PSI - Issue 64

Cevdet Enes Cukaci et al. / Procedia Structural Integrity 64 (2024) 531–538 Cukaci and Soyoz/ Structural Integrity Procedia 00 (2024) 000 – 000

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for the purposes of the calculation, the bending stiffness is a constant value along the cable's length. The boundary conditions at both ends of the cable as hinge boundary conditions are assumed as (0, ) = 0 , ( , ) = 0 , ′′(0, ) = 0 , ′′( , ) = 0 (3) Ignoring the axial vibration of the cable and assuming the boundary conditions, the equation evolves into = 4 2 ( ) 2 − 2 2 2 (4) In the equation, m stands for the mass per unit length of the cable, and l e represents the cable's effective length. f n denotes the natural frequency corresponding to the cable's n th vibration mode. The tension within the cable can be estimated by considering these factors along with the cable's bending stiffness. If the bending stiffness of the cable is assumed to be small, then the equation can be written as = 4 2 ( ) 2 (5) According to the above formula, since the natural frequency of the cable in each vibration mode is used as shown in Fig. 2(b), measurement errors and errors caused by cable sagging and non-linear bending stiffness can be minimized. In longer cables, the estimation of cable tensions can be more accurate, since the effect of the bending stiffness of the cables is less (Russell & Lardner, 1998; Zui et al., 1996). The vibration method is firstly used to estimate the cable tension by using the vibration response under four seismic events and three hours before them from accelerometers. The acceleration data is transformed into power spectral densities (PSDs) in the frequency domain using Fast Fourier Transform (FFT). The cable's natural frequencies for each vibration mode are identified by locating the peak frequencies. From these natural frequencies and the cable's characteristics, the cable tensions are estimated through the vibration method.

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Fig. 2 (a) simply-supported cable model; (b) linear regression for tension estimation in vibration method.

3. Vision-based dynamic response calculation of cable A Nikon D750 DSLR camera with a CMOS sensor is utilized to measure dynamic response of the stay cables in this study. This camera records video at a resolution of 1920x1080 pixels and a rate of 60 frames per second. The lens focal length is set between 50 mm and 200 mm, with manual focus to ensure clarity. Known that natural frequencies of the bridge cables are typically below 10 Hz, the camera's frame rate is adequate for capturing frequencies up to 30 Hz without aliasing issues according to the Nyquist criterion. In the vision-based analysis, the first video frame is selected as a reference template. A region of interest (ROI) in the reference frame is defined to reduce computational load, and this window is used in analyzing subsequent images. A quadratic polynomial function corrects pixel positions for geometric distortions due to movement and displacement, enabling subpixel scale calculations of cable displacement. Video frames are then converted into binary by utilizing the contrast between the cable and its background with the intensity of the light. A specific row within the ROI is manually selected in the first frame. This edge point's position is then tracked along the successive to determine the cable's displacement. The displacement responses are transformed into power spectra using the FFT. The cable tensions are then estimated by analyzing these peaks in the spectra for the natural frequencies and shape conditions of each cable. Calibration or scale factor calculation is not required since these transformations do not affect the location of these peaks in the frequency domain (Feng & Feng, 2018). The algorithm during displacement response calculation of each cable is shown in Fig. 3.