# Analysis of Vertical Stiffness Characteristics Based on Spoke Shape of Non-Pneumatic Tire
## 4 Conclusions  
In this study, the vertical stiffness characteristics depending on shape change in shape of spokes of the same structure were compared and analyzed via finite element analysis assuming a quasi-static state. The tree branch structure was adopted for the non-pneumatic tire spoke, and a fillet-applied model, asymmetric-spoke division model, and symmetric-spoke division model were used for the analysis. The static analysis results were compared  
Figure 11: Static structural analysis results with respect to force.  
Figure 12: Comparison of vertical stiffness and total deformation of four models.  
with the reference model obtained from Kumho Tire Co., Inc., and the vertical stiffness characteristics based on the change in the spoke shape were analyzed. Consequently, the following conclusions were drawn.  
1. Compared with the reference model, the total deformation was lower by approximately 1.56% for the fillet-applied model, higher by 4.09% for the asymmetric-spoke division model, and higher by approximately 6.43% for the symmetric-spoke division model.
2. Compared with the reference model, the maximum principal stress was higher by approximately 11.13% for the fillet-applied model, 5.34% for the asymmetric-spoke division model, and 9.81% for the symmetric-spoke division model.
3. Compared with the reference model, the directional deformation was lower by approximately 4.13% for the fillet-applied model, higher by 0.53% for the asymmetric-spoke division model, and higher by 2.53% for the symmetric-spoke division model.
4. The fillet-applied model with the smallest deformation indicated the highest vertical stiffness at 226,078 N/m, whereas the symmetric-spoke division model with the greatest deformation indicated the lowest vertical stiffness at 211,405 N/m.
5. When applying the fillet, the maximum principal stress was relatively large but the directional deformation decreased. Consequently, the vertical stiffness of the non-pneumatic tire increased.
6. When the spoke was divided, the resistance against the load in the longitudinal direction decreased, deformation increased, vertical stiffness decreased, and the stress acting on the spoke increased.  
The findings of this study indicate that the vertical rigidity of a non-pneumatic tire can be secured if a fillet with a minimum size, which does not impair the performance of the tire, is applied at the location where concentrated stress occurs in the spoke. This can ultimately improve the operating stability of a non-pneumatic tire.  
Based on the analysis results of this study, a detailed validation will be conducted in the future via an experiment using a prototype non-pneumatic tire.  
[PERSON] and [PERSON] conceived and designed the data analysis, analyzed the data, and wrote the paper; [PERSON] performed the analysis and produced the analytic data; [PERSON] researched the paper and proposed the direction; All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Ethical review and approval are not applicable for this study not involving humans or animals.  
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.  
The data presented in this study are available on request from the corresponding author.  
We would like to express our deep gratitude to Kumho Tire Co., Inc. for supporting this research.  
The authors declare no conflict of interest.