  We develop a model and numerical method to study the large-amplitude flutter
of rectangular membranes (of zero bending rigidity) that shed a trailing
vortex-sheet wake in a three-dimensional (3D) inviscid fluid flow. We apply
small initial perturbations and track their decay or growth to large-amplitude
steady state motions. For 12 combinations of boundary conditions at the
membrane edges we compute the stability thresholds and the subsequent
large-amplitude dynamics across the three-parameter space of membrane mass
density, pretension, and stretching rigidity. With free side edges we find good
agreement with previous 2D results that used different discretization methods.
We find that the 3D dynamics in the 12 cases naturally form four groups based
on the conditions at the leading and trailing edges. The deflection amplitudes
and oscillation frequencies have scalings similar to those in the 2D case. The
conditions at the side edges, though generally less important, may have small
or large qualitative effects on the membrane dynamics -- e.g. steady versus
unsteady, periodic versus chaotic, or the variety of spanwise curvature
distributions -- depending on the group and the physical parameter values.
