Bifurcation and chaos analysis of a gear-shaft-bearing system considering tooth-bearing backlash nonlinearity and time-varying mesh stiffness

Wei Liu; Ying Cui; Qiang Wang; Hanlong Cai; Weimin Ding

doi:10.59400/adecp3693

Bifurcation and chaos analysis of a gear-shaft-bearing system considering tooth-bearing backlash nonlinearity and time-varying mesh stiffness

Wei Liu College of Automobile and Traffic Engineering, Heilongjiang Institute of Technology, Harbin 150050, China
Ying Cui College of Automobile and Traffic Engineering, Heilongjiang Institute of Technology, Harbin 150050, China
Qiang Wang College of Automobile and Traffic Engineering, Heilongjiang Institute of Technology, Harbin 150050, China
Hanlong Cai Ningbo Donly Transmission Equipment Co., LTD, Ningbo 315000, China
Weimin Ding Ningbo Donly Transmission Equipment Co., LTD, Ningbo 315000, China

Article ID: 3693

DOI: https://doi.org/10.59400/adecp3693

Keywords: gear transmission system; multiple backlash; nonlinear; bifurcation; TVMS

Abstract

The bifurcation and chaos of gear-shaft-bearing system considering tooth-bearing backlash nonlinearity and time-varying mesh stiffness (TVMS) are investigated through employing shafting element method. In our previous work, the dynamic system merely considered the simple nonlinear backlash and completely ignored the case that tooth-bearing backlash were introduced simultaneously. This work primarily concentrates on the investigation of nonlinear dynamics of gear-shaft-bearing system with tooth-bearing backlash. Initially, TVMS is calculated and simulated to validate Y.Cai result. The coupling relationship for the tooth backlash, TVMS and time-varying center distance is taken into account. Accordingly, the expression of TVMS considering tooth backlash and time-varying center distance is decuced theroetically. Based on the shaft element method, the whole gear dynamic model is established mathematically with the help of the shaft element, gear mesh element and bearing element. Bifurcation diagrams of the system are investigated and compared for three cases, i.e., only tooth backlash, no coupling backlash and coupling backlash. Furthermore, time-domain responses, phase diagrams, and Poincare maps of central components with different speed conditions are analyzed. Ultimately, the illustrative parametric studies are further scrutinized to exhibit the nonlinear dynamics. The results show that chaotic motion region of gear system becomes wider while coupling the tooth-bearing backlash nonlinearity. With the speed condition increase, the gear system exhibits significant nonlinear behavior, i.e., periodic motion and chaotic motion. This work is extremely helpful for the optimization design and vibration reduction of gear system.

Published

2025-10-23

How to Cite

Liu, W., Cui, Y., Wang, Q., Cai, H., & Ding, W. (2025). Bifurcation and chaos analysis of a gear-shaft-bearing system considering tooth-bearing backlash nonlinearity and time-varying mesh stiffness. Advances in Differential Equations and Control Processes, 32(4). https://doi.org/10.59400/adecp3693

Download Citation

Issue

Vol. 32 No. 4 (2025)

Section

Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]Mélot A, Benaïcha Y, Rigaud E, et al. Effect of gear topology discontinuities on the nonlinear dynamic response of a multi-degree-of-freedom gear train. Journal of Sound and Vibration. 2022; 516: 116495. doi: 10.1016/j.jsv.2021.116495

[2]Debeurre M, Grolet A, Cochelin B, et al. Finite element computation of nonlinear modes and frequency response of geometrically exact beam structures. Journal of Sound and Vibration. 2023; 548: 117534. doi: 10.1016/j.jsv.2022.117534

[3]Ning Z, Bai Z. Nonlinear dynamics of space manipulator system considering planetary gear joints. Mechanism and Machine Theory. 2025; 214: 106136. doi: 10.1016/j.mechmachtheory.2025.106136

[4]Pizzolante F, Battarra M, Mucchi E, et al. Combining the asymptotic numerical method with the harmonic balance method to capture the nonlinear dynamics of spur gears. Mechanical Systems and Signal Processing. 2024; 214: 111384. doi: 10.1016/j.ymssp.2024.111384

