Elastically driven filaments subjected to animating compressive follower
forces provide a synthetic way to mimic the oscillatory beating of active
biological filaments such as eukaryotic cilia. The dynamics of such active
filaments can, under favorable conditions, exhibit stable time-periodic
responses that result due to the interplay of elastic buckling instabilities,
geometric constraints, boundary conditions, and dissipation due to fluid drag.
In this paper, we use a continuum elastic rod model to estimate the critical
follower force required for the onset of the stable time-periodic flapping
oscillations in pre-stressed rods subjected to fluid drag. The pre-stress is
generated by imposing either clamped-clamped or clamped-pinned boundary
constraints and the results are compared with those of clamped-free case, which
is without pre-stress. We find that the critical value increases with the
initial slack--that quantifies the pre-stress, and strongly depends on the type
of the constraints at the boundaries. The frequency of oscillations far from
the onset, however, depends primarily on the magnitude of the follower force,
not on the boundary constraints. Interestingly, oscillations for the
clamped-pinned case are observed only when the follower forces are directed
towards the clamped end. This finding can be exploited to design a mechanical
switch to initiate or quench the oscillations by reversing the direction of the
follower force or altering the boundary conditions
