Symmetry breaking and chaos-induced imbalance in planetary gears

The goal of the present paper was a complete analysis of the dynamic scenario of planetary gears. A lumped mass two-dimensional model is adopted; the model takes into account: time-varying stiffness; nonsmooth nonlinearity due to the backlash, i.e., teeth contact loosing; and bearing compliance. The time-varying meshing stiffness is evaluated by means of a nonlinear finite element model, which allows an accurate evaluation of global and local teeth deformation. The dynamic model is validated by comparisons with the most authoritative literature: linear natural frequencies and nonlinear response. The dynamic scenario is analyzed over a reasonable engineering range in terms of rotation speed and torque. The classical amplitude-frequency diagrams are accompanied by bifurcation diagrams, and for specific regimes, the spectral and topological properties of the response are discussed. Periodic, quasiperiodic and chaotic regimes are found; nonsmooth bifurcations lead period one to period two trajectories. It is found that the bearing compliance can influence the natural frequencies combination magnifying the modal interactions due to internal resonances and greatly enlarging the chaotic regions. It is evidenced that the chaotic response indices a symmetry breaking in the dynamical systems. The physical consequence is that the planetary gearbox under investigation, which is perfectly balanced for each position, can suffer of a big dynamic imbalance when chaotic regimes take place; such imbalance gives rise to alternate and unexpected high-level stresses on bearings.