Estimating the standard deviation of eigenvalue distributions for the nonlinear free-vibration stochastic dynamics of cable networks

Cross-ties are used on cable-stayed bridges for mitigating wind-induced stay vibration. The system obtained by connecting two adjacent stays with a transverse cross-tie is the basic component of an “in-plane cable network”. Failures in the restrainers of cable networks have motivated investigations focusing on the nonlinear dynamics of cable networks. In these problems, the nonlinearity in the restoring force transferred by the cross-tie is used to simulate the behavior at incipient failure. The “equivalent linearization method” can be used to reduce the system of nonlinear dynamic equations to an equivalent eigenvalue/eigenvector problem, which is solved algebraically as a function of a reference amplitude parameter. Since the value of the initial vibration amplitude during aeroelastic vibration is affected by uncertainties, simulations based on stochastic processes must be considered. The resulting random nonlinear eigenvalue problem can be solved with an implementation of the standard stochastic approximation (SA). We propose here a novel numerical procedure, based on a “layered” SA algorithm, to estimate not only the mean but the standard deviation of the eigenvalue distribution as well. This method is computationally efficient and can accurately evaluate the average and standard deviation of the random eigenvalue (frequency), mode by mode. Therefore, we have now enabled a more complete and numerically efficient characterization of the frequency probability distribution.