# The Numerical Study for the Effect of Stiffness Matching on Wheel-Rail Curve Squeal Noise
## 5 Conclusions  
In this study, a comprehensive analysis of wheel-rail curve squeal noise is conducted by establishing a three-dimensional solid finite element model of the wheel-rail system using finite element software ABAQUS 2022. The investigation focuses on the stability analysis of the system under train slip conditions in small radius curved lines. The primary objectives are to delve into the mechanism, location, and influencing factors of wheel-rail curve squeal noise. The methodology employed employs modal coupling theory and the complex eigenvalue analysis method to calculate the complex modes of the wheel-rail system. Subsequently, unstable modes are extracted, and their modal characteristics are thoroughly examined and discussed. Lastly, the impact of wheel-rail stiffness matching on the propensity for curve squeal is explored, and approaches to mitigate curve squeal are proposed from this perspective. This comprehensive research leads to the following specific conclusions:  
1. In comparison to the resonant modes of the wheel and rail, as well as the unstable modes within the wheel-rail system, it is evident that the frequencies of these unstable modes closely align with the resonant frequencies of the wheel and rail. Furthermore, the mode shapes of these unstable modes bear a striking resemblance to the resonant mode shapes of the wheel and rail. Specifically, the wheel predominantly exhibits axial vibrations along the 0-pitch-circle axial mode, while the rail primarily experiences vertical bending vibrations. Notably, the axial vibration of the wheel interacts with the vertical vibration of the rail, resulting in the occurrence of frictional self-excited vibrations.
2. When considering the amplitude of complex mode shapes concerning the occurrence of wheel-rail curve squeal noise, it becomes evident that the noise primarily originates from vibrations occurring at three key positions: the rim, railhead, and rail foot.
3. The wheel-rail stiffness exerts a significant influence on curve squeal noise. Analyzing its impact from the perspective of wheel-rail stiffness matching offers valuable insights into controlling curve squeal noise effectively. When the difference in elastic modulus between the rail and wheel is excessively large or small, it tends to increase the likelihood of squeal noise occurrence. The optimal ratