单自由度齿轮传动系统安全盆侵蚀与分岔

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振动与冲击 ›› 2020, Vol. 39 ›› Issue (2) : 123-131.

论文

单自由度齿轮传动系统安全盆侵蚀与分岔

苟向锋1,2，韩林勃1,2，朱凌云1,2，石建飞1,2

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Erosion and bifurcation of the safe basin for a single-degree-of-freedom spur gear system

GOU Xiangfeng1,2，HAN Linbo1,2，ZHU Lingyun1,2，SHI Jianfei1,2

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摘要

齿轮啮合过程中的振动幅值超过一定限度，会造成系统稳定性的破坏，影响其安全特性。为研究齿轮传动系统的安全特性，以单自由度直齿圆柱齿轮传动系统为研究对象，数值计算了随着综合传递误差波动系数和时变刚度波动幅值增大时系统在考察区域内的安全盆，分析了系统的安全盆侵蚀与分岔过程。结合系统相图、Poincaré映射图分析了系统在考察区域内的运动转迁过程，并计算了系统的最大Lyapunov指数图(TLE)及分岔点的Floquet乘子；借助多初值分岔图进一步分析了系统安全盆侵蚀与分岔现象及演变规律。结果表明：一定范围内随着参数的增大，系统不同运动的吸引域演变导致安全盆的侵蚀，系统的安全性逐渐变差；在考察区域内，系统运动分岔和周期跳跃引起安全盆的分岔；周期跳跃使原运动部分初值的安全盆产生新的安全盆，另一部分初值的安全盆保持不变，而倍化分岔和鞍结分岔则使原安全盆全部分岔为新的安全盆。研究结果可为直齿圆柱齿轮系统的参数设计和选择提供指导，为保证齿轮传动系统的安全服役提供理论依据。

Abstract

In order to study the vibration-boundedness and the safety characteristics of gear transmission systems, a single-degree-of-freedom spur gear transmission system was taken as a studied object.The safety basin of the system in the inspection area was analyzed with the variation of the fluctuant coefficient of comprehension-transmission error and the fluctuant amplitude of time-varying stiffness.The erosion and bifurcation process of the safe basin were analyzed.Based on the system phase diagram and Poincaré map, the motion bifurcations process of the system in the investigation area was analyzed, and the top Lyapunov exponent (TLE) and the Floquet multiplier at the bifurcation point were calculated, and the evolution of the system safety basin erosion and bifurcation was further studied by means of the multi-initial bifurcation diagram.The results show that, with the increase of the parameters in a certain range, the evolution of different motions of the system leads to the erosion of safe basin, and the safety of the system becomes worse; in the investigation area, the bifurcation of the safety basin is caused by the bifurcation and periodic jumping of the system, and the periodic jumping will transform a part of the original safe basin into a new safe basin, while the other part of the original safety basin remains unchanged, and the doubling bifurcation and saddle knot bifurcation makes the original whole safety basin become a new safe basin.The results provide a guidance for the design and selection of spur gear system parameters, and provide a theoretical basis for ensuring the safety service of gear transmission systems.

导出引用

苟向锋1,2，韩林勃1,2，朱凌云1,2，石建飞1,2. 单自由度齿轮传动系统安全盆侵蚀与分岔[J]. 振动与冲击, 2020, 39(2): 123-131

GOU Xiangfeng1,2，HAN Linbo1,2，ZHU Lingyun1,2，SHI Jianfei1,2. Erosion and bifurcation of the safe basin for a single-degree-of-freedom spur gear system[J]. Journal of Vibration and Shock, 2020, 39(2): 123-131

