Local spectral submanifold modeling of fundamental and higher-order periodic responses in flexible bevel gear dynamical systems

Driven by time-varying mesh parameters and backlash nonlinearity, gear systems exhibit multi-periodic responses. However, the high dimensionality of full-order models impedes efficient local-branch tracking and dynamical-structure identification. A local spectral submanifold (SSM) framework is therefore proposed for flexible gear systems. Initially, the flexibility of shaft and gear is characterized using Timoshenko beam theory and component mode synthesis (CMS). A flexible bevel gear model is formulated by coupling these substructures with bearing elements and nonlinear mesh elements incorporating time-varying mesh stiffness, transmission error, and backlash. Through nonlinear frequency sweeps, bifurcation analysis, and Poincaré sections, 1 T, 2 T, and 3 T periodic branches and local modeling intervals are identified, where T denotes the meshing period. Subsequently, a unified non-autonomous state-space interface is constructed between the dynamic model and local SSM. Local trial spaces and closure models are then established for 1 T responses with one recurring Poincaré point, 2 T responses with two alternating points, and 3 T responses with a three-point cycle. Results indicate that all models satisfy periodic closure conditions and yield locally stable periodic solutions, with forward and backward sweeps exhibiting excellent agreement. The 2 T and 3 T models preserve their alternating and cyclic state sequences. Finally, the approach is validated via rolling inspection, modal and gearbox vibration tests. The study reveals that as periodic order increases, local SSM modeling shifts from single-section parameterization to multi-phase coordination, providing a methodological foundation for low-dimensional characterization and stable tracking of multi-periodic branches in flexible gear dynamics.