Subharmonic bifurcation and multi-solution coexistence mechanism in fractional-order piecewise smooth impact systems under different resonance conditions

This paper systematically investigates the evolutionary law of the existence of subharmonic solutions for a two-degree-of-freedom piecewise-smooth vibro-impact vibration system across four resonance types. Via Hamiltonian normalization, the original system is transformed into a Hamiltonian form, and fractional-order memory damping and nonlinear stiffness are integrally incorporated into the analytical construction of the Melnikov function, which extends the applicability of the existing Melnikov theory to multi-factor coupled systems. Combined with the subharmonic solution bifurcation theorem, the criteria for the existence of solutions are obtained, and a unified analytical framework covering subharmonic resonance, double subharmonic resonance, super-harmonic resonance, and double super-harmonic resonance is established. Theoretical determination of the number of subharmonic solutions, the coexistence of multiple solutions, and the deterministic evolutionary characteristics of the solutions are realized through verification of the zero distribution of the Melnikov function and the Jacobian determinant. We quantify how excitation amplitude, fractional order, nonlinear stiffness, and linear damping regulate solution coexistence by altering Melnikov zeros. The results show that the system can exhibit multi-solution coexistence characteristics, corresponding to multiple coexisting periodic orbits. Double subharmonic resonance presents a nested trajectory structure, whose time-series responses satisfy fixed time-series frequency ratios. Under super-harmonic resonance and double super-harmonic resonance, the system violates the Jacobian non-vanishing criterion, admitting no nearby subharmonic solutions. In this paper, the variation law of subharmonic solutions under the regulation of excitation amplitude is verified by the visualization of phase trajectories and time-series responses, which provides a theoretical basis for the dynamic design and vibration control of piecewise-smooth mechanical systems.