## Appendix B Mathematical Expressions of the Terms and Coefficients in Paragraph 3  
\[\Pi=\frac{1}{2}\int\limits_{0}^{\mathrm{L_{M_{S}}}}\mathrm{GJ_{X_{2} \mathrm{Ms}}}\bigg{(}\frac{\partial\varphi_{2}}{\partial\mathrm{x}}\bigg{)}^{2} \mathrm{dx}+ \tag{10}\] \[+\frac{1}{2}\bigg{[}\mathrm{k_{G\mathrm{T}}}\varphi_{1}^{2}+ \mathrm{k_{T\mathrm{T}}}(\varphi_{2}(0,t)-\varphi_{1})^{2}+\mathrm{k_{B \mathrm{T}}}(\varphi_{3}-\varphi_{2}(\mathrm{L_{M\mathrm{S}}},t))^{2}+\mathrm{ k_{G\mathrm{B}}}\varphi_{3}^{2}\bigg{]},\]  
\[ \begin{array}{c}\mathrm{T}=\int\limits_{0}^{\mathrm{L_{M_{S}}}}\frac{1}{2} \rho\mathrm{J_{X_{2}\mathrm{Ms}}}\bigg{(}\frac{\partial\varphi_{2}}{\partial t }\bigg{)}^{2}\mathrm{dx}+\frac{1}{2}\mathrm{I_{X_{2}\mathrm{GT}}}\bigg{(} \varphi_{1}^{\bullet}\bigg{)}^{2}+\frac{1}{2}\mathrm{I_{0\mathrm{IMs}}}\bigg{(} \frac{\partial\varphi_{2}(0,t)}{\partial t}\bigg{)}^{2}+\\ +\frac{1}{2}\mathrm{I_{0\mathrm{IMs}}}\bigg{(}\frac{\partial\varphi_{2}( \mathrm{L_{M\mathrm{S}}},t)}{\partial t}\bigg{)}^{2}+\frac{1}{2}\mathrm{I_{X_{ 2}\mathrm{GB}}}\bigg{(}\varphi_{3}^{\bullet}\bigg{)}^{2},\end{array}  \tag{11}\]  
\[ \begin{array}{c}\mathrm{W}=\int_{0}^{\mathrm{L_{M_{S}}}}\{[\mathrm{M_{C_{B} }}]\Delta(x)+[\mathrm{M_{W}}]\Delta(x-\mathrm{L_{M\mathrm{S}}})\}\varphi_{2} (x,t)\mathrm{dx}=\\ =[\mathrm{M_{C\mathrm{B}}}]\varphi_{2}(0,t)+[\mathrm{M_{W}}]\varphi_{2}( \mathrm{L_{M\mathrm{S}}},t),\end{array}  \tag{12}\]  
\[ \begin{array}{c}\mathrm{A_{T\mathrm{T}}}=0.5\frac{\mathrm{R_{T\mathrm{T}}}} {\mathrm{L_{M\mathrm{S}}}}\tan\beta_{1}\tan^{2}\frac{\beta_{1}}{2},\mathrm{A_ {B\mathrm{T}}}=0.5\frac{\mathrm{R_{B\mathrm{T}}}}{\mathrm{L_{M\mathrm{S}}}} \tan\beta_{2}\tan^{2}\frac{\beta_{2}}{2},\end{array}  \tag{13}\]  
\[ \begin{array}{c}\varphi_{2}(0,t)=\varphi_{1}(t)+\frac{\mathrm{R_{T\mathrm{T} }}}{\mathrm{L_{M\mathrm{S}}}}\tan\beta_{1}\tan^{2}\frac{\beta_{1}}{2}\cos(3 \varphi_{1}),\\ \varphi_{3}(t)=\varphi_{2}(\mathrm{L_{M\mathrm{S}}},t)+\frac{\mathrm{R_{B \mathrm{T}}}}{\mathrm{L_{M\mathrm{S}}}}\tan\beta_{2}\tan^{2}\frac{\beta_{2}}{2 }\cos(3\varphi_{2}(\mathrm{L_{M\mathrm{S}}},t)),\end{array}  \tag{14}\]  
\[ \begin{array}{c}\varphi_{2}(x,t)=\sum\limits_{n}\varphi_{2n}\cos\Big{(} \frac{\omega_{n}x}{c}-\Theta_{1n}\Big{)}\cos(\omega_{n}t-\Theta_{2n}),\\ \mathrm{c}=\sqrt{\frac{G}{\rho}},\omega_{n}=\frac{n\pi c}{\mathrm{L_{M\mathrm{S }}}},n=1,2,3,\dots\end{array}  \tag{15}\]  
\[ \begin{array}{c}\chi_{\mathrm{GTn}}=\frac{0.5(\mathrm{I_{T\mathrm{T}}}+ \mathrm{I_{T\mathrm{T}}})\rho\mathrm{L_{T}}\sin^{2}\beta_{1}}{\mathrm{I_{X_{2} \mathrm{GT}}}}\mathrm{I_{X_{2}\mathrm{GT}}}\mathrm{I_{\mu}}\mathrm{I_{1}} =\frac{\chi_{\mathrm{GTn}}}{1-\frac{\chi_{\mathrm{GTn}}\rho\mathrm{A_{T}}}{ \mathrm{I_{X_{2}\mathrm{GT}}}}},\\ \mathrm{a_{2}}=\Big{(}\frac{\mathrm{L_{M\mathrm{T}}}}{\mathrm{L_{T}}}\Big{)}^{2 }\frac{\mathrm{I_{X_{2}\mathrm{GT}}}}{\mathrm{I_{X_{2}\mathrm{AT}}}\rho \mathrm{A_{T}}}\frac{\chi_{\mathrm{GTn}}}{1+\Big{(}\frac{\mathrm{L_{M\mathrm{ T}}}}{\mathrm{L_{T}}}\Big{)}^{2}\Big{[}\frac{\mathrm{I_{X_{2}\mathrm{GT}}}}{\mathrm{I_{X_{2} \mathrm{AT}}}\rho\mathrm{A_{T}}}-1\Big{]}},\\ \chi_{\mathrm{GBn}}=\frac{0.5(\mathrm{I_{B\mathrm{T}}}+\mathrm{I_{B\mathrm{T}}}) \rho\mathrm{L_{B}}\sin^{2}\beta_{2}}{\mathrm{I_{X_{2}\mathrm{GB}}}}\mathrm{a _{3}}=\frac{\chi_{\mathrm{GBn}}}{1-\frac{\chi_{\mathrm{GBn}}\rho\mathrm{A_{T}} }{\mathrm{I_{X_{2}\mathrm{GB}}}}},\\ \mathrm{a_{4}}=\Big{(}\frac{\mathrm{L_{M\mathrm{B}}}}{\mathrm{L_{B}}}\Big{)}^{2 }\frac{\mathrm{I_{X_{2}\mathrm{GB}}}}{\mathrm{I_{X_{2}\mathrm{AB}}}\mathrm{I_{ \mu}}\mathrm{I_{1}+\Big{(}\frac{\mathrm{L_{M\mathrm{B}}}}{\mathrm{L_{M \mathrm{B}}}}\Big{)}^{2}}\Big{[}\frac{\mathrm{I_{X_{2}\mathrm{GB}}}}{\mathrm{I_{X_{2 }\mathrm{AB}}}\mathrm{I_{\mu}}\mathrm{I_{1}}}-1\Big{]}},\end{array}  \tag{16}\]  
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