Softening/hardening dynamics of beam resting on generalized nonlinear foundation with a linear stiffening effect

An asymptotic study of softening/hardening dynamics of beam on a generalized nonlinear foundation is presented, by focusing on its linear stiffening effect on quadratic/cubic nonlinearities. It is found that, though linear stiffening simultaneously weakens both quadratic and cubic nonlinearities, they decrease at different rates, leading to complex competing dynamics in sequel, such as enhanced hardening/weakened softening, weakened hardening, and quenched nonlinearity in the limit, among which softening–hardening transition dynamics is of particular interest when quadratic and cubic nonlinearities ‘balance’ with each other (i.e., a vanishing effective nonlinear coefficient). Close to transition, standard third-order perturbation fails and the predicted responses turn invalid in mediate/high amplitude ranges, which critically depends on transition inclination when transversally crossing the line of vanishing effective nonlinear coefficient. In parallel, linear stiffening effects on softening/hardening dynamics are also investigated from a methodology viewpoint, i.e., it reduces differences between discretization and direct perturbations, with the former becoming more reliable by an increased linear stiffness.