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Article History

Received: 8 October 2023

Accepted: 29 September 2024

First Online: 16 June 2025

Declarations

:

: Shuai Mo and Yingxin Zhang contributed equally to this manuscript, and Shuai Mo and Yingxin Zhang are co-first authors of the article. We declare that we have no conflicts with other people or organizations that can inappropriately influence our work.

: According to Newton’s second law, the torsional vibration equation of pinion and gear can be expressed Eq. ().Based on the assumption of this paper, the tooth pair #1 was de-meshing in the single tooth meshing interval, and the meshing force and friction force were assumed to be 0, which was expressed as Eqs. (, ). The Eq. () has been valid in the meshing process.In Eq. (), the two equations were subtracted and simplified.where, xn = (Rbpθp − Rbgθg)cosα was the newly introduced degree of freedom in the simplification process, which represented the equivalent displacement on the meshing line, and the occurrence condition of the positive meshing state was: xn > D(t). The meshing force of the positive meshing was composed of the stiffness term and the damping term, which was expressed as Eq. ().Similarly, the torsional vibration equation of pinion and gear under reverse impact was:According to the same steps to simplify:The occurrence condition of reverse impact state was: xn <  − D(t), and the reverse meshing force can be expressed as:For the de-meshing case, the absolute rotation equations for pinion and gear were:There was no meshing and friction. After simplification, Eq. () was obtained.The conditions for the occurrence of de-meshing were − D(t) < xn < D(t).The positive meshing equation Eq. (), the reverse impact equation Eq. () and the de-meshing equation Eq. () were integrated into Eq. ().where, h(t,x) was the meshing state function and f(x,D(t)) was the backlash function, which were expressed as Eqs. (, ).The backlash D(t) was taken as the dimensionless scale of displacement, the excitation frequency ωn was the dimensionless scale of time, the equivalent mass me was taken as the dimensionless scale of mass, and the average meshing stiffness Kavg was taken as the dimensionless scale of stiffness. The quantity in Eq. () was dimensionless processed.After simplification, the nonlinear dynamic model of gear was obtained.where, the dimensionless meshing state function h(τ,x) and the backlash function f(x,D(τ)) were expressed as