The present research
uses multi-timing regular perturbations in asymptotic expansions to analyze a
certain differential equation having a cubic-quintic nonlinearity. The
differential equation contains slowly-varying explicitly time-dependent
coefficients as well as some small parameters upon which asymptotic expansions
are initiated. The formulation is seen to be typical of a certain mass-spring
arrangement (with geometric imperfection), trapped by a loading history that is
explicitly time-dependent and slowly varying, but continuously decreasing in
magnitude, while the restoring force on the spring has a cubic-quintic
nonlinearity. The dynamic buckling load
of the elastic model structure is determined analytically and is related to the
corresponding static buckling load. To the level of the accuracy retained, it
is observed that the dynamic buckling load depends, among others, on the value
of the first derivative of the loading function evaluated at the initial time.
All results are asymptotic and implicit in the load amplitude.
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