## 2 Methodology
### Dynamic Model for Free Vibration  
Flugge shell theory provides a basis for the dynamic model of the shield tunnel lining [13]. The governing differential equations of the shield tunnel lining are as follows. Each equation represents one of the three principal directions. \[ \begin{split} R\frac{\partial^{2}u}{\partial x^{2}}+\frac{1-\mu} {2R}\frac{\partial^{2}u}{\partial\theta^{2}}+\frac{1+\mu}{2}\frac{\partial^{2 }v}{\partial x\partial\theta}-\mu\frac{\partial w}{\partial x}+\frac{h^{2}}{12 }\bigg{(}\frac{1-\mu}{2R^{3}}\frac{\partial^{2}u}{\partial\theta^{2}}+\frac{ \partial^{3}w}{\partial x^{3}}-\frac{1-\mu}{2R^{2}}\frac{\partial^{3}w}{ \partial x\partial\theta^{2}}\bigg{)}\\ -\rho R\frac{\big{(}1-\mu^{2}\big{)}}{E}\frac{\partial^{2}u}{ \partial t^{2}}=0\end{split}  \tag{1}\]  
\[ \begin{split} R\frac{1+\mu}{2}\frac{\partial^{2}u}{\partial x \partial\theta}+R^{2}\frac{1-\mu}{2}\frac{\partial^{2}v}{\partial x^{2}}+ \frac{\partial^{2}v}{\partial\theta^{2}}-\frac{\partial w}{\partial\theta}+ \frac{h^{2}}{12}\bigg{[}\frac{3(1-\mu)}{2}\frac{\partial^{2}v}{\partial x^{2 }}+\frac{3-\mu}{2}\frac{\partial^{3}w}{\partial x^{2}\partial\theta}\bigg{]} \\ -\rho R^{2}\frac{\big{(}1-\mu^{2}\big{)}}{E}\frac{\partial^{2}v}{ \partial t^{2}}=0\end{split}  \tag{2}\]  
\[ \begin{split} R\mu\frac{\partial u}{\partial x}+\frac{\partial v }{\partial\theta}-w\frac{h^{2}}{12}\bigg{(}R^{2}\frac{\partial^{4}w}{\partial x ^{4}}+\frac{2\partial^{4}w}{\partial x^{2}\partial\theta^{2}}+\frac{\partial^ {4}w}{R^{2}\partial\theta^{4}}\bigg{)}\\ -\frac{h^{2}}{12}\bigg{(}R\frac{\partial^{3}u}{\partial x^{3}}- \frac{1-\mu}{2R}\frac{\partial^{3}u}{\partial x\partial\theta^{2}}+\frac{3- \mu}{2}\frac{\partial^{3}v}{\partial x^{2}\partial\theta}+\frac{w}{R^{2}}+ \frac{2\partial^{2}w}{R^{2}\partial\theta^{2}}\bigg{)}\\ +\rho R^{2}\frac{\big{(}1-\mu^{2}\big{)}}{E}\frac{\partial^{2}w}{ \partial t^{2}}=0\end{split}  \tag{3}\]  
Symbols \(u\), \(v\) and \(w\) represent the axial \(x\), circumferential \(\theta\), and radial \(r\) displacements, as shown in Figure 1. The radius of the midplane is \(R\), the lining thickness is \(h\), the material density is \(\rho\), [PERSON]'s ratio is \(\mu\), and the Young modulus is \(E\). The displacements of the shell have the following forms of wave propagation [33]. \[ \begin{split} u&=\sum_{n=0}^{\infty}\sum_{m=1}^{ \infty}\widetilde{U}\cos n\theta e^{i(\omega t-k_{mn}x)}\\ v&=\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}\widetilde {V}\sin n\theta e^{i(\omega t-k_{mn}x)}\\ w&=\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}\widetilde {W}\cos n\theta e^{i(\omega t-k_{mn}x)}\end{split}  \tag{4}\]  
Symbols \(k_{nm}\) and \(\omega\) represent the axial wavenumber and circular frequency. The circumferential mode number is \(n\), the half-wavenumber in the axial direction is \(m\), the time factor is \(e^{i\omega t}\). Symbols \(\widetilde{U}\), \(\widetilde{V}\), and \(\widetilde{W}\) represent the axial, circumferential, and radial displacement amplitudes. Furthermore, the clamped-clamped boundary condition provides the axial wavenumber as \(k_{nm}=\frac{(2n+1)\pi}{2L}\), and \(L\) is the lining length [34]. After that, substituting the waveform solution into Equations (1)-(3) provides three equations in the amplitudes of \(\widetilde{U}\), \(\widetilde{V}\), and \(\widetilde{W}\), which can be presented in matrix form. \[\left[ \begin{array}{ccc}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\end{array} \right]\left[ \begin{array}{ccc}\widetilde{U} \\ \widetilde{V}\\ \widetilde{W}\end{array} \right]=\left[ \begin{array}{ccc}0\\ 0\\ 0\end{array} \right] \tag{5}\]  
This study sets the dimensionless frequency to \(\Omega=\omega R\sqrt{\frac{\rho(1-\mu^{2})}{E}}\), and the dimensionless axial wavenumber to \(\lambda=k_{nm}R\). Then, the components of the matrix \([A_{3\times 3}]\) are  
\[