## 4 Validation and Discussion of the Results  
In order to demonstrate the correctness and validity of the proposed model, a numerical simulation and experimental measurement of the vibration characteristics of the MRCS are carried out in this paper. The MR samples are manufactured from stainless steel alloy wires with a diameter of 0.12 mm. The mechanical properties of 304 stainless steel are shown in Table 4.  
In the experiments in this paper, MRCSs are suspended by a soft rubber band in order to simulate the free boundary conditions at both ends of the MRCS, as shown in Figure 11a. Due to how the MRCS is almost free of forces in the horizontal direction, its boundary conditions can be approximately considered free in this direction. The force signal and the acceleration signal generated by the force hammer hitting the MRCS are collected by the force sensor and acceleration sensor and transmitted to the B&K data processing terminal for modal analysis. Finally, the natural frequency and the damping ratio of the MRCS are obtained by analyzing the obtained data through the B&K signal processing system. The experimental modeling work in this paper was carried out by the B&K data acquisition system, and the MRCSs were discretized into a polyhedral shell structure, as \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Grades** & **Yield Strength (N/mm2)** & **Young’s Modulus (Gpa)** & **Density (kg/m3)** & **Poisson’s Ratio** \\ \hline
06 Cr19 ni10 & 205 & 193 & 7930 & 0.285 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mechanical properties of stainless steel alloy wire. Figure 10: Effect of \(k_{i}\) (\(i\) =1, 2, 3, 4) on the natural frequency and modal loss factor of the MRCS under 0.8% preload (\(m\) = 2, \(n\) = 1): (**a**) Effect of \(k_{1}\) and \(k_{3}\) on the natural frequency; (**b**) Effect of \(k_{1}\) and \(k_{3}\) on the modal loss factor; (**c**) Effect of \(k_{2}\) and \(k_{4}\) on the natural frequency; (**d**) Effect of \(k_{2}\) and \(k_{4}\) on the modal loss factor. shown in Figure 11c. The characteristic parameters of the MRCSs are shown in Table 5. MRCSs are divided into nine groups of nodes in the circumferential direction and nine groups of nodes in the length direction. Table 6 shows the comparison between the theoretical values and the experimental results which were consistent, indicating that the calculated results are correct. The comparison of their modes is shown in Figure 12. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Vibration Type**} & \multicolumn{3}{c}{**Natural Frequency**} & \multicolumn{3}{c}{**Damping Ratio**} \\ \cline{2-5}  & **Theoretical** & **Experimental** & **Error (Hz)** & **Theoretical** & **Experimental** & **Error (\%)** \\  & **Value (Hz)** & **Value (Hz)** & **Value (\%)** & **Value (\%)** & **Error (\%)** \\ \hline \(m=1,n=2\) & 88.3 & 85.512 & 2.79 & 0.738 & 1.08 & 0.342 \\ \(m=3,n=3\) & 248 & 235.058 & 12.942 & 1.389 & 1.03 & 0.359 \\ \(m=2,n=4\) & 449.779 & 452.344 & 2.565 & 2.32 & 1.72 & 0.6 \\ \(m=4,n=4\) & 459.263 & 461 & 1.74 & 1.8 & 1.4 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of experimental results. Figure 11: MRCS and B&K analysis system: (**a**) modal test system; (**b**) hammer test of MRCS; (**c**) test model of B&K system. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Characteristic Parameters**} & **Numerical Value** \\ \hline \hline \multicolumn{2}{c}{Diameter of metal wire (mm)} & 0.3 \\ \multicolumn{2}{c}{Metal coil diameter (mm)} & 3 \\ \multicolumn{2}{c}{Density (\(\mathrm{kg/m^{3}}\))} & 3172 \\ \multicolumn{2}{c}{Young’s modulus (GPa)} & \multicolumn{1}{c}{5.2 \(\times\) 10\({}^{6}\)} \\ \multicolumn{2}{c}{Loss factor (\%)} & 22.609 \\ \multicolumn{2}{c}{Shell thickness (mm)} & \(1.5\times 10^{-2}\) \\ \multicolumn{2}{c}{Cylindrical shell diameter (mm)} & \(6.85\times 10^{-2}\) \\ \multicolumn{2}{c}{Cylindrical shell length (mm)} & \(6.20\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Characteristic parameters of MRCS samples.