## 5 Conclusions  
This paper starts from the dynamic stiffness theory, and uses the W-W algorithm as the basis to calculate and analyze the influence of the cable boundary conditions on the cable frequency with the tension-bending ratio \(\xi\leq 45\) using programming language. At the same time, this paper describes the boundary conditions with the boundary condition coefficients, analyzes the influence of each boundary condition on the cable frequency, and extends it to the engineering practice in reverse and puts forward the discussion of determining the boundary conditions with the cable parameters, which provides the cornerstone for the subsequent research. This paper has the following main conclusions.  
1. The analysis of the results of this paper shows that the influence of the stiffness ratio \(\lambda_{AB}\) of the upper and lower boundary of the bridge cable on the cable frequency is negligible, and only the influence of the change of the boundary coefficient on the cable frequency needs to be considered.
2. The boundary condition coefficient is used to describe the cable boundary condition with a certain accuracy, and the actual boundary condition coefficient in this paper can be used to replace the simple and fixed support conditions in the theory. When the cable vibrates transversely and the displacement stiffness of the lateral support at both ends is infinite, the boundary condition coefficient of both ends of the cable reaches 0.4 from 0, and the two ends of the cable change from simple support to fixed support, that is, the rotational stiffness of the cable is \(k^{\theta}\in\left[0,\frac{2\sqrt{H\cdot EI}}{3}\right]\), which replaces the case of judging the boundary condition by \(k^{\theta}\) in the theory.
3. In actual engineering, the boundary conditions of the cable can be determined from the cable parameters. Both ends of the cable with \(l\geq 45\sqrt{\frac{EI}{H}}\) can be calculated by using simple support boundary; when \(l\leq 15\sqrt{\frac{EI}{H}}\), the boundary conditions at both ends of the cable shall be treated as a fixed support boundary; when \(15\sqrt{\frac{EI}{H}}\leq l\leq 45\sqrt{\frac{EI}{H}}\), the boundary at both ends of the cable shall be treated as an elastic complex boundary.  
Through the analysis of the dynamic characteristics of the cable--from defining the boundary conditions of the cable with the boundary condition coefficient to the correct application of the cable boundary conditions in practical engineering--this paper verifies the accuracy with the literature and measured data, which can determine the cable boundary conditions on the basis of the cable parameters, providing a reference basis for the study of the dynamic characteristics of the cable and the health monitoring of the actual cable.  
Conceptualization, D.D. and F.H.; methodology, X.L.; software, X.L.; validation, D.D., X.L. and F.H.; formal analysis, X.L.; investigation, X.L.; resources, D.D.; data curation, D.D., X.L. and F.H.; writing--original draft preparation, X.L.; writing--review and editing, D.D. and X.L.; visualization, X.L.; supervision, D.D. and F.H.; project administration, D.D.; funding acquisition, D.D. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Data Availability Statement: Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.  
This work was supported by the Xinjiang University Tianshan Scholar Distinguished Professor Research Start-up Fund (grant numbers 620312327).  
The authors declare no conflict of interest.