参考文献

[1] THOMPSON J M T. Transient basins: a new tool for designing ships against capsize[C]// IUTAM Symp. Dynamics of Marine Vehicles & Structures in Waves, Brunel University, June. 1990: 325-331.
[2] RAINEY R C T, THOMPSON J M T. The transient capsize diagram: a new method of quantifying stability in waves[J]. Journal of Ship Research, 1991, 35: 58-62.
[3] THOMPSON J M T, MCROBIE F A. Indeterminate bifurcations and the global dynamics of driven oscillators[C]// European Nonlinear Oscillations Conf. Hamburg, Aug. 1993: 107- 128.
[4] NEVES M A S, RODRíGUEZ C A, MERINO J A, et al. Control of unstable ship motions using anti-rolling tanks: erosion of safe basins[J]. ASME International Conference on Ocean, 2010, 119-125.
[5] FREITAS MST DE, VIANA RL, GREBOGI C. Erosion of the safe basin for the transversal oscillations of a suspension bridge[J]. Chaos Solitons & Fractals, 2003, 18(4): 829-841.
[6] LENCI S, REGA G. Optimal control of homoclinic bifurcation: theoretical treatment and practical reduction of safe basin erosion in the Helmholtz oscillator[J]. Journal of Vibration and Control, 2003, 9 (3-4): 281-315.
[7] 葛根, 竺致文, 许佳. 形状记忆合金梁在简谐和白噪声联合激励下的混沌及安全盆侵蚀现象[J]. 振动与冲击, 2012, 31(23): 1-5.
GE Gen, ZHU Zhiwen, XU Jia. Chaos and fractal boundary of safe basin of a shape memory alloy beam subjected to simple harmonic and white noise excitations[J]. Journal of Vibration and Shock, 2012, 31(23): 1-5.
[8] 尚慧琳, 宋书峰, 文永蓬. 时滞位置反馈对一类静电微传感器的吸合不稳定的控制研究[J]. 振动与冲击, 2016, 35(4):81-86.
SHANG Huilin, SONG Shufeng, WEN Yongpeng. Controlling pull-in instability of a typical electrostatically actuated microsensor with time-delay position feedbacks[J]. Journal of Vibration and Shock, 2016, 35(4): 81-86.
[9] RONG Haiwu, WANG Xiangdong, LUO Qizhi, et al. Bifurcation of safe basins and chaos in nonlinear vibroimpact oscillator under harmonic and bounded noise excitations[J]. Journal of Applied Mathematics, 2014(2014):1-10.
[10] 张媛, 秦勇, 邢宗义,等. 基于LMD-PCA-LSSVM的滚动轴承安全域估计和状态辨识方法[J]. 振动与冲击, 2013, 32(20):172-178.
ZHANG Yuan, QIN Yong, XING Zongyi, et al. Safety region estimation and state identification of rolling bearings based on LMD-PCA-LSSVM[J]. Journal of Vibration and Shock, 2013, 32(20):172-178.
[11] 张媛, 秦勇, 邢宗义,等. 基于EMD-PCA-LSSVM方法的滚动轴承安全域估计和状态辨识[J]. 高技术通讯, 2013, 23(5):525-532.
ZHANG Yuan, QIN Yong, XING Zongyi, et al. Safety region estimation and state identification of rolling bearings based on LMD-PCA-LSSVM[J]. High Technology Letters, 2013, 23(5): 525- 532.
[12] GAN Chunbiao. Noise-induced chaos and basin erosion in softening Duffing oscillator[J]. Chaos Solitons & Fractals, 2005, 25(5): 1069-1081.
[13] NAYFEH A H, SANCHEZ N E. Bifurcations in a forced softening duffing oscillator[J]. International Journal of Non- Linear Mechanics, 1989, 24(6):483-497.
[14] WEI Duwei, ZHANG Bo, QIU Dongyuan, et al. Effect of noise on erosion of safe basin in power system[J]. Nonlinear Dynamics, 2010, 61(3):477-482.
[15] 韩清凯. 故障转子系统的非线性振动分析与诊断方法[M]. 科学出版社,2010.
[16] GOU Xiangfeng, ZHU Lingyun, CHEN Dailin. Bifurcation and chaos analysis of spur gear pair in two-parameter plane[J]. Nonlinear Dynamics, 2015, 79(3): 2225- 2235.