## 2 Basic Theory of Cable Dynamics Analysis
### Method Validation  
The method in this paper is verified by combining the analysis process of elastic boundary conditions considering beam axial forces in the literature [18], and the frequency calculation Equation (9) shown as Equation (23) in this literature is used to calculate the frequency values of the cable under the boundary conditions in this paper. From the dimensionless parameter \(k=\frac{k^{6}l}{EI}\) in this paper, the rotational stiffnesses \(k^{6}\) of the boundary conditions at the both ends of the corresponding cable can be determined. Taking \(k\) as 1.0, 10, 100, and \(10^{3}\), respectively, the corresponding boundary condition coefficient values can be obtained. Taking a cable as an example, its material characteristics are: free length of the cable \(l=13.5\) m, modulus of elasticity \(E\!=\!2.0\times 10^{11}\) N/m\({}^{2}\), bending moment of inertia \(I\!=\!2.45\times 10^{-7}\) m\({}^{4}\), line mass per unit length \(m=14.3\) kg/m, and cable force \(H=2.63\times 10^{5}\) N.  
\[\begin{array}{l}\alpha\left(\alpha^{2}+k_{1}k_{3}\right)\left(\alpha^{2}+k _{2}k_{4}\right)\\ -[2(p^{2}+\alpha^{2})\left(\alpha^{2}k_{1}k_{2}-k_{3}k_{4}\right)+p\alpha^{2} (k_{1}k_{3}+k_{2}k_{4})-p\alpha^{2}(3\alpha^{2}+4p^{2})+pk_{1}k_{3}k_{2}k_{4} ]\sin r_{1}\mathrm{sinh}r_{2}\\ +\alpha[2\alpha^{2}(k_{1}k_{4}+k_{2}k_{3})-(\alpha^{2}-k_{1}k_{3})(\alpha^{2} -k_{2}k_{4})+2p^{2}(k_{1}+k_{2})(k_{3}+k_{4})]\cos r_{1}\cosh r_{2}\\ -\sqrt{\sqrt{p^{2}+\alpha^{2}}-p}\Big{\{}\sqrt{p^{2}+\alpha^{2}}[-(p\alpha^{2 }+k_{3}k_{4})(k_{1}+k_{2})+\big{(}2p^{2}-k_{1}k_{2}p+\alpha^{2})(k_{3}+k_{4} )]\\ -(p^{2}+\alpha^{2})[(k_{1}k_{2}-2p)(k_{3}+k_{4})-\alpha^{2}(k_{1}+k_{2})]\Big{\}} \sin r_{1}\cosh r_{2}\\ +\sqrt{\sqrt{p^{2}+\alpha^{2}}-p}\Big{\{}\sqrt{p^{2}+\alpha^{2}}[-(p\alpha^{2 }+k_{3}k_{4})(k_{1}+k_{2})+\big{(}2p^{2}-k_{1}k_{2}p+\alpha^{2})(k_{3}+k_{4} )]\\ +(p^{2}+\alpha^{2})[(k_{1}k_{2}-2p)(k_{3}+k_{4})-\alpha^{2}(k_{1}+k_{2})]\Big{\}} \cos r_{1}\mathrm{sinh}r_{2}=0\\ \text{where, }2p=Hl^{2}/(EI)\text{, }\alpha=\omega\sqrt{ml^{4}/(EI)}\text{, }r_{1}=\sqrt{\sqrt{p^{2}+\alpha^{2}}+p}\text{, }r_{2}=\sqrt{\sqrt{p^{2}+\alpha^{2}}-p}\text{, }k_{1}=\frac{k_{2}^{6}l}{EI}\text{, }k_{2}=\frac{k_{2}^{6}l}{EI}\text{, }k_{3}=\frac{k_{2}^{6}l}{EI}\text{, }k_{4}=\frac{k_{2}^{6}l}{EI}\text{.}\end{array} \tag{23}\]Using the dynamic stiffness method and the W-W algorithm, the frequency value of the cable under the corresponding boundary condition coefficient is calculated by programming and compared with the frequency calculated by using the elastic boundary condition formula in the literature. The calculation results are shown in Table 1. In order to verify the accuracy of the method for calculating the cable frequencies and the definition of the boundaries by the boundary condition coefficients, the relative errors between the calculated values in this paper and the calculated values in the literature were analyzed, and each relative error values are included in Table 1.  
From the calculation results and error analysis in Table 1, it can be seen that through the comparison of calculation results, the maximum relative error between the existing literature and the theory of this paper in calculating the order frequencies of cables under different boundary conditions is no more than 1.5%. The error analysis shows that the results calculated by the method of this paper are very close to the order frequencies calculated by the literature, that is, under the premise that the accuracy of the calculation method of the literature has been verified, the method of this paper can be used to accurately calculate the modal frequencies of each order of the cable under each boundary condition, which greatly verifies the accuracy of the theoretical analysis and calculation method of this paper providing an accurate theoretical basis for further calculation of the cable.