Dynamic Characteristic Analysis of a Single-stage Spur Gear System in Backlash-frequency Plane
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摘要: 为揭示含间隙的齿轮系统分岔、啮合冲击、脱啮和动载系数间的耦合关系，在间隙与频率参数平面内以单级齿轮系统的动力学模型为研究对象，用CPNF法求解其周期/冲击、脱啮占空比、齿背啮合比、动载系数伪彩图。研究表明：间隙和频率耦合下系统表现出复杂的综合动力特性，随频率和间隙变化系统经倍化、Hopf、激变、擦切、鞍结分岔由周期运动通向混沌；分岔和啮合冲击突变导致脱啮占空比、齿背啮合比、动载系数阶跃性增大；在系统共振频率和小间隙范围内因混沌、拟周期运动和双边冲击的影响其脱啮、齿背接触、动载系数达到极值。Abstract: In order to reveal the coupling relationship among bifurcation, gear meshing impact, tooth mesh apart, tooth back-side contact and dynamic load coefficient (DLC) of the nonlinear gear system, the pseudo color maps of meshing impact and period, duty cycle of the mesh apart (DC1), duty cycle of the tooth back-side contact (DC2) and DLC for a single-stage spur gear system were obtained by CPNF (continue Poincaré-Newton-Floquet) in dimensionless backlash- frequency plane. The results indicate that the system exhibits complex comprehensive dynamic characteristics under the coupling of clearance and frequency. With the change of backlash and frequency, the periodic motion of the system turns to chaos by the way of period-doubling, Hopf, crisis, grazing and saddle node bifurcation. The abrupt change of bifurcation and meshing impact leads to step increase of DC1, DC2 and DLC. DC1, DC2 and DLC reach the extreme value due to the influence of chaos, quasi periodic motion and bilateral impact within the scope of the resonance frequency and small gap. These dynamic characteristics transition laws can provide the theoretical reference for the optimization of the gear structure parameter.
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表 1 CPNF算法的分岔类型判别
考查点 参数(b, Ω) 追踪周期/Pi Floquet乘子λ |λmax| 分岔类型 A (0.1, 1.330) P1 (-1.049 2, -0.773 6, 0.421 4±0.808 0i, 0.695 7, 0.732 6) 1.049 2 倍化 B (0.1, 1.527 5) P1 (0.526 0±0.529 3i, -0.105 0±0.915 0i, -1.001 5, -0.820 7) 1.001 5 倍化 C (0.2, 1.390 0) P4 (0.068 3±1.272 9i, 0.144 0±0.681 5i, 0.060 2±0.127 3i) 1.274 7 Hopf D (0.2, 1.345 0) P4 (1.055 6, 0, 310 4±0.614 2i, -0.307 8±0.426 5i, 0.059 1) 1.055 6 鞍结 
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