## 2 Methodology
### Dynamic Model for Forced Vibration  
Equation (10) shows the state of external excitation \(p\) for time-harmonic normal point load. Symbols \(\delta(x)\) and \(\delta(\theta)\) are Dirac delta functions. Symbol \(F\) represents the normal harmonic point load amplitude.  
\[p(x,\theta,t)=F\frac{\delta(x)\delta(\theta)}{R}e^{i\omega t} \tag{10}\]  
Fourier transformation on the axial, circumferential, radial displacement and excitation changes the positive and negative relationship between axial wavenumber and the time factor. Its matrix form is as follows. The linear combination of these spatially harmonic components offers the total displacements resulting from the harmonic point load,  
\[\left[\begin{array}{ccc}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\end{array}\right]\left[\begin{array}{c}\widetilde{U}\\ \widetilde{V}\\ \widetilde{W}\end{array}\right]=\frac{(1-\mu^{2})R^{2}}{Eh}\left[\begin{array} []{c}0\\ 0\\ \widetilde{P}\end{array}\right] \tag{11}\]  
where harmonic stress \(\widetilde{P}\) for particular circumferential mode number \(n\) is to be deduced from Equation (10) as  
\[\widetilde{P}=\left\{\begin{array}{cc}F/2\pi R,&n=0\\ F/\pi R,&n\geq 1\end{array}\right. \tag{12}\]  
This study sets matrix \([C_{3\times 3}]\) to be the inverse of matrix \([A_{3\times 3}]\). Equation (11) becomes  
\[\left[\begin{array}{c}\widetilde{U}\\ \widetilde{V}\\ \widetilde{W}\end{array}\right]=\frac{\widetilde{P}(1-\mu^{2})R^{2}}{Eh} \left[\begin{array}{c}C_{13}\\ C_{23}\\ C_{33}\end{array}\right] \tag{13}\]  
The inverse Fourier transformation can produce spatial-domain axial \(u_{x}\), circumferential \(v_{x}\), and radial \(w_{x}\) displacement as follows:  
\[\left[\begin{array}{c}u_{x}\\ v_{x}\\ w_{x}\end{array}\right]=\frac{\widetilde{P}(1-\mu^{2})R^{2}}{Eh}\left[\begin{array} []{c}\int_{-\infty}^{+\infty}C_{13}e^{i\frac{\lambda}{R}t}d\lambda\\ \int_{-\infty}^{+\infty}C_{23}e^{i\frac{\lambda}{R}t}d\lambda\\ \int_{-\infty}^{+\infty}C_{33}e^{i\frac{\lambda}{R}t}d\lambda\end{array}\right] \tag{14}\]Equation (14) can be calculated using the residue theorem [11]. It can compute the integral form of Equation (14) by finding the singularity of the function. Then, the equations for the displacements of the coupled system in the form of the residual method are  
\[\left[\begin{array}{c}u_{x}\\ v_{x}\\ w_{x}\end{array}\right]=\frac{\widetilde{P}\left(1-\mu^{2}\right)R^{2}}{Eh}\left[ \begin{array}{c}\sum_{k=1}^{k}\text{Res}[\text{C}_{13}(\lambda_{s})]\\ \sum_{k=1}^{k}\text{Res}[\text{C}_{23}(\lambda_{s})]\\ \sum_{k=1}^{k}\text{Res}[\text{C}_{33}(\lambda_{s})]\end{array}\right] \tag{15}\]  
Symbol \(k\) is the roots' number of Equation (9) and \(\text{Res}[\text{C}_{3\times 1}(\lambda_{s})]\) is the residue of the integrand. The linear combination of spatially harmonic components yields the total displacements induced by a harmonic point load. The summation of an infinite number of circumferential modes is impossible. Therefore, achieving satisfactory convergence becomes the determining criterion for the number of modes. Also, the time-harmonic displacement response can be obtained through the inverse Fourier transform of the summation of circumferential modes in the wavenumber domain. The total time-harmonic displacements are as follows. Displacements \(\widetilde{U}_{n}\), \(\widetilde{V}_{n}\), and \(\widetilde{W}_{n}\) correspond to a circumferential mode \(n\) from Equation (11) by applying the harmonic point load as in Equation (12).  
\[\left[\begin{array}{c}u\\ v\\ w\end{array}\right]=\frac{1}{2\pi R}\int_{-\infty}^{+\infty}\left(\left[ \begin{array}{c}\widetilde{U}_{0}\\ 0\\ \widetilde{W}_{0}\end{array}\right]+\sum_{n=1}^{\infty}\left[\begin{array}{c} \widetilde{U}_{n}\cos n\theta\\ \widetilde{V}_{n}\sin n\theta\\ \widetilde{W}_{n}\cos n\theta\end{array}\right]\right)e^{i\frac{\lambda}{R}t}d \lambda e^{i\omega t} \tag{16}\]