## 6 Conclusions  
An accurate FEM with a relative error of about 1% for deflection and 29% for acceleration compared with the data from the SHM system in modeling a train passing case of a rail-cum-road cable-stayed bridge was established based on the ANSYS software to analyze the vibration behavior of the bridge and the cable. To comprehend the vibration behavior of the cable under the double-end excitations, the partial differential equations were derived by neglecting the bending stiffness and numerically solved using MATLAB software [33]. The moving loads method was adopted to simulate the train, and the cable vibration under the action of train loads was analyzed in detail. Some conclusions can be summarized, as follows:  
* The cable will generate parametric vibration under the end axial displacement excitation and forced vibration under the end vertical displacement excitation. The numerical vibration model established through simplifying the two forms of excitation agreed well with the multi-segment rod finite element model analysis results by less than 1% relative errors for its first four frequencies. In practical applications, the numerical model is more straightforward to use than the finite element model and can quickly calculate the vibration response of any position of the cable for the symmetrical bridge structures;
* Under the action of the high-speed train, the displacement and acceleration amplitude of the cable near the train load side were larger; the maximum displacement amplitude appeared on the longest cable near the middle of the main span, and the acceleration amplitude of the cable in the main span was almost the same; and the cables on the side spans of the bridge were relatively less affected by the train load;
* While high-speed trains pass over the bridge with a velocity of over 200 km per hour, the excitation generated on the cable end will make the cable vibrate, and the cable vibration frequency will be close to the dynamic frequency excited by the train wheel. Therefore, it is necessary to avoid the train passing over the bridge at certain velocities to avoid cable resonance.
* As the PDEs of the cable vibration were derived with four assumptions, the numerical model in this paper can be only used for slender cables on which the bending and torsional stiffness can be ignored. Further work will be conducted on the fatigue assessment of the cable based on its vibration character studies in this paper.  
Conceptualization, H.Z. and F.Y.; methodology, A.L.; software, H.Z. and F.Y.; validation, Z.F., H.Z. and F.Y.; formal analysis, Z.F. and H.Z.; investigation, F.Y. and Z.F.; resources, H.Z.; data curation, F.Y. and H.Z.; writing--original draft preparation, F.Y. and H.Z.; writing--review and editing, F.Y.; visualization, F.Y.; supervision, A.L.; project administration, A.L.; funding acquisition, H.Z., Z.F. and A.L. All authors have read and agreed to the published version of the manuscript.  
Figure 10: The displacement time history of the end excitation and midpoint of the cable.  
Funding: This research was supported by the National Natural Science Foundation of China (Grants. 52008099 and 52008202) and the Natural Science Foundation of Jiangsu Province (Grant. BK20200369). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Due to the nature of this research, participants of this study did not agree for their data to be shared publicly, so supporting data are not available. Conflicts of Interest: The authors declare no conflict of interest. Abbreviations