Nonlinear random vibration of damaged cable systems under Gaussian white noise excitations

Cable structures commonly experience structural damage due to material aging, adverse environmental conditions, and other various factors like overload. Accordingly, accurate prediction of dynamical response and analysis of their vibrational characteristics of damaged cable systems are crucial for achieving effective health monitoring and formulating appropriate repair strategies in cable bridge engineering. In this paper, the nonlinear random vibration of the damaged cable system is studied. First of all, the mathematical model for the damaged cable system under random excitation is formulated, encompassing the determination of the static configuration of the damaged stay cable and the derivation of the nonlinear stochastic dynamic equations governing the in-plane and out-of-plane motions of the damaged cable system. Subsequently, by adopting the radial basis function neural network (RBFNN) method, the reduced Fokker-Planck-Kolomogrov (FPK) equation associated with the discrete stochastic system of damaged cable system is solved to yield the steady-state probability density function (PDF) of the system. Finally, a specific cable example is studied for illustration. Some trends of the steady-state probability distribution of the PDF and the mean value (MV) under different parameters are examined, respectively. The results show that an increase in damage severity and range significantly amplifies the in-plane displacement response, but little affects the out-of-plane motion. The influence of damage location can be almost negligible. The MVs of in-plane displacement exhibit some linear trends with variations in damage severity and range. Additionally, the Monte Carlo simulations (MCS) data is utilized for validating the accuracy of the RBFNN results. The study can provide a practical approach for assessing the healthy operational condition of cable systems during monitoring processes.