一类齿轮传动系统的簇发振荡现象研究
Study on the Bursting Oscillations Phenomenon of a Kind of Gear Transmission System

DOI: 10.12677/OJAV.2021.94016, PP. 153-165

Keywords: 非光滑齿轮传动系统，分岔，簇发振荡
Non-Smooth Gear Transmission System, Bifurcation, Bursting Oscillation

Full-Text   Cite this paper   Add to My Lib

Abstract:

本文以带有一齿侧间隙的非光滑齿轮传动系统为研究对象，系统被非光滑分界面分成不同的子系统，当激励频率远小于系统固有频率时，会产生诸如簇发振荡等特殊行为。将整个激励项看做慢变参数，激励系统转化为广义自治系统也即快子系统，分析快子系统平衡点的稳定性以及分岔条件，并运用快慢分析法和转换相图揭示了簇发振荡的动力学机理。考虑无量纲一齿侧间隙、啮合频率、啮合刚度和波动幅值对齿轮传动的影响，借助分岔图、相图、庞加莱映射以及最大李雅普诺夫指数来分析其系统的动力学机制。研究结果发现无量纲啮合频率、啮合刚度和波动幅值对系统有一定的影响，研究结果可以为齿轮传动系统安全设计提供参考。
Taking a non-smooth gear transmission system with a toothed clearance as the research object, the system is divided into different subsystems by non-smooth interfaces. When the excitation frequency is far less than the natural frequency of the system, special behaviors such asbursting oscillation will occur. The whole excitation is regarded as a slow parameter, and the excitation system is transformed into a generalized autonomous system, that is, a fast subsystem. The stability and bifurcation conditions of the equilibrium point of the fast subsystem are analyzed, and the dynamic mechanism of bursting oscillation is revealed by using the fast and slow analysis method and transformation phase diagram. Considered the impact of dimensionless toothed clearance, meshing frequency, meshing stiffness and fluctuation amplitude on the gear transmission, the bifurcation mechanism and the bursting oscillation of the system are based on bifurcation diagrams, Lyapunov exponents and phase portraits, poincare maps. The results show that the dimensionless meshing frequency, meshing stiffness and fluctuation amplitude have some effect on the system. The research results can provide reference for the safety design of gear transmission system.

