Influence of nonlinear damping and softening stiffness on the dynamic behavior of a two-degree-of-freedom quarter-car model

This study investigates the nonlinear dynamics of a two-degree-of-freedom quarter-car suspension model incorporating softening cubic suspension stiffness, Rayleigh-type nonlinear damping, and hardening tire characteristics under harmonic road excitation. A stiff time-integration scheme is employed, and the formulation is validated through comparison with a linear Amesim model and benchmarked against a nonlinear single-degree-of-freedom counterpart, confirming the correctness of the implementation and the essential role of the unsprung mass. Nonlinear analyses based on frequency response curves, stroboscopic bifurcation diagrams, Poincaré sections, basins of attraction, and largest Lyapunov exponents reveal resonance detuning and coexistence of multiple attractors that cannot be captured by reduced-order models. The inclusion of the unsprung mass leads to a pronounced downward shift of the primary resonance (approximately 9%) and a modification of the higher-frequency wheel-hop resonance, together with an earlier onset of chaotic dynamics (around f ≈ 0.70 instead of f ≈ 0.75), arising from nonlinear modal coupling between the sprung and unsprung subsystems. A parametric investigation further shows that the softening stiffness coefficient α predominantly governs the location of the comfort-critical primary resonance, whereas the nonlinear damping coefficient η has a secondary influence on resonance locations and produces a measurable shift (on the order of 5%) in the onset of chaos by enhancing stability. The second resonance remains largely controlled by tire stiffness and unsprung-mass dynamics, exhibiting only minor sensitivity to suspension nonlinearities. Overall, the results demonstrate that nonlinear 2DOF models are necessary for accurately capturing coupled resonance behavior, multistability, and nonlinear stability phenomena relevant to suspension design.