State augmentation method for multimode nonstationary vibrations of long-span bridges under extreme winds

Stochastic dynamic analysis frequently relies on the assumption of time independence of linear systems and the stationarity of stochastic excitations, facilitating a variety of engineering studies. Nevertheless, these assumptions may not consistently remain valid, particularly in cases of structural vibrations induced by nonstationary extreme winds, and can lead to inaccurate predictions. The excitations in these scenarios have notable nonstationary characteristics because of the unstable nature of the flow. In addition, when aeroelastic forces are considered, the combined aerodynamic-mechanical system transforms into a linear time-varying system with aerodynamic damping and stiffness that change over time. In this work, a state augmentation approach for computing the multimode vibrations of a long-span bridge under nonstationary wind conditions is presented. The methodology integrates both nonstationary turbulence-induced forces and unsteady motion-induced forces. The coupling between motion-induced forces and bridge vibrations renders the system damping and stiffness matrices both time-varying and asymmetric; this results in complex-valued modes and coupled dynamics that cannot be adequately captured by a single-degree-of-freedom (SDOF) model. The proposed multi-degree-of-freedom (MDOF) approach is a stochastic calculus-based method that avoids complex modal analysis. The statistical moments of all orders for the responses of the MDOF systems are derived via Itô's formula and the stars and bars approach. Compared with existing approaches, the new approach is both reliable and efficient, demonstrating its potential for accurate and efficient analysis of nonstationary vibrations in complex engineering systems.