Flow-induced vibration of an elastically mounted prolate spheroid: dynamic branches and modal competition

Understanding fluid-elastic instabilities in slender bodies is crucial for predicting and controlling flow-induced vibration (FIV) in engineering and biological systems. The FIV of a prolate spheroid with an aspect ratio of $\epsilon = 3$ , a mass ratio of $m^* = 3$ and a damping ratio of $\zeta = 0$ , elastically mounted in a uniform flow at ${\textit{Re}} = 600$ , are investigated using direct numerical simulations and a reduced-order model (ROM). As reduced velocity $U_r$ increases, five vibration states emerge: quasi-steady (QS), periodic (PM), large-amplitude chaotic (LAC), quasi-periodic (QP) and small-amplitude chaotic (SAC) modes. These mode regimes form two categories of response branches, namely synchronised branches (SB) and desynchronised branches (DB). In SB, three synchronisation mechanisms are identified, i.e. conventional lock-in (CLI), secondary-component lock-in (SCLI) and superharmonic lock-in (SHLI), corresponding to PM, LAC and QP modes, respectively. In contrast, DB comprises two types. The flow-dominated desynchronised (FDD) branch corresponds to QS mode, where flow instability dominates while structural vibrations remain weak. The dual-mode competition desynchronised (DMCD) branch corresponds to the SAC mode, where fluid and structural instabilities coexist but fail to synchronise. Analysis of wake dynamics identifies spanwise, transverse and high-frequency spanwise shedding patterns that are closely correlated with vibration regimes. The overlap of the response branches produces three distinct hysteresis zones, emphasising the sensitivity of spheroidal FIV to initial conditions and its inherently path-dependent behaviour. Dynamic mode decomposition (DMD) and an ERA-based ROM, which together resolve mode-specific spatial structures and frequency evolution, show that the vibration dynamics is governed by a persistently unstable wake mode (WM) and a structural mode (SM) whose stability alternates across branches. This clarifies the resulting sequence of vibration modes and provides insight into how different branches compete and transition.