Fractional-order viscoelastic damping and chaos suppression in a bio-inspired toe-like-structure vibration isolator

This paper investigates nonlinear vibration responses and chaos suppression in a bio-inspired vibration isolator with toe-like structures and fractional-order viscoelastic damping. A single-degree-of-freedom dynamic model is established by representing the geometry-induced restoring force of the toe-like structure with a quintic polynomial. A Caputo fractional derivative is introduced to characterize memory-dependent damping in compliant components. Under the steady-state harmonic-balance approximation, the fractional-order damping term is decomposed into equivalent viscous damping and stiffness-correction components, providing a physically interpretable integer-order representation for nonlinear vibration analysis. The equilibrium topology of the unperturbed system is shown to be governed by the effective stiffness, leading to pitchfork bifurcations and transitions among monostable, bistable and tristable configurations. In the bistable regime, explicit homoclinic orbits are obtained, and a Melnikov-based criterion is derived to predict the onset of chaotic vibration. Representative numerical simulations, including bifurcation diagrams, phase portraits with Poincaré-section samples and time histories, together with supplementary transition responses, confirm the analytical prediction for both integer- and fractional-order cases. The results show that fractional-order damping reshapes the instability regions in the excitation-amplitude domain, while reducing the excitation amplitude or tuning the fractional-order parameters can suppress chaotic vibration responses. These results establish a fractional-memory-based mechanism for regulating nonlinear instability and provide a parameter-tuning route for improving the dynamic stability of bio-inspired vibration isolation systems.