## 3 Effect of Boundary Conditions on Dynamic Characteristics of Cable  
### Effect of Changing Boundary Conditions on Frequency  
(1) Ideal state equality boundary  
In the ideal state, the main cable stiffness and stiffening beam stiffness are considered to be nearly the same, that is, assuming that \(v_{A}\) and \(v_{B}\) are 0, 0.2, 0.4, 0.6, and 0.8, respectively, the rotational stiffness at both ends of the corresponding boundary becomes larger from 0, and the boundary conditions at both ends of the cable evolve gradually from simple to solid support. The cable parameters are shown in Table 2 for five cables of different lengths arbitrarily. Using the cable frequency calculation method in this paper, the frequency variation of each cable under different boundary condition coefficients is studied and plotted as a curve, as shown in Figure 2.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(\eta=\frac{k^{gl}}{\varGamma}\)} & \multirow{2}{*}{\(k_{A}^{0}=k_{B}^{0}\)} & \multirow{2}{*}{\(v_{A}=v_{B}\)} & \multicolumn{2}{c}{**First-Order Frequency (Hz)**} & \multirow{2}{*}{\(\delta_{1}\)} & \multicolumn{2}{c}{**Second-Order Frequency (Hz)**} & \multirow{2}{*}{\(\delta_{2}\)} \\ \cline{5-5} \cline{5  
From Figure 2, it can be obtained that in the process of changing the boundary condition coefficient, the change trend of each order frequency of cables with different tension-bending ratios \(\xi\) is almost the same but with the increase in the tension-bending ratio \(\xi\), the change of each order frequency of cables from simple support to solid support is less obvious. When the tension-bending ratio is \(\xi\geq 15\), the change of frequency of cables with boundary conditions is more obvious. During the change of boundary condition coefficient, when \(\upsilon_{A}=\upsilon_{B}=0.4\), with the increase in the boundary condition coefficient at both ends, each order frequency of each cable basically remains the same. At this time, both ends of the boundary have tended to solid support, that is, when the boundary condition coefficient is 0.4, the upper and lower boundaries of the cable can be treated as the solid support boundary. (2) Actual non-equal boundary  
In actual engineering, the constraints of suspension bridge main cable and stiffening beam on the cable, or arch bridge, arch rib, and tie beam on the cable are different. Therefore, a material influence coefficient \(\lambda_{AB}\) is introduced, where \(\lambda_{AB}\) is the ratio of the stiffness of the main cable to the stiffening beam of a suspension bridge (or the ratio of the stiffness of the arch rib to the stiffness of the tie beam of an arch bridge), referred to as the stiffness  
Figure 2: Variation of each order frequency of each cable under equal boundary conditions: (**a**) to (**f**) denote the variation of the first to sixth order frequencies of each cable under equal boundary conditions with different boundary condition coefficients, respectively. ratio, i.e., \(\upsilon_{A}=\lambda_{AB}\upsilon_{B}\). Reinforced concrete and steel are the main materials used for stiffened girders in suspension bridges, but the materials can vary from plain concrete to steel. The flexural stiffness of plain concrete and reinforced concrete stiffening beam is less than the stiffness of the main cable with strong tensile reserves, and the impact of the flexural stiffness of this kind of stiffening beams is only of secondary importance to the structural behavior of the whole bridge, while the flexural stiffness of the steel stiffening beam is significantly greater than the flexural stiffness of the main cable made of steel strands. For the now commonly used parallel steel wire main cable, the research [19] shows that its internal structure is closest to that of a semi-parallel steel wire cable, and its flexural stiffness is 0.37 times that of the steel strand.