Abstract This study presents a unified analytical–numerical framework for the El Borhamy–Rashad–Sobhy equation with Duffing-type nonlinearity, augmented by an external harmonic forcing term. The used application in the study is motivated by the electromechanical dynamics of salient-pole synchronous machines. Starting from an energy-based formulation, the harmonic balance method is used to obtain closed-form frequency-response relations. The local stability of the periodic orbit is further quantified via a Floquet-based test derived from the linear variational equation, yielding stability-classified frequency-response curves. While the method of multiple scales captures primary, superharmonic, and subharmonic resonance conditions and the associated stability boundaries are analyzed. Beyond the weakly nonlinear regime, numerical bifurcation analysis is performed to trace a period-doubling route to chaos, supported by Poincaré sections, the largest Lyapunov exponent, and time-series diagnostics. The system is further coupled with linear and nonlinear piezoelectric energy-harvesting branches, to demonstrate that, the large-amplitude responses enhance harvested power and the cubic stiffness can be tuned for bandwidth optimization. Overall, the results bridge perturbation theory with nonlinear time-domain diagnostics, providing design-relevant insights for broadband energy harvesters and electromechanical systems operating under parametric excitation and external forcing.
El-Borhamy et al. (2026) studied this question.