We revisit a previous study (2024) of the first author, which developed a set of
pragmatic asymptotic approximations for the lowest eigenvalue of a boundary-value
problem arising in the divergence-buckling of an axially moving thin elastic web.
Although the proposed formulae showed very good agreement with numerical results,
they were primarily tailored to a regime where the perturbation parameter is only
moderately small. In the present work, we undertake a more detailed and
mathematically rigorous investigation of the underlying bifurcation problem. In doing
so, we show that the mode structure assumed in the earlier analysis corresponds,
under appropriate modifications, to the second eigenmode. We then derive a new
asymptotic description of the critical mode and its associated eigenvalue, which
more accurately reflects its analytical structure in the singular limit. The
resulting asymptotic approximations for both the first and second modes are
validated against numerical computations, demonstrating their accuracy and
offering further insight into the asymptotic behaviour of this class of stability
problems.
Keywords
matched asymptotics, elastic stability, bifurcations,
boundary layers, thin-plate theory
