NEIGHBORING PERIODIC MOTION AND ITS IDENTIFICATION FOR SPUR GEAR PAIR WITH SHORT-PERIOD ERRORS
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摘要: 齿轮副中的齿距偏差等短周期误差使系统出现复杂的周期运动, 影响齿轮传动的平稳性. 将该类复杂周期运动定义为近周期运动, 采用多时间尺度Poincaré映射截面对其进行辨识. 为研究齿轮副的近周期运动, 引入含齿距偏差的直齿轮副非线性动力学模型, 并计入齿侧间隙与时变重合度等参数. 采用变步长4阶Runge-Kutta法数值求解动力学方程, 由所提出的辨识方法分析不同参数影响下系统的近周期运动. 根据改进胞映射法计算系统的吸引域, 结合多初值分岔图、吸引域图与分岔树状图等研究了系统随扭矩与啮合频率变化的多稳态近周期运动. 研究结果表明, 齿轮副中的短周期误差导致系统的周期运动变复杂, 在微观时间尺度内, 系统的Poincaré映射点数呈现为点簇形式, 系统的点簇数与实际运动周期数为宏观时间尺度的Poincaré映射点数. 短周期误差导致系统在微观时间尺度内的吸引子数量增多, 使系统运动转迁过程变复杂. 合理的参数范围及初值范围可提高齿轮传动的平稳性. 该辨识与分析方法可为非线性系统中的近周期运动研究奠定理论基础.Abstract: The pitch deviation and other errors as the short-period error in spur gear pair lead to complex periodic motion which affects the transmission stationarity of gear systems. The complex periodic motion is defined as the neighboring periodic motion. It is identified by using the multi-time scale Poincaré mapping sections. A nonlinear dynamics model of the spur gear pair with pitch deviation is introduced in order to study the neighboring periodic motion of the gear pair. Backlash, time-varying contact ratio and other parameters are considered. The dynamics model is numerically calculated by the variable step 4-order Runge-Kutta method. The proposed identification method is used to analyze the neighboring periodic motion of the system under different parameters. The information of attractors and the basin of attraction in the state plane can be obtained by the improved cell mapping theory. The multi-stable neighboring periodic motions of the system under the variation of torque and meshing frequency are investigated by typical nonlinear dynamics analysis methods, such as multi-initial values bifurcation diagrams, phase diagrams, Poincaré maps, basin of attraction and bifurcation dendrogram. Results show that the short-period error in the spur gear pair leads to the complex periodic motion of the system. In the micro time scale, Poincaré mapping points of the system show the form of point clusters. The number of point clusters and actual motion period of the system are the number of Poincaré mapping points in the macro time scale. The short-period error leads to the increase of the number of attractors in the micro time scale which makes the motion transition process of the system more complex. The reasonable range of parameters and initial values can improve the transmission stationarity of the gear pair. The identification and analysis methods provide a theoretical basis for the study of the neighboring periodic motion in nonlinear systems.
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Key words:
- neighboring periodic motion /
- gear pair /
- nonlinear dynamics /
- multi-scale time /
- multi-stable motion
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图 4 随刚度波动幅值变化的直齿轮副多时间尺度分岔图
Figure 4. Bifurcation diagrams of the spur gear pair with different time scales via stiffness fluctuation amplitude
图 5 随扭矩变化的直齿轮副多时间尺度分岔图
Figure 5. Bifurcation diagrams of the spur gear pair with different time scales via torque
图 6 随啮合频率变化的直齿轮副多时间尺度分岔图
Figure 6. Bifurcation diagrams of the spur gear pair with different time scales via meshing frequency
图 7 系统随扭矩变化的多初值分岔图及TLE谱
Figure 7. Multi-initial values bifurcation diagrams and TLE spectrums of system via F
图 10 系统随啮合频率变化的多初值分岔图及TLE谱
Figure 10. Multi-initial values bifurcation diagrams and TLE spectrums of system via ω
图 12 系统随啮合频率变化的相图与Poincaré映射图
Figure 12. Phase portraits and Poincaré maps of system via ω
表 1 齿轮参数
Table 1. Parameters of a spur gear pair
Parameters Pinion Gear precision grade 6 GM 6 GM module/mm 5 5 number of teeth 21 26 
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表 2 主动轮齿距偏差值(Np =21)
Table 2. Pitch deviation values of the pinion (Np =21)
a 1 2 3 4 5 6 7 ΔFpt1(a)
/μm0 1.94 3.7 5.09 5.9 6.3 5.9 a 8 9 10 11 12 13 14 ΔFpt1(a)
/μm5.09 3.7 1.94 0 −1.94 −3.7 −5.09 a 15 16 17 18 19 20 21 ΔFpt1(a)
/μm−5.9 −6.3 −5.9 −5.09 −3.7 −1.94 0
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表 3 从动轮齿距偏差值(Ng =26)
Table 3. Pitch deviation values of the gear (Ng =26)
b 1 2 3 4 5 6 7 ΔFpt2(b)
/μm0 1.74 3.37 4.7 5.9 6.6 6.9 b 8 9 10 11 12 13 14 ΔFpt2(b)
/μm6.8 6.3 5.39 4.11 2.57 0.87 −0.87 b 15 16 17 18 19 20 21 ΔFpt2(b)
/μm−2.57 −4.11 −5.39 −6.3 −6.8 −6.9 −6.6 b 22 23 24 25 26 ΔFpt2(b)
/μm−5.9 −4.79 −3.37 −1.74 0
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表 4 从动轮齿距偏差值(Ng =13)
Table 4. Pitch deviation values of the gear (Ng =13)
b 1 2 3 4 5 6 7 ΔFpt2(b)
/μm0 3.5 6.06 7 6.06 3.5 0 b 8 9 10 11 12 13 ΔFpt2(b)
/μm−3.5 −6.06 −7 −6.06 −3.5 0
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