## 3 Numerical Calculation and Discussion of Results
### Effect of the Preload State and Axial Boundary Conditions  
The proposed model is used in this section to calculate the intrinsic frequency and modal loss factor of the MRCS under different boundary conditions at both ends. The volume fractions utilized in the calculations were obtained from X-ray tomography tests conducted by [PERSON] et al. [15], who provided volume fractions for different preload states. Pre-strain \(\delta/h_{0}=0\), 0.8%, 1.6%, and 2.5% were used in this study to describe the pre-stressed state, where \(h_{0}\) is the thickness of the shell before deformation. The IHP model is demonstrated to be valid within the range of strain \(\varepsilon\leq\) 2.5% by [PERSON] et al., and all calculations in this paper fall within this valid range. The volume fraction fitting results for different pre-strain states are summarized in Table 2, and the fits obtained in this research are compared to the experimental results of [PERSON] et al. [15], illustrated in Figures 4 and 5. The resulting root mean square error has a mean value of 5 \(\times\) 10\({}^{-3}\), satisfying the requirements of this study. The material parameters of the MRCS are determined using the IHP model [12]: \(E_{a}\) = 4.437 MPa, \(E_{b}\) = 2.2069 Mpa, \(\mu\) = 0.088, \(G_{a}\) = 1.1042 Mpa, \(G_{b}\) = 2.3617 Mpa, \(L\) = 20 \(R\), \(h\) = 0.05 \(R\), and \(k_{1}\) = \(k_{2}\) = 0. Since MR has a small Poisson's ratio, which is close to zero with or without pre-compression [16], this paper ignores the variation of [PERSON]'s ratio and sets it to \(\mu\) = 0.088. Finally, Figure 6 illustrates the intrinsic frequencies and modal loss factors of the MRCS without pre-compression under different boundary conditions at both ends. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Vibration** & **Present** & **Zhang [36]** & **FEM** \\ \hline
**Parameters** & & & \\ \hline \(m=1\), \(n=2\) & 269.108 & 268.49 & 269.71 \\ \(m=2\), \(n=2\) & 270.855 & 272.58 & 276.95 \\ \(m=1\), \(n=3\) & 760.599 & 758.3 & 765.17 \\ \(m=2\), \(n=3\) & 774.53 & 770.67 & 762.721 \\ \(m=1\), \(n=4\) & 1456.99 & – & 1473.5 \\ \(m=2\), \(n=4\) & 1459.04 & – & 1483.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Validation of method validity. Figure 3: Error of the natural frequency and modal loss factor: (a) convergence of natural frequencies; (b) convergence of the modal loss factor. In Figure 6, the intrinsic frequencies and modal loss factors of the MRCS with different boundary conditions are presented. Specifically, at low circumferential wave numbers, the inherent frequencies for different two-end boundary conditions vary. As the circumferential wave number increases, the inherent frequencies converge, indicating a weakening influence of two-end boundary conditions on the inherent frequencies. In contrast, the modal loss factor converges to different values based on the boundary conditions at both ends. With the increase of n, the damping and damping capacity of the MRCS under the \(C\)-\(C\) and \(S\)-\(S\) boundary conditions tend to be the same, while the MRCS under the \(F\)-\(F\) boundary conditions exhibits a different damping and damping capacity. Figure 7 demonstrates the effects of different pre-strain states for the same two-end boundary conditions on the natural frequency and modal loss factor of the MRCS. The findings reveal increasing pre-strain results in an increase in the natural frequency of the MRCS, while the modal loss factor exhibits a decreasing trend. This aligns with the expectation that the number of contact points of the wire inside the MRCS increases with a pre-strain, leading to a higher stiffness and natural frequency. Additionally, the percentage of dissipated energy within the MRCS decreases, contributing to a lower modal loss factor. Notably, the peak point of the modal loss factor under the \(F\)-\(F\) boundary condition shifts gradually to higher-order vibrations as the pre-displacement increases.