  In this study, we present a global Fourier mode decomposition framework for
unsteady fluid-structure interaction. We apply the framework to isolate and
extract the aeroelastic modes arising from a coupled three-dimensional
fluid-membrane system. The proposed framework is employed to decompose the
physical variables in the fluid and structural domains into frequency-ranked
aeroelastic modes in a unified way. We observe the frequency synchronization
between the vortex shedding and the structural vibration via mode decomposition
analysis. We examine the role of flexibility in the aeroelastic mode selection
and perform a systematic comparison of flow features among a rigid wing, a
rigid cambered wing and a flexible membrane. With the aid of our mode
decomposition technique, we find that the dominant structural mode exhibits a
chordwise second and spanwise first mode at different angles of attack. The
structural natural frequency corresponding to this mode is estimated using an
approximate analytical formula. By examining the dominant frequency of the
coupled system, we find that the dominant membrane vibrational mode is selected
via the frequency lock-in between the dominant vortex shedding frequency and
the structural natural frequency. From the fluid modes and the mode energy
spectra at $\alpha=20^\circ$ and $25^\circ$, the aeroelastic modes
corresponding to the non-integer frequency components lower than the dominant
frequency are found to be associated with the bluff body vortex shedding
instability. The non-periodic aeroelastic response observed at higher angles of
attack are related to the interaction between aeroelastic modes caused by the
frequency lock-in and the bluff-body-like vortex shedding.
