty}^{+\infty}\frac{H_{i}^{(2)}(k_{ r}r)}{k_{r}\Big{[}\mathrm{d}H_{i}^{(2)}(k_{r}r)/\mathrm{d}(k_{r}r)\Big{]}_{r=a}} \widetilde{f}_{ij}(\tau)\cdot{\rm e}^{-\mathrm{j}\tau z}\mathrm{d}\tau \tag{16}\]Directional conductance along any direction can be calculated using the nine conductivity functions described above. Let \(\sigma_{i}(a,\,\varphi_{i},\,z_{i})\) be the coordinates of the cylindrical coordinate system (\(i=1,\,2,\,3,\,4\)) in the coupling point between each vibration isolation support and the thin cylindrical shell in the system of Figure 1, then the point and transfer conductance \(c_{ij}\) in Equation (7), according to its definition, can be calculated by the following equation:  
\[ \begin{array}{c}c_{ij}=Y_{\perp}\big{(}\sigma_{i},\sigma_{j}\big{)}=\mathrm{ j}\omega\big{[}Y_{ww}(\sigma_{i},\sigma_{j})\cdot\cos\varphi_{i}+Y_{ww}( \sigma_{i},\sigma_{j})\cdot\sin\varphi_{j}\big{]}\sin\varphi_{i}+\mathrm{j} \omega\\ \big{[}Y_{vv}(\sigma_{i},\sigma_{j})\cdot\cos\varphi_{j}+Y_{vw}(\sigma_{i}, \sigma_{j})\cdot\sin\varphi_{j}\big{]}\cos\varphi_{i}\end{array}