## Appendix A
 \tag{18}\]  
where  
\[\begin{split} u_{7}&=\int_{-\infty}^{+\infty}\! \begin{bmatrix}2u_{1}{}^{2}(z-\tau)u_{3}(z-\tau)+2u_{1}{}^{2}(z-\tau)u_{5}(z- \tau)+u_{2}{}^{2}(z-\tau)  
\[(\rho Av^{2}-\lambda AG-G_{p})\frac{d^{2}q_{1}}{dz^{2}}+(-cv)\frac{dq_{1 }}{dz}+k_{1}q_{1}+\lambda AG\frac{d\psi_{1}}{dz}=A_{6}\delta(z), \tag{108}\] \[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\] \[(\rho Av^{2}-\lambda AG-G_{p})\frac{d^{2}q_{2}}{dz^{2}}+(-cv)\frac {dq_{2}}{dz}+(-\omega^{2}\rho A+k_{1})q_{2}+\lambda AG\frac{d\psi_{2}}{dz}=A_{4 }\delta(z)\] \[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{2}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {2}}{dz}+(-\omega^{2}\rho I+\lambda AG+k_{f})\psi_{2}-\lambda AG\frac{dq_{2}}{ dz}=0.\]  
The Fourier transform is defined as follows  
\[\hat{q}_{j}(\xi)=\int_{-\infty}^{+\infty}q_{j}(z)e^{-i\xi z}dz, \quad q_{j}(z)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\hat{q}_{j}(\xi)e^{i\xi z }d\xi, \tag{109}\] \[\hat{\psi}_{j}(\xi)=\int_{-\infty}^{+\infty}\psi_{j}(z)e^{-i\xi z }dz,\quad\hat{\psi}_{j}(z)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\hat{\psi}_{j }(\xi)e^{i\xi z}d\xi,\quad j=1,2,\]  
By applying a Fourier transform to both sides of each equation and then rearranging, we can derive the following:  
\[[(-\rho Av^{2}+\lambda AG+G_{p})\xi^{2}-cvi\xi+k_{1}]\hat{q}_{1}+ \lambda AGi\xi\hat{\psi}_{1}=A_{6}, \tag{110}\] \[[(EI-\rho Iv^{2})\xi^{2}-c_{f}vi\xi+k_{f}+\lambda AG]\hat{\psi}_{1 }-\lambda AGi\xi\hat{q}_{1}=0,\] \[[(-\rho Av^{2}+\lambda AG+G_{p})\xi^{2}-cvi\xi+(k_{1}-\omega^{2} \rho A)]\hat{q}_{2}+\lambda AGi\xi\hat{\psi}_{2}=A_{4},\] \[[(EI-\rho Iv^{2})\xi^{2}-c_{f}vi\xi+(k_{f}+\lambda AG-\omega^{2} \rho I)]\hat{\psi}_{2}-\lambda AGi\xi\hat{q}_{2}=0.\]  
Equation (110) is a system of linear equations concerning \(\hat{q}_{1,2}\) and \(\hat{q}_{1,2}\); solving it yields  
\[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\]0 \[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\]1 \[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\]2 \[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\]3  
\[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\]4 \[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\]5 \[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\]6 \[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\]7 \[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\]8 \[(\rho Iv^{2}-EI)\frac{d^{2}\psi_{1}}{dz^{2}}+(-vc_{f})\frac{d\psi_ {1}}{dz}+(\lambda AG+k_{f})\psi_{1}-\lambda AG\frac{dq_{1}}{dz}=0,\]9 \[(\rho Av^{2}-\lambda AG-G_{p})\frac{d^{2}q_{2}}{dz^{2}}+(-cv)\frac {dq_{2}}{dz}+(-\omega^{2}\rho A+k_{1})q_{2}+\lambda AG\frac{d\psi_{2}}{dz}=A_{4 }\delta(z)\]0 \[(\rho Av^{2}-\lambda AG-G_{p})\frac{d^{2}q_{2}}{dz^{2}}+(-cv)\frac {dq_{2}}{dz}+(-\omega^{2}\rho A+k_{1})q_{2}+\lambda AG\frac{d\psi_{2}}{dz}=A_{4 }\delta(z)\]1 \[(\rho Av^{2}-\lambda AG-G_{p})\frac{d^{2}q_{2}}{dz^{2}}+(-cv)\frac {dq_{2}}{dz}+(-\omega^{2}\rho A+k_{1})q_{2}+\lambda AG\frac{d\psi_{2}}{dz}=A_{4 }\delta(z)\]2By carrying out an inverse Fourier transform across both sides of each equation within Equation (A19), we derive the following results:  
\[