# Variational Mode Decomposition Based Time-Varying Force Identification of Stay Cables
###### Abstract  
Stay cables are important structural members of cable-stayed bridges, which play a significant role in the health monitoring and assessment of cable-stayed bridges. The in-service cable force, which varies from the effects of vehicle load, wind load and other environmental factors, may cause fatigue damage in stay cables. Traditional force identification methods can only calculate the time-average cable force instead of the instantaneous force. A novel method has been proposed in this paper for identifying time-varying cable tension based on the variational mode decomposition (VMD) method. This recent method decomposes signals and adaptively estimates instantaneous frequency combined with the Hilbert-Huang transform method. In the proposed study, the time-varying modal frequencies were identified from stay cable acceleration data, and then the time-varying cable tension was identified by the relationship between cable tension and identified fundamental frequency. Scaled and full-scale models of stay cables were implemented successively to illustrate the validity of the proposed method. The results showed that the variational mode decomposition (VMD) method has a good effect on identifying the time-varying cable forces, even the sudden changes in cable force. According to the cable force identification results, the maximum error was 8.4%, which meets the actual application of time-varying cable force measurements. An on-site test was also implemented to monitor the cable force during a construction period, and the results showed that the proposed method can provide accurate real-time results for evaluation and decision-making. variational mode decomposition; stay cables; time-varying force; cable force identification; hilbert-huang transform +
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Many studies on the methods of cable force identification have been carried out [1; 2; 3; 4; 5]. Currently, cable force identification methods mainly include the static force-based method, which directly measures the cable force of a stay cable by embedding a tension sensor at the end of the stay cable's anchoring; the vibration-based method [1; 6; 7], which calculates cable force by the cable force-fundamental frequency formula; the method in which vibration frequency is calculated by accelerations of the stay cable using fast Fourier transform; and the magnetic flux-based method [8; 9], which uses the correspondence between magnetic flux and cable force. For any type of stay cable method, several sets of tests under stress and temperature have been performed in the laboratory to establish the relationship between magnetic flux, structural stress and temperature, which can then be used to identify the cable force of a specific material. For existing bridges, it is not possible to install a tension sensor during the service period, and it is cost-consuming to use the magnetic flux-based method. In recent years, the vibration-based method has been widely studied, which has the advantages of inexpensiveness and convenience. Currently, the calculation models of cable forces based on vibration [10] are mainly divided into four categories: tense string model theory, simple-supported beam model theory, fixed-supported beam theory, and complex boundary model theory. The tension string theory equates the stay cable to a tensionless string without stiffness and sag, and a vibration differential equation is established to calculate the cable force:  
\[F-4 mL^{2}\left(\frac{f_{n}}{n}\right)^{2}=0 \tag{1}\]  
where \(F\) is the cable force, \(L\) is the length of the stay cable, \(f_{n}\) is the \(n\)th vibration frequency of the stay cable, \(n\) is the order of the vibration mode, and \(m\) is the mass per unit cable length. Therefore, as long as the vibration frequency of the stay cable is obtained, the cable force value can be calculated. However, in practical applications, stay cables have a certain stiffness and sag.