External Periodic Excitation Control of Gear Transmission Systems for Safety-Attraction Basin Erosion and Bifurcation

In response to the instability problem caused by the coupling of strong nonlinear factors such as time-varying mesh stiffness and backlash in high-speed heavy-duty gear systems, an external periodic excitation control strategy was introduced, to establish and numerically solve a dynamic model for single-degree-of-freedom gear transmission systems. With the cell mapping method, the effects of control parameters on the erosion and bifurcation transition process of the system safety-attraction basin, as well as the evolution law of the attraction domain proportion p, were quantitatively analyzed. Based on the Floquet multiplier analysis, a quantitative mapping relationship between the doubling coefficient, the excitation amplitude, and the bifurcation threshold, was established. Combined with the system phase diagram and the Poincaré mapping diagram, the stable control mechanism realized through reconstruction of the phase space topology under external excitation, was revealed, and the control mechanism of key control parameters on the global stability transition of the system was quantitatively elucidated. The results shows that, the low-frequency excitation can easily induce high period attractors, leading to motion boundary instability; the high frequency excitation can trigger safety-attraction basin erosion and bifurcation, where the P3S attractor is stable and the P2S attractor undergoes inverse doubling bifurcation to transition to the P1S single period safe orbit; and the reverse excitation amplitude will destroy the system stability, while increasing the forward excitation amplitude can accelerate the stabilization process, ultimately achieving full coverage of the P1S attraction domain. The research provides a theoretical support for vibration suppression, parameter optimization, and safety design of gear transmission systems.