## 4 Application Discussion  
### Discussion  
In the cable performance analysis, the cable boundary conditions have a great influence on the dynamic characteristics of the cable, while the cable stiffness ratio has a negligible effect on them. Since only the axial force of the cable is considered, the conclusions of the studied cable-related boundary conditions of suspension bridges can be extended to arch bridge booms and other bridge cables. In order to make the research results better applied to the engineering practice, further derivation is needed. Then, the actual boundary conditions of relevant parameter cables are inversely obtained by using the defined range of boundary condition coefficients, which is convenient for the study of dynamic characteristics of cables in practical engineering. Previous studies have shown that the effect of cable boundary conditions on the cable gradually decreases as the tension-bending ratio increases [14]. When the tension-bending ratio is \(\xi\geq 45\), the influence of the boundary conditions at both ends of the cable on the cable frequency can be neglected, but when the tension-bending ratio is \(\xi\leq 45\), the calculation of the cable must consider the influence of the actual boundary conditions. Therefore, it can be concluded that when the tension-bending ratio is \(\xi\geq 45\), the two ends of the cable with the tension-bending ratio of \(l\geq 45\sqrt{\frac{EI}{H}}\) obtained from the definition of the tension-bending ratio can be calculated with simple support or fixed support. However, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Modal Order**} & \multirow{2}{*}{**Measured Value**} & \multicolumn{2}{c}{\(v_{\text{B}}=0.4\)} & \multirow{2}{*}{\(\delta_{1}\)**(\%)**} & \multirow{2}{*}{\(\delta_{2}\)**(\%)**} \\ \cline{3-3} \cline{5-6}  & & \(\lambda_{AB}=0.3\) & & & \\ \hline
1 & 3.0586 & 3.1097 & 3.1100 & 1.67\% & 1.68\% \\
2 & 6.1406 & 6.2367 & 6.2372 & 1.56\% & 1.57\% \\
3 & 9.2344 & 9.3979 & 9.3988 & 1.77\% & 1.78\% \\
4 & 12.3867 & 12.61 & 12.6114 & 1.80\% & 1.81\% \\
5 & 15.6797 & 15.8898 & 15.8912 & 1.34\% & 1.35\% \\
6 & 19.0195 & 19.2521 & 19.2539 & 1.22\% & 1.23\% \\ \hline \hline \end{tabular} Note: where \(\delta_{1}\) is the relative error between the calculated and measured values at \(\lambda_{AB}=0.3\), \(\delta_{2}\) is the relative error between the calculated and measured values at \(\lambda_{AB}=1\), and the relative error is \(\delta=\left|\frac{\text{Calculated value}-\text{Measured value}}{\text{Measured value}}\right|\times 100\%\).  
\end{table} 
Table 3: Comparison analysis of measured and calculated frequencies. due to the convenience of the actual calculation, it is recommended to give preference to the simple support boundary. This paper mainly analyzes the cable with the tension-bending ratio of \(\xi\leq 45\). The research results show that when the cable boundary condition coefficient is \(v_{A}=0.4\), i.e., \(k^{\theta}=\frac{2\sqrt{H.EI}}{3}\), the two ends of the cable can be treated as the solid support boundary and the cable frequency changes more obviously with the boundary condition when the cable tension-bending ratio is \(\xi\geq 15\) obtained from the above analysis. Therefore, substituting \(v_{A}=0.4\), \(\xi\!=15\), and \(k^{\theta}=\frac{2\sqrt{H.EI}}{3}\) into Equation (20) can get \(l=15\sqrt{\frac{EI}{H}}\), that is, the boundary conditions of both ends of the cable with \(l\leq 15\sqrt{\frac{EI}{H}}\) shall be treated as a solid support boundary. However, in the case of the simple and solid support, it is recommended that the boundary of both ends of the cable with \(15\sqrt{\frac{EI}{H}}\leq l\leq 45\sqrt{\frac{EI}{H}}\) shall be treated as the elastic complex boundary.