[5]Cao Y, Yao H, Li H, et al. Torsional vibration dynamics of a gear-shafting system attaching a nonlinear energy sink. Mechanical Systems and Signal Processing. 2022; 176: 109172. doi: 10.1016/j.ymssp.2022.109172

[6]Ren Z, Li J, Wang K, et al. Nonlinear dynamic analysis of a coupled lateral-torsional spur gear with eccentricity. Journal of Vibroengineering. 2016; 18(7): 4776–4791. doi: 10.21595/jve.2016.17067

[7]Hu Y-H, Du Q-G, Xie S-H. Nonlinear dynamic modeling and analysis of spur gears considering uncertain interval shaft misalignment with multiple degrees of freedom. Mechanical Systems and Signal Processing. 2023; 193: 110261. doi: 10.1016/j.ymssp.2023.110261

[8]Wei J, Zhang A, Wang G, et al. A study of nonlinear excitation modeling of helical gears with Modification: Theoretical analysis and experiments. Mechanism and Machine Theory. 2018; 128: 314–335. doi: 10.1016/j.mechmachtheory.2018.06.005

[9]Shi J, Gou X, Jin W, et al. Multi-meshing-state and disengaging-proportion analyses of a gear-bearing system considering deterministic-random excitation based on nonlinear dynamics. Journal of Sound and Vibration. 2023; 544: 117360. doi: 10.1016/j.jsv.2022.117360

[10]Xiang L, Zhang Y, Gao N, et al. Nonlinear dynamics of a multistage gear transmission system with multi-clearance. International Journal of Bifurcation and Chaos. 2018; 28(03): 1850034. doi: 10.1142/S0218127418500347

[11]Pan W, Li X, Wang L, et al. Nonlinear response analysis of gear-shaft-bearing system considering tooth contact temperature and random excitations. Applied Mathematical Modelling. 2019; 68: 113–136. doi: 10.1016/j.apm.2018.10.022

[12]Zhang HB, Wang R, Chen ZK et al. Nonlinear dynamic analysis of a gear-rotor system with coupled multi-clearance. Journal of Vibration and Shock. 2015; 34: 144–150. Available online: https://jvs.sjtu.edu.cn/EN/Y2015/V34/I8/144

[13]Aihara T, Sakamoto K. Theoretical analysis of nonlinear vibration characteristics of gear pair with shafts. Theoretical and Applied Mechanics Letters. 2022; 12(2): 100324. doi: 10.1016/j.taml.2022.100324

[14]Wang S, Zhu R. Nonlinear dynamic analysis of GTF gearbox under friction excitation with vibration characteristics recognition and control in frequency domain. Mechanical Systems and Signal Processing. 2021; 151: 107373. doi: 10.1016/j.ymssp.2020.107373

[15]Sun Z, Chen S, Hu Z, et al. Vibration response analysis of a gear-rotor-bearing system considering steady-state temperature. Nonlinear Dynamics. 2022; 107(1): 477–493. doi: 10.1007/s11071-021-07024-8

[16]Gill-Jeong C. Analysis of the nonlinear behavior of gear pairs considering hydrodynamic lubrication and sliding friction. Journal of Mechanical Science and Technology. 2009; 23(8): 2125–2137. doi: 10.1007/s12206-009-0623-x

[17]Lu CR, Li YN, Dou ZC, et al. Nonlinear vibration characteristics of bending torsional coupling of gear rotor-bearing system. Journal of Vibration engineering. 2018; 31: 238–244.

