## 3 Numerical Calculation and Discussion of Results
### Effect of the Pasternak Elastic Foundation  
This section discusses the influence of the Winkler foundation stiffness \(k_{i}\) (\(i=1,3\)) and Shear subgrade modulus \(k_{i}\) (\(j=2,4\)) on the natural frequency and modal loss factor. The Pasternak elastic foundation is applied to the interior of the MRCS, resulting in \(k_{1}=k_{2}=0\), and the MRCS is in an internal pretension state. Conversely, when the Pasternak elastic foundation is applied externally to the MRCS, \(k_{3}=k_{4}=0\), and the MRCS is in an externally preloaded state. Figures 8 and 9 illustrate the impact of \(k_{1}\) and \(k_{2}\) on the natural frequency of MRCS in the external support state. The results indicate that the natural frequency of MRCS increases with an increase in \(k_{1}\), with a more pronounced effect when \(k_{1}<10^{9}\) N/m. However, the effect of \(k_{2}\) on the natural frequency of MRCS is relatively weak. When \(m<30\), an increase in \(k_{2}\) leads to an increase in natural frequency, while when \(30<m<35\), the effect of \(k_{2}\) is similar to that of artificial spring technology. When \(k_{2}<10^{6}\), the shear deformation of the MRCS can be regarded as unconstrained. Conversely, when \(k_{2}>10^{9}\) \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Preload**} & \multirow{2}{*}{\(n\)} & \multicolumn{3}{c}{**Natural Frequency**} & \multicolumn{3}{c}{**Modal Loss Factor**} \\ \cline{3-8}  & & _C-C_ & S-S & _F-F_ & _C-C_ & _S-S_ & _F-F_ \\ \hline \multirow{8}{*}{Internal preload} & 1 & 37.342 & 18.182 & 4.502 & 0.052 & 0.208 & 0.163 \\  & 2 & 45.455 & 32.393 & 12.896 & 0.140 & 0.262 & 0.153 \\  & 3 & 54.907 & 45.920 & 22.743 & 0.215 & 0.281 & 0.122 \\  & 4 & 60.380 & 53.575 & 32.947 & 0.188 & 0.265 & 0.098 \\  & 5 & 67.262 & 62.236 & 43.175 & 0.162 & 0.226 & 0.084 \\  & 6 & 75.153 & 71.428 & 53.338 & 0.138 & 0.186 & 0.076 \\  & 7 & 83.731 & 80.917 & 63.425 & 0.118 & 0.152 & 0.072 \\  & 8 & 92.774 & 90.595 & 73.447 & 0.100 & 0.125 & 0.072 \\  & 9 & 102.137 & 100.406 & 83.412 & 0.086 & 0.104 & 0.074 \\  & 10 & 111.725 & 110.319 & 93.322 & 0.074 & 0.087 & 0.077 \\ \hline \multirow{8}{*}{External preload} & 1 & 37.341 & 18.180 & 4.500 & 0.052 & 0.208 & 0.163 \\  & 2 & 45.451 & 32.388 & 12.893 & 0.140 & 0.262 & 0.153 \\  & 3 & 54.904 & 45.916 & 22.738 & 0.215 & 0.281 & 0.122 \\  & 4 & 60.376 & 53.570 & 32.941 & 0.188 & 0.265 & 0.098 \\  & 5 & 67.256 & 62.229 & 43.167 & 0.162 & 0.226 & 0.084 \\  & 6 & 75.145 & 71.419 & 53.328 & 0.138 & 0.186 & 0.076 \\  & 7 & 83.721 & 80.906 & 63.414 & 0.118 & 0.152 & 0.072 \\  & 8 & 92.762 & 90.582 & 73.435 & 0.100 & 0.125 & 0.072 \\  & 9 & 102.123 & 100.392 & 83.398 & 0.086 & 0.104 & 0.074 \\  & 10 & 111.709 & 110.303 & 93.306 & 0.074 & 0.087 & 0.077 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Validation of method validity (\(m=1\)). Figure 6: The natural frequencies and modal loss factors of the MRCS under various two-end boundary conditions in the un-preloaded state (\(m=1\)): (**a**) natural frequency; (**b**) modal loss factor. the shear deformation of the MRCS is fixed. When \(10^{6}<k_{2}<10^{9}\), the Pasternak elastic foundation can be regarded as imposing arbitrary radial boundary conditions on the MRCS. It is worth noting that both \(k_{1}\) and \(k_{2}\) have effective ranges, and numerical instability will occur in the calculation if \(k_{1}\) and \(k_{2}\) are too high. Figure 10 compares the natural frequency and modal loss factors of the MRCS in internal and external bracing states. The results indicate that \(k_{1}\) and \(k_{3}\) have the same effect on the natural frequency and modal loss factor of the MRCS, while \(k_{2}\) and \(k_{4}\) have the same effect on the natural frequency and modal loss factor of the MRCS. With an increase in \(k_{i}\) (\(i\) = 1, 2, 3, 4), the natural frequency increases, while the modal loss factor decreases and tends to be stable.