Asymmetric Hill Equations Arising from Bilinear Oscillators with State-Dependent Parametric Excitation

Abstract

This study presents an analytical and numerical investigation of bilinear oscillators subjected to piece- wise parametric excitation, where both the excitation amplitude and damping are state-dependent. The governing equations are formulated as asymmetric Hill’s equations with asymmetric parametric excitation. The method of averaging is employed to derive the corresponding slow-flow equations, whose stability is analyzed and validated against direct numerical simulations. Closed-form expressions for the transition curves are developed, providing insight into instability boundaries for primary and subharmonic resonance conditions (1 : 1, 2 : 1, 3 : 1 and 4 : 1). The analysis is further extended to a physically relevant system—a cantilever beam with a breathing crack—demonstrating the equivalence between the asymmetric Hill model and structures with state-dependent stiffness and excitation. The proposed framework establishes an efficient approach for predicting dynamic instability in engineering systems exhibiting piecewise or state-dependent nonlinearities, such as beams and rotors with breathing cracks.