# Variational Mode Decomposition Based Time-Varying Force Identification of Stay Cables
## 5 Conclusions  
In this article, a new method based on VMD was established to identify the time-varying force of stay cables. VMD is a new signal decomposition method, which searches for the optimal solution of the constrained variational model established by the specified center frequency to achieve adaptive signal decomposition. Due to its advantages, VMD has been used by researchers in the field of damage diagnostics and signal decomposition. The advantage of the Hilbert-Huang transform (HHT) is that it can find the instantaneous frequency of the signal and can give a physical meaning to the instantaneous frequency of the signal. This paper analyzed the theory and implementation steps of the VMD algorithm, and HHT transformation was also integrated to show that both methods can be used to calculate the instantaneous frequency of each order mode. Then, a scaled  
Figure 16: Acceleration signal, vibration spectrum and time-varying force of the A15 cable in the time interval of 16:00–17:00 on 11 October 2019. Figure 17: Acceleration signal, vibration spectrum and time-varying force of the A15 cable in the time interval of 09:00–09:30 on 12 October 2019.  
stay cable model, a full-scale stay cable model and an on-site bridge experiment were conducted to validate the proposed method. The VMD and HHT methods were applied to the vibration acceleration signal of the stay cables, the instantaneous frequency of the first-order vibration mode of the stay cable was obtained, and the instantaneous cable force based on the subinstant frequency was calculated and compared with the real cable force. The results show that the method in this paper can successfully identify the time-varying cable force even when the mutation amplitude is more than 15%, when using one accelerometer without extra information. For the initial vibration interval, the signal contains less information so that the initial value of the iteration has greater impact, which may cause fluctuations. The errors of the scaled model and the full-scale model in the stationary stage are both below 6%, and the average error is also below 5%; there are fluctuations at the mutation point, the average error of the scaled experiment mutation is 21.702%, and the average error of the full-scale experiment mutation is 5.718%, but all of them can identify the mutation point very well, indicating that the time-varying cable force identification method based on the variational mode decomposition method has a better identification effect for stable signals, and the error at the mutation intervals is relatively large but can be identified as discontinuity. In general, the method in this paper can identify the time-varying cable force well in the experiments and the on-site test, the algorithm in this paper can be used to identify time-varying cable forces in actual engineering. Further research should be conducted focus on more on-site bridge tests for the validation and improvement of the time-varying cable force identification methods. Conceptualization, G.W. ; Methodology, S.H. ; Validation and formal analysis, B.D. ; Investigation, J.F. and Y.H. ; Data resources, H.W. and X.Z. All authors have read and agreed to the published version of the manuscript. This research is funded by the National Natural Science Foundation of China (Grant No. 51525801), Graduate Research and Innovation Projects of Jiangsu Province, China (No. KYCX17_0120) and the Shanxi Provincial Transportation Science and Technology Project (Grant No. 2019-1-19; 2019-2-4. This research is studying the cable force identification of bridges, the IRB review of this research would not be required. The permission of all materials including sensor data, images, videos, etc are granted to be used by researchers for scientific publications. Not applicable. The authors greatly appreciate the financial support from the National Natural Science Foundation of China (Grant No. 51525801), Graduate Research and Innovation Projects of Jiangsu Province, China (No.