## 2 Basic Theory of Cable Dynamics Analysis
### Solution of Dynamic Characteristics  
The above frequency Equations (14) and (20) can be used to calculate the dynamic characteristics of bridge cables. Obviously, both forms of frequency equations are complex transcendental equations, and ordinary algorithms for solving them will encounter difficulties, high computational effort, and low accuracy, or even loss of modalities. Therefore, in this paper, the Wittrick-Williams (W-W) algorithm [16] is chosen for the calculation of cable frequencies, which can solve the cable frequency equation accurately and efficiently.  
The W-W algorithm is not directly used for frequency calculation, but obtains the number of modal frequencies of the structure less than the trial frequency \(\omega^{*}\) by counting, so that the upper and lower bounds of arbitrary order frequencies can be determined. Finally, the frequency solution of arbitrary accuracy can be obtained by combining the dichotomous method or [PERSON]'s method. Since this method allows the dynamic stiffness matrix to maintain the Sturm sequence properties, it can ensure that no roots are missed during the solution of the equation, and its stability and correctness have been proved theoretically, which is an advantage that most analytical and approximate solution methods do not have [16]. The key step in applying the W-W algorithm to calculate the structural modal frequency problem is to determine the modal frequency count \(J\) of the structure, which represents the number of modal frequencies of the structure less than \(\omega^{*}\):  
\[J=J_{0}+s\Big{\{}\mathbf{K}^{\Delta}(\omega^{*})\Big{\}}=J_{0}+J_{K} \tag{21}\]  
where, \(J_{K}\) is the count related to the overall dynamic stiffness matrix of the structure, which is equal to the number of negative elements on the main diagonal of the triangular matrix \(\mathbf{K}^{\Delta}(\omega^{*})\); \(\mathbf{K}^{\Delta}(\omega^{*})\) is the upper triangular matrix formed by the Gauss elimination of the dynamic stiffness matrix \(\mathbf{K}(\omega^{*})\) of the structure; \(s\{\cdot\}\) is the count symbol; and \(J_{0}\) is the count of the solid end frequency of the structure, which is equal to the number of solid end frequencies of the structure less than the trial frequency \(\omega^{*}\). The solid end frequency of the structure is the modal frequency after the boundary conditions of the structure are all solidified, which may be called the solid end structure corresponding to the original structure.  
After calculating the solid end frequency count \(J_{0}(\omega^{*})\) of the structure under the trial frequency \(\omega^{*}\), the number \(J(\omega^{*})\) of modal frequencies of the structure less than the trial frequency \(\omega^{*}\) can be calculated according to Equation (21). Thereafter, after first roughly determining the upper and lower bounds \(\omega_{l}\) and \(\omega_{u}\) of the modal frequency of that order, the \(i\)-th order modal frequency of the structure can be calculated, so that it satisfies:  
\[J(\omega_{l})\leq i-1,J(\omega_{u})\geq i \tag{22}\]  
The \(i\)-th order modal frequency of the structure satisfies \(\omega_{i}\in[\omega_{l},\omega_{u}]\), and thereafter the dichotomous method [17] can be used to approach the true frequency \(\omega_{i}\) by continuously adjusting the values of the upper and lower bounds. When \(\omega_{l}-\omega_{u}\leq Tol(1+\omega_{u})\) is met, the modal frequency within the allowable error range \(Tol\) can be obtained. In this paper, the method is used to write a program to solve the modal frequencies of each order of the cable.