## Appendix A
\begin{split}& q_{1}=\frac{A_{6}}{2\pi}\int_{-\infty}^{+\infty} \frac{Q_{6}\xi^{2}+Q_{7}\xi+Q_{8}}{Q_{1}\xi^{4}+Q_{2}\xi^{3}+Q_{3}\xi^{2}+Q_{4} \xi^{2}+Q_{5}}e^{i\xi z}d\xi,\\ &\psi_{1}=\frac{A_{6}}{2\pi}\int_{-\infty}^{+\infty}\frac{ \lambda AGi\xi}{Q_{1}\xi^{4}+Q_{2}\xi^{3}+Q_{3}\xi^{2}+Q_{4}\xi+Q_{5}}e^{i\xi z }d\xi,\\ & q_{2}=\frac{A_{4}}{2\pi}\int_{-\infty}^{+\infty}\frac{Q_{6}\xi^ {2}+Q_{7}\xi+Q_{12}}{Q_{1}\xi^{4}+Q_{2}\xi^{3}+Q_{9}\xi^{2}+Q_{10}\xi+Q_{11}} e^{i\xi z}d\xi,\\ &\psi_{2}=\frac{A_{4}}{2\pi}\int_{-\infty}^{+\infty}\frac{ \lambda AGi\xi}{Q_{1}\xi^{4}+Q_{2}\xi^{3}+Q_{9}\xi^{2}+Q_{10}\xi+Q_{11}}e^{i \xi z}d\xi.\end{split} \] (A21)  
Following the previously described computational steps, the expressions for \(w_{1}\) and \(w_{2}\) can be deduced  
\[ \begin{split} w_{1}=&-k_{3}[q_{3}(z)+q_{4}(z)(e^{i \omega t}+e^{-i\omega t})+q_{5}(z)(e^{i2\omega t}+e^{-i2\omega t})+q_{6}(z)(e^{ i3\omega t}+e^{-i3\omega t})],\\ w_{2}=& 3{k_{3}}^{2}[q_{7}+q_{8}(e^{i\omega t}+e^{-i \omega t})+q_{9}(e^{i2\omega t}+e^{-i2\omega t})+q_{10}(e^{i3\omega t}+e^{-i3 \omega t})\\ &+q_{11}(e^{i\omega t\omega t}+e^{-i\omega t})+q_{12}(e^{i5\omega t }+e^{-i5\omega t})],\end{split} \] (A22)  
where  
\[ \begin{split}& q_{3}=\int_{-\infty}^{+\infty}[q_{1}^{3}(z-\tau)+6q _{1}(z-\tau)q_{2}^{2}(z-\tau)]\eta_{0}(\tau)d\tau,\\ & q_{4}=\int_{-\infty}^{+\infty}[3q_{2}^{3}(z-\tau)+3q_{1}^{2}(z- \tau)q_{2}(z-\tau)]\eta_{1}(\tau)d\tau,\\ & q_{5}=\int_{-\infty}^{+\infty}3q_{1}(z-\tau)q_{2}^{2}(z-\tau) \eta_{2}(\tau)d\tau,\\ & q_{6}=\int_{-\infty}^{+\infty}q_{2}^{3}(z-\tau)\eta_{3}(\tau)d \tau,\end{split} \]  
\[ \begin{split}& q_{7}=\int_{-\infty}^{+\infty}\left[q_{1}^{2}(z- \tau)q_{3}(z-\tau)+2q_{2}^{2}(z-\tau)q_{3}(z-\tau)+2q_{2}^{2}(z-\tau)q_{5}(z- \tau)\right]\eta_{0}(\tau)d\tau,\\ & q_{8}=\int_{-\infty}^{+\infty}\left[\begin{matrix}q_{1}^{2}(z -\tau)q_{4}(z-\tau)+2q_{1}(z-\tau)q_{2}(z-\tau)q_{3}(z-\tau)+q_{2}^{2}(z-\tau)q _{6}(z-\tau)\\ +2q_{1}(z-\tau)q_{2}(z-\tau)q_{5}(z-\tau)+3q_{2}^{2}(z-\tau)q_{4}(z-\tau) \end{matrix}\right]\eta_{1}(\tau)d\tau,\\ & q_{9}=\int_{-\infty}^{+\infty}\left[\begin{matrix}q_{1}^{2}(z -\tau)q_{5}(z-\tau)+q_{2}^{2}(z-\tau)q_{3}(z-\tau)+2q_{2}^{2}(z-\tau)q_{5}(z- \tau)\\ +2q_{1}(z-\tau)q_{2}(z-\tau)q_{4}(z-\tau)+2q_{1}(z-\tau)q_{2}(z-\tau)q_{6}(z- \tau)\end{matrix}\right]\eta_{2}(\tau)d\tau,\\ & q_{10}=\int_{-\infty}^{+\infty}\left[\begin{matrix}q_{1}^{2}(z -\tau)q_{6}(z-\tau)+2q_{1}(z-\tau)q_{2}(z-\tau)q_{5}(z-\tau)\\ +q_{2}^{2}(z-\tau)q_{4}(z-\tau)+2q_{2}^{2}(z-\tau)q_{6}(z-\tau)\end{matrix} \right]\eta_{3}(\tau)d\tau,\\ & q_{11}=\int_{-\infty}^{+\infty}\left[q_{2}^{2}(z-\tau)q_{5}(z- \tau)+2q_{1}(z-\tau)q_{2}(z-\tau)q_{6}(z-\tau)\right]\eta_{4}(\tau)d\tau,\\ & q_{12}=\int_{-\infty}^{+\infty}q_{2}^{2}(z-\tau)q_{6}(z-\tau)\eta_{5}(\tau)d \tau,\end{split} \]  
\(\eta_{j}\), \(j=0,1,2,3,4,5\), are determined according to the following equations  
\[ \begin{split}&(\rho Av^{2}-\lambda AG-G_{p})\frac{d^{2}\eta_{j}}{dz ^{2}}+(-cv)\frac{d\eta_{j}}{dz}+(-(j\omega)^{2}\rho A+k_{1})\eta_{j}+\lambda AG \frac{d\phi_{j}}{dz}=\delta(z),\\ &(\rho Iv^{2}-EI)\frac{d^{2}\phi_{j}}{dz^{2}}+(-vc_{f})\frac{d\phi_ {j}}{dz}+(-(j\omega)^{2}\rho I+\lambda AG+k_{f})\phi_{j}-\lambda AG\frac{d\eta_{ j}}{dz}=0.\end{split} \] (A23)The above calculation process can continue, but in practice, it is usually only necessary to compute the first few Adomian polynomials to achieve the desired accuracy.