Bifurcation of multi-stable behaviors in a two-parameter plane for a non-smooth nonlinear system with time-varying parameters

Shi, Jianfei; Gou, Xiangfeng; Zhu, Linsheng

doi:10.1007/s11071-020-05510-z

Cited by 13 publications

(2 citation statements)

References 37 publications

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“…Zhu et al [68] studied the influence of multi-stable response on the safety characteristics of the spur gear pair by nonlinear dynamics and safety basin methods. Shi et al [69] analyzed the bifurcation behavior of multi-stable behaviors in a twoparameter plane for a spur gear system. Mo et al [70] studied the global dynamic behavior evolution of non-orthogonal face gear-bearing transmission system based on cell-to-cell method.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

Bai,

Qiu,

Shi

et al. 2023

Nonlinear Dyn

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Gear disengaging, back-side tooth contact or poor dynamic behavior during operating leads to dynamic instability in planetary gear trains (PGTs). A novel nonlinear dynamic model of PGTs with internal and external gear pairs considering multi-state engagement induced by backlash and contact ratio is established. An improved time-varying meshing stiffness model including temperature stiffness is analytically derived. The time-varying meshing stiffness with temperature effect, friction, backlash, time-varying pressure angle, and time-varying friction arm are integrated into the dynamic model of PGTs. Multi-state engaging behavior is efficiently identified by constructing different Poincaré mappings. A method to calculate dynamic instability is proposed in the time-domain trace. The intrinsic relationship between multi-state engaging and dynamic instability is investigated via multi-section bifurcation plots and phase trajectory topology. The global dynamic instability is revealed based on the bifurcation and evolution of coexistence behavior under the parameter-state synergy. The results show that the multistate engagement is heavily depending on bifurcation and phase trajectory topology, which whereby affects the dynamic instability. Two special phenomena, complete and incomplete bifurcations, are discovered under parameter-state synergy. Complete bifurcation causes global instability and incomplete bifurcation results in local instability and yields coexistence responses. Incomplete bifurcation brings about new bifurcation branches.

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Section: Introductionmentioning

confidence: 99%

Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

Bai,

Qiu,

Shi

et al. 2023

Nonlinear Dyn

Self Cite

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“…It is worth noting that we can only solve the system by approximating analytical and numerical methods, including multiple time scales [3] and average methods [4]. Of course, in addition to the development of research methods, many scholars have studied the dynamic behavior of van der Pol oscillators under the influence of time delay [5,6] and non-smooth oscillators [7].…”

Section: Introductionmentioning

confidence: 99%

Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Chen¹,

Qian²

2021

Mathematics

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In this paper, the normal form and central manifold theories are used to discuss the influence of two-degree-of-freedom coupled van der Pol oscillators with time delay feedback. Compared with the single-degree-of-freedom time delay van der Pol oscillator, the system studied in this paper has richer dynamical behavior. The results obtained include: the change of time delay causing the stability switching of the system, and the greater the time delay, the more complicated the stability switching. Near the double Hopf bifurcation point, the system is simplified by using the normal form and central manifold theories. The system is divided into six regions with different dynamical properties. With the above results, for practical engineering problems, we can perform time delay feedback adjustment to make the system show amplitude death, limit loop, and so on. It is worth noting that because of the existence of unstable limit cycles in the system, the limit cycle cannot be obtained by numerical solution. Therefore, we derive the approximate analytical solution of the system and simulate the time history of the interaction between two frequencies in Region IV.

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Evolutionary mechanism of safety performance for spur gear pair based on meshing safety domain

Zhu

Gou

et al. 2021

Nonlinear Dyn

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Bifurcation of multi-stable behaviors in a two-parameter plane for a non-smooth nonlinear system with time-varying parameters

Cited by 13 publications

References 37 publications

Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Evolutionary mechanism of safety performance for spur gear pair based on meshing safety domain

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