# Variational Mode Decomposition Based Time-Varying Force Identification of Stay Cables
###### Abstract
Bridge experiments [11; 12] which ignored bending stiffness were implemented to identify the cable force according to the tension string theory, and the error between the measured value of the cable force and true value remained within 5%. The simple-supported beam model equates the stay cable to a horizontal beam that hinges at one end and is simply supported at another end. Stay cables usually have stiffness, so they can be considered as simply supported beams. According to this theory, the cable force can be calculated according to the following formula:  
\[F-4 mL^{2}\left(\frac{f_{n}}{n}\right)^{2}+EI\frac{n^{2}\pi^{2}}{L^{2}}=0 \tag{2}\]  
where \(EI\) refers to the bending stiffness of the stay cable, and the meanings of other symbols are the same as in Equation (1). Related studies [13; 14] proposed several practical simplified formulas according to this theory and the measured natural frequency of the stay cable. The measured natural frequency according to their theory was only 3% different from the theoretical value. Compared with the simple-supported beam model, the fixed-supported beam model uses fixed-support constraints at both ends. Specifically, the boundary conditions are different. Due to the clamped constraint, the displacement and rotation angle at both ends is 0. The cable force calculation formula is shown in Equations (3)-(6):  
\[2\sigma e[1-\cos(\delta l)\cos(\epsilon l)]+\left(\epsilon^{2}-\delta^{2} \right)\sin(\delta l)\sinh(\epsilon l)=0 \tag{3}\]  
\[\delta l=\frac{\xi}{\sqrt{2}}\sqrt{\sqrt{1+\left(\frac{2\pi_{1}^{2}}{\xi} \varphi_{n}\right)^{2}-1}} \tag{4}\]\[el=\frac{\xi}{\sqrt{2}}\sqrt{\sqrt{1+\left(\frac{2\alpha_{1}^{2}}{\xi} \varphi_{n}\right)^{2}}+1} \tag{5}\] \[\xi=\frac{F}{EI}l\qquad\varphi_{n}=\frac{f}{\frac{\alpha_{1}}{2 \pi l^{2}\sqrt{\frac{EI}{m}}}} \tag{6}\]  
where \(l\) is the length of the stay cable, \(\varphi_{n}\) and \(F\) can be iteratively calculated according to the given \(\alpha_{1}=4.730\) and initial value \(\xi\). The meanings of other symbols are the same as in Equation (1). In practical applications, the boundary conditions of stay cables are more complicated [15; 16] and may be between simple-supported and fixed-supported. Related studies [17; 18] established a unified and simplified model of stay cables under elastic boundary conditions. Unlike the previous three models, the elastic boundary can withstand a certain bending moment and a certain angle by which the bending moment can be determined. The calculation formula is shown in Equation (7):  
\[F-\frac{4m\pi^{2}l^{2}f_{n}^{2}}{(n\pi-\theta_{n})^{2}\Big{(}1+\beta^{2}(n\pi- \theta_{n})^{2}\Big{)}}=0 \tag{7}\]  
where \(l\) is the length of the stay cable, and \(\theta_{n}\) and \(\beta\) are parameters related to boundary conditions and the meanings of other symbols are the same as in Equation (1). The above theories and vibration models provide formulas for vibration-based cable force measurement methods, but there are still few methods for measuring the time-varying cable force. [PERSON] et al. [19] used the Kalman filter algorithm to estimate the time-varying cable force according to the measured acceleration data with extra wind speed data measured on site. [PERSON] et al. [20] proposed an algorithm called complexity pursuit to identify time-varying cable tension forces according to acceleration data. This method requires two independent acceleration data points from two accelerometers on a cable. [PERSON] et al. [21] proposed a sparse time-frequency decomposition method [22; 23; 24] that calculates the time-varying cable force by estimating the instantaneous frequency. Aiming at time-varying cable force identification, this present study leverages the advantages of variational mode decomposition (VMD) and the Hilbert-Huang transformation (HHT) theories to propose a satisfactory method, and a single accelerometer without extra information such as wind speed was used for more convenient application.