  We investigate the dynamics of membranes that are held by freely-rotating
tethers in fluid flows. The tethered boundary condition allows periodic and
chaotic oscillatory motions for certain parameter values. We characterize the
oscillations in terms of deflection amplitudes, dominant periods, and numbers
of deflection extrema along the membranes across the parameter space of
membrane mass density, stretching modulus, pretension, and tether length. We
determine the region of instability and the small-amplitude behavior by solving
a nonlinear eigenvalue problem. We also consider an infinite periodic membrane
model, which yields a regular eigenvalue problem, analytical results, and
asymptotic scaling laws. We find qualitative similarities among all three
models in terms of the oscillation frequencies and membrane shapes at small and
large values of membrane mass, pretension, and tether length/stiffness.
