## 5 Conclusions  
In this study, a dynamics model of the PRVT based on the Lagrangian method was established. The gear meshing stiffness, damping, and the influence of machining errors on dynamic torsional stiffness were considered in this model. Through error impact analysis, the main conclusions are summarized as follows:  
(1) By considering parameters such as gear time-varying meshing stiffness, bearing stiffness, disc spring stiffness, damping, and transmission error, a dynamic model of the PRVT coupled with the two-stage transmission is established based on the Lagrangian equation.  
(2) Based on the dynamics model, the impact of machining errors on dynamic torsional stiffness was investigated. The results show that the crank distribution circle radius error of the crankshaft (the slope is 1000 N/arcmin) and the distribution circle radius error of the crankshaft bearing hole on the carrier (the slope is 3500 N/arcmin) will cause the peak-to-peak torsional stiffness to increase, and the latter is 3.5 times the former. It can be found that the distribution circle radius error of the crankshaft bearing hole on the carrier has a greater impact on torsional stiffness than the crank distribution circle radius error of the crankshaft. In contrast, the crank angle error of the crankshaft (the slope is \(-3500\) N/arcmin) will cause the peak-to-peak torsional stiffness to decrease. To improve the whole machine's stability, it is necessary to control the error within a reasonable range.  
(3) The rotation angle error of the crankshaft bearing hole on the No. 1 carrier and the crankshaft bearing hole rotation error has no significant effect on the peak-to-peak value of torsional stiffness (the slope is 0 N/arcmin). Therefore, to reduce the whole machine's processing cost, it is recommended to properly address these machining errors. When the slopes of the torsional stiffness average are 0 N/arcmin, the machining errors researched in the paper have no significant effect on the torsional stiffness.  
(4) For different machining errors, the cyclic variation in the curve and its influence on torsional stiffness is small. The impact of the distribution circle radius error of the crankshaft bearing hole on the carrier on dynamic torsional stiffness is 3 times that of the crank distribution circle radius error of the crankshaft. The impact of the rotation angle error of the crankshaft bearing hole on the No. 1 carrier on dynamic torsional stiffness is 4.17 times that of the distribution circle radius error of the crankshaft bearing hole, and 12.5 times that of the crankshaft bearing hole rotation error.  
(5) For further research, it is highly recommended to consider the dynamic mesh stiffness and subsurface stress at the same time for gear reducers to further analyze the variational patterns of the torsional stiffness.  
Methodology, L.C.; data curation, L.C. and Y.H.; writing--original draft preparation, L.C., Y.H. and N.U.; writing--review and editing, L.Z., Y.J. and Z.L. All authors have read and agreed to the published version of the manuscript.  
This research work was supported by the National Key Research and Development Program of China under Grant No. 2020 YFB2009601.  
The authors would like to thank the editors and the reviewers for their comments on an earlier draft of this article.  
The authors declare no conflict of interest.