References

[1]	Chang-Jian, C.-W. and Chang, S.M. (2011) Bifurcation and Chaos Analysis of Spur Gear Pair with and without Non-linear Suspension. Nonlinear Analysis: Rear World Applications, 12, 979-989. https://doi.org/10.1016/j.nonrwa.2010.08.021
[2]	Chang-Jian, C.-W. (2010) Strong Nonlinearity Analysis for Gear-Bearing System under Nonlinear Suspension-Bifurca- tion and Chaos. Nonlinear Analysis: Rear World Applica-tions, 11, 1760-1774. https://doi.org/10.1016/j.nonrwa.2009.03.027
[3]	Farshidianfar, A. and Saghafi, A. (2014) Global Bifurcation and Chaos Analysis in Nonlinear Vibration of Spur Gear Systems. Nonlinear Dynamics, 75, 783-806. https://doi.org/10.1007/s11071-013-1104-4
[4]	Gou, X.F., Zhu, L.Y. and Chen, D.L. (2015) Bifurcation and Chaos Analysis of Spur Gear Pair in Two-Parameter Plane. Nonlinear Dynamics, 79, 2225-2235. https://doi.org/10.1007/s11071-014-1807-1
[5]	Hotait M.A. and Kahraman, A. (2013) Experiments on the Rela-tionship between The Dynamic Transmission Error and the Dynamic Stress Factor of Spur Gear Pairs. Mechanism and Machine Theory, 70, 116-128. https://doi.org/10.1016/j.mechmachtheory.2013.07.006
[6]	Liu, H.X., Wang, S.M. and Guo, J.S. (2011) Solution Domain Boundary Analysis Method and Its Application in Parameter Spaces of Nonlinear Gear System. Chinese Journal of Mechanical Engineering, 24, 507-514.
[7]	Xiang, L., Jia, Y. and Hu, A.J. (2016) Bifurcation and Chaos Analysis for Multi-Freedom Gear-Bearing System with Time-Varying Stiffness. Applied Mathematical Modelling, 40, 10506-10520. https://doi.org/10.1016/j.apm.2016.07.016
[8]	李大鹏, 江海燕, 丁虎. 含张紧轮和单向离合器的轮-带系统稳态响应[J]. 振动工程学报, 2017, 30(2): 194-201.
[9]	Shi, J.F., Gou, X.F. and Zhu, L.Y. (2018) Bifurcation and Erosion of Safe Basin for a Spur Gear System. International Journal of Bifurcation and Chaos, 28, Article ID: 1830048. https://doi.org/10.1142/S0218127418300483
[10]	Gou, X.F., Zhu, L.Y. and Qi, C.J. (2018) Bifurcation and Cha-os Analysis of a Gear-Rotor-Bearing System. Journal of Theoretical and Applied Mechanics, 56, 585-599. https://doi.org/10.15632/jtam-pl.56.3.585
[11]	Wei, S., Chu, F.L., Ding, H. and Chen, L.Q. (2021) Dynamic Analysis of Uncertain Spur Gear Systems. Mechanical Systems and Signal Processing, 150, Article ID: 107280. https://doi.org/10.1016/j.ymssp.2020.107280
[12]	Rinzel, J. (1985) Bursting Oscillation in an Excitable Mem-Brane Model. In: Sleeman, B.D. and Jarvis, R.J., Eds., Ordinary and Partial Differential Equations, Vol. 1151, Springer, Berlin, Heidelberg, 304-316. https://doi.org/10.1007/BFb0074739
[13]	Rush, M. and Rinzel, J. (1994) Analysis of Bursting in a Thalamic Neuron Model. Biological Cybernetics, 71, 281-291. https://doi.org/10.1007/BF00239616
[14]	Qu, Z.F., Zhang, Z.D., Peng, M. and Bi, Q.S. (2018) Non-Smooth Bursting Analysis of a Filippov-Type System with Multiple-Frequency Excitations. Pramana, 91, Article No. 72. https://doi.org/10.1007/s12043-018-1644-8
[15]	Zhang, R., Peng, M., Zhang, Z.D. and Bi, Q.S. (2018) Bursting Oscillations as well as the Bifurcation Mechanism in a Non-Smooth Chaotic Geomagnetic Field Model. Chinese Physics B, 27, Article ID: 110501. https://doi.org/10.1088/1674-1056/27/11/110501
[16]	Qu, R., Wang, Y., Wu, G.Q., Zhang, Z.D. and Bi, Q.S. (2018) Bursting Oscillations and the Mechanism with Sliding Bifurcations in a Filippov Dynamical System. International Journal of Bifurcation and Chaos, 28, Article ID: 1850146. https://doi.org/10.1142/S0218127418501468
[17]	Wang, Z.X., Zhang, Z.D. and Bi, Q.S. (2019) Relaxation Os-cillations in a Nonsmooth Oscillator with Slow-Varying External Excitation. International Journal of Bifurcation and Chaos, 29, Article ID: 1930019. https://doi.org/10.1142/S0218127419300192
[18]	Huang, L., Wu, G.Q., Zhang, Z.D. and Bi, Q.S. (2019) Fast-Slow Dynamics and Bifurcation Mechanism in a Novel Chaotic System. International Journal of Bifurcation and Chaos, 29, Article ID: 1930028. https://doi.org/10.1142/S0218127419300283
[19]	Wang, Z.X., Zhang, Z.D. and Bi, Q.S. (2020) Bursting Oscilla-tions with Delayed C-Bifurcations in a Modified Chua’s Circuit. Nonlinear Dynamics, 100, 2899-2915. https://doi.org/10.1007/s11071-020-05676-6

Full-Text