Probabilistic Identification Method of Cable Force Considering Uncertain and Missing Modal Frequencies

Vibration-based cable force identification has been widely applied in engineering practice. The order of modes is typically required in the identification, but it may not be always available, e.g., due to uncontrolled excitations. Aiming at resolving this challenge and quantifying the uncertainty associated with the estimated cable force, a maximum likelihood estimation (MLE) approach is proposed in this paper following the two-stage system identification framework. The first stage involves the MLE of modal frequencies and quantifying their uncertainty in terms of the observed Fisher information matrix. The second stage regards the identified frequencies as measured data, and the MLE is further adopted considering the relationship between the cable force, flexural stiffness, and modal frequencies as an implicit and nonlinear equation. A semianalytical approach is then developed to iteratively maximize the likelihood function and satisfy the equality constraint, yielding an efficient implementation. The identification uncertainty of cable force is obtained by an analytical formula propagating the identification uncertainty of modal frequencies. The effectiveness of the proposed method is validated through numerical simulations and field test data, showing a good performance in estimating forces for long and short cables in the case of missing frequencies. The proposed method does not require information on the order of modal frequencies and is applicable to both the long and short cables, besides its capability in efficient calculation and uncertainty quantification, making it a practical and robust method for cable force identification.