ON A CERTAIN TWO-SMALL-PARAMETER CUBIC-QUINTIC NON-LINEAR DIFFERENTIAL EQUATION HAVING SLOWLY-VARYING COEFFICIENTS WITH APPLICATION TO DYNAMIC BUCKLING

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.