Refined hardening/softening dynamic analysis close to internal resonance for sagged cables

Hardening/softening (H/S) dynamics of continuous structures have long been investigated by referring to an effective nonlinear coefficient associated with modulation/averaged equation (derived by perturbation technique). Close to internal resonance, however, this traditional formulation becomes unreliable: the effective coefficient may diverge or become ill-posed, leading to misleading H/S predictions. Being confined to a weakly damping scenario, the current paper aims to develop a refined perturbation framework that explicitly incorporates internal resonance thereby removing this nominal singularity. By focusing on a typical quadratic–cubic system (a sagged cable), two distinct types of crossing-singularity are revealed. Namely, type-I for a generalized H/S transition in which the response exhibits a hardening-to-softening or softening-to-hardening transition, while type-II for a singularity crossing without H/S transition, being hardening-to-hardening or softening-to-softening. Although the main work is currently developed for a typical cable model, the framework can be meaningfully extended to other quadratic–cubic structures involved with internal resonance.