## 6 Conclusions  
This study establishes an efficient mechanical model for the torsional stiffness of the leaf spring torsional vibration damper by integrating the Euler-Bernoulli beam theory with the geometric nonlinearity observed during spring deformation. Subsequently, the research investigates the impact of clamping groove arc radius, leaf spring thickness and length, and red copper gasket length on torsional stiffness properties. The principal conclusions are as follows:  
1. The analytical outcomes from the mechanical model proposed in this article exhibit a high level of concordance with the results obtained through finite element analysis, validating the reliability of the model. The model can not only compute the torsional stiffness of the damper but also facilitate the study of how design parameters affect the torsional stiffness characteristics, thereby assisting in the design of dampers. Figure 16: Influence of length of red copper gasket on torsional stiffness of damper. Figure 17: Influence of length of red copper gasket on the torsional stiffness of damper under the same torsion angle. 2. The damper's torsional stiffness and nonlinear features undergo significant influence from the clamping groove arc radius. Specifically, nonlinear traits become more pronounced with increasing arc radius. This serves to prevent damage and failure due to excessive torsion angles within the damper. 3. The length of the leaf spring significantly impacts the damper's torsional stiffness. Longer leaf springs result in reduced stiffness within the leaf spring group, sharply diminishing the damper's torsional stiffness and displaying pronounced nonlinear characteristics. Leaf spring thickness minimally affects the damper's nonlinear characteristics but strongly influences its torsional stiffness. Decreasing the thickness of the leaf spring's thin side leads to a reduction in the damper's torsional stiffness. 4. The length of the red copper gasket minimally affects the damper's torsional stiffness nonlinearity but moderately influences its stiffness. 5. During the design of leaf spring torsional vibration dampers, attention must be given to the impact of clamping groover radius, spring length, and spring thickness on torsional stiffness and its nonlinear characteristics. If researchers observe a significant deviation in torsional stiffness from the anticipated value during the design phase, their priority should be adjusting the spring length, followed by reasonable adjustments to other parameters. Additionally, the impact of the red copper gasket length on torsional stiffness can be taken into account, but there is no need to consider its influence on the nonlinear characteristics of torsional stiffness. Conceptualization, Z.T. and C.S. ; methodology, Z.T. and C.S. ; software, G.L. ; validation, G.L. and C.C. ; formal analysis, G.L. ; investigation, Y.Z. ; resources, Z.T. ; data curation, G.L. ; writing--original draft preparation, G.L. ; writing--review and editing, Z.T. and C.S. ; visualization, G.L. ; supervision, Z.T. and C.S. ; project administration, Z.T. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The data used to support the findings of this study are available from the corresponding author upon request. The data are not publicly available due to privacy. The authors declare no conflicts of interest. The term \"The nonlinear characteristics of the torsional stiffness\" means the slope of torsional stiffness curve (as shown in Figures 10, 12, 14 and 16) changes with the increase in torsion angle. The functions in deflection expressions (19) and (20) are as follows.