Full-field identification of coupled vortex-induced vibration in nonlinear sagged stay cables with spatially dense measurements via physics-constrained deep learning

Vortex-induced vibrations (VIVs) in long-span stay cables often induce substantial undesirable amplitudes of vibration and exhibit intricate nonlinear characteristics. These complexities typically stem from the interaction between the fluid and structure, the cable’s geometric nonlinearity, and the coupling in both the in-plane (IP) and out-of-plane (OP) directions. The first-principle modeling and analysis of long-span cables’ VIV face significant challenges due to incomplete knowledge of complex fluid-structure interactions and the pronounced nonlinearity in cables. In this study, we develop a physics-constrained deep-learning framework, leveraging the full-field, spatially dense measurements, for robust and interpretable identification of the complex nonlinear cable VIV behaviors. Specifically, the deep learning network integrates an autoencoder for low-dimensional nonlinear feature projection from noisy, spatially dense full-field measurements; and a normalizing flow neural architecture with physics-constrained loss functions to accurately identify nonlinear normal modes (NNMs) and reconstruct high-spatial-dimensional invariant manifolds. This neural architecture, termed full-field physics-constrained normalizing flow ( f PNF), strictly adheres to fundamental dynamic principles and represents spatially nonlinear dynamic behaviors through interpretable, high-spatial-dimensional invariant manifolds. The f PNF is tested using numerically simulated nonlinear cable VIV responses of a derived nonlinear wake oscillator model. It is found that the f PNF robustly identifies NNMs and spatially-spanned invariant manifolds from full-field, very noisy measurements and accurately reconstructs the high-dimensional VIV response, even in the case of coupled VIV in the IP and OP directions of the cable. These identified spatially-dense nonlinear mode shapes and invariant manifolds enable effective analysis and visualization of spatially nonlinear dynamic behaviors along the cable’s length and in both IP and OP directions. Notably, it enables the findings of the presence of internal resonance between the IP and OP directions and further elucidates the nonlinear characteristics across the cable span in both vibrational directions. In addition, a parametric study is conducted, indicating that the identification and reconstruction accuracy improves gracefully with the increasing number of spatial measurement points. Finally, the applicability of this method for complex cable VIV is also discussed.