Deep learning for predicting frequency lock-in of a freely vibrating sphere

In this paper, we present a deep learning-based reduced-order model (DL-ROM) for the stability prediction of unsteady three-dimensional fluid-structure interaction systems. The proposed DL-ROM has the format of a nonlinear state-space model and employs a recurrent neural network with long short-term memory (LSTM) cells. We consider a canonical fluid-structure system of an elastically mounted sphere coupled with the incompressible fluid flow in a state-space format. Specifically, we develop a nonlinear data-driven coupling for predicting the unsteady forces and the vortex-induced vibration (VIV) lock-in of the freely vibrating sphere in a transverse direction. We design an input-output relationship as a temporal sequence of force and displacement datasets for a low-dimensional approximation of the fluid-structure system. Based on the prior knowledge of the VIV lock-in process, the input function contains a range of frequencies and amplitudes, which significantly affects the system realization ability and enables an efficient DL-ROM without the need for a massive training dataset for the low-dimensional modeling. Once trained, the network provides a nonlinear mapping of input-output dynamics that can predict the coupled fluid-structure dynamics for a longer horizon via the feedback process. By integrating the LSTM network with the eigensystem realization algorithm (ERA), we construct a data-driven state-space model for the reduced-order stability analysis. We investigate the underlying mechanism and stability characteristics of VIV of a freely vibrating sphere via an eigenvalue selection process. Using the reduced-order eigenvalue analysis, we characterize the vibrating sphere-wake lock-in phenomenon and demonstrate that the lock-in responds at preferred vibration frequencies. To understand the frequency lock-in mechanism, we study the eigenvalue trajectories for a range of the reduced oscillation frequencies and the mass ratios. Consistent with the full-order simulations, the frequency lock-in branches are accurately captured by the combined LSTM-ERA procedure. The proposed DL-ROM aligns with the development of physics-based digital twin of engineering systems involving moving boundaries and fluid-structure interactions. Published under an exclusive license by AIP Publishing.