[18]Liu W, Shi K, Tupolev V, et al. Nonlinear dynamics of a two-stage planetary gear system with sliding friction and elastic continuum ring gear. Journal of Mechanical Science and Technology. 2022; 36(1): 77–85. doi: 10.1007/s12206-021-1206-8

[19]Chen B, Cui Z, Jiang H. Producing negative active stiffness in redundantly actuated planar rotational parallel mechanisms. Mechanism and Machine Theory. 2018; 128: 336–348. doi: 10.1016/j.mechmachtheory.2018.06.002

[20]David BS. Geared rotor dynamic methodologies for advancing prognostic modeling capabilities in rotary-wing transmission systems [PhD thesis]. University of Virginia; 2008. Available online: https://www.proquest.com/docview/304438408

[21]Huang K, Yi Y, Xiong Y, et al. Nonlinear dynamics analysis of high contact ratio gears system with multiple clearances. Journal of the Brazilian Society of Mechanical Sciences and Engineering. 2020; 42(2): 98. doi: 10.1007/s40430-020-2190-0

[22]Cai Y. Simulation on the rotational vibration of helical gears in consideration of the tooth separation phenomenon (a new stiffness function of helical involute tooth pair). Journal of Mechanical Design. 1995; 117(3): 460–469. doi: 10.1115/1.2826701

[23]Xiang, L., Gao, N., Hu, A., 2018. Dynamic analysis of a planetary gear system with multiple nonlinear parameters. Journal of Computational and Applied Mathematics. 327, 325–340. DOI: https://doi.org/10.1016/j.cam.2017.06.021

[24]Wang, J., Wu, Z., Wang, H., et al., 2025. Analysis of influence of thermal tooth backlash on nonlinear dynamic characteristics of planetary gear system. Nonlinear Dynamics. 113(1), 289–311. DOI: https://doi.org/10.1007/s11071-024-10160-6

[25]Mo, S., Wang, Z., Tang, X., et al., 2024. Nonlinear dynamics of planetary gearboxes containing cracks. Nonlinear Dynamics. 112(19), 17007–17031. DOI: https://doi.org/10.1007/s11071-024-09942-9

Early fault signals of the rolling bearing in the rotor are weak and present the characteristics of non-periodic and non-stationary; it is more difficult to carry out fault diagnosis on it. In this regard, this paper proposes a weak rolling bearing fault diagnosis algorithm based on whale optimization algorithm, simplistic geometry mode decomposition, and maximum correlated kurtosis deconvolution (WOA-SGMD-MCKD). Firstly, the vibration signal of the rotor platform is obtained, and the Symmetric Geometric Mode Decomposition (SGMD) is used to reconstruct the vibration signal. To obtain the best decomposition effect of the SGMD and overcome modal aliasing, the Whale Optimization Algorithm (WOA) is used to optimize the embedding dimension. Secondly, for the reconstructed vibration signal, the Maximum Correlated Kurtosis Deconvolution (MCKD) is used to extract its impulse component, and the WOA is used to optimize the filter length and deconvolution period of the MCKD so that the frequency envelope spectrum of the vibration signal can be obtained, which can provide the basis for the fault diagnosis of rolling bearings. Finally, the effectiveness and feasibility of the algorithm proposed are verified by a non-periodic and non-stationary simulation platform and rotor maneuvering platform in this paper.

Numerical solution of a 3D mathematical model for the progression of tumor angiogenic factor in a tissue

In this work, the movement of tumor angiogenic factor in a three-dimensional tissue is obtained by the Method of Lines. This method transforms a partial differential equation into a system of ordinary differential equations together with the initial and boundary conditions. The more the number of lines is increased, the more the accuracy of the method increases. This method results in very accurate numerical solutions for linear and non-linear problems in contrast with other existing methods. We present Matlab-generated figures, which are the movement of tumor angiogenic factor in porous medium and explain the biological importance of this progression. The computer codes are also provided.

Mathematical modelling and controllability analysis of fractional order coal mill pulverizer model

This paper investigates the controllability of nonlinear dynamical systems and their applications, with a focus on fractional-order systems and coal mill models. A novel theorem is proposed, providing sufficient conditions for controllability, including constraints on the steering operator and nonlinear perturbation bounds. The theorem establishes the existence of a contraction mapping for the nonlinear operator, enabling effective control strategies for fractional systems. The methodology is demonstrated through rigorous proof and supported by an iterative algorithm for controller design. Additionally, the controllability of a coal mill system represented as a nonlinear differential system, is analyzed. The findings present new insights into the interplay of fractional dynamics and nonlinear systems, offering practical solutions for real-world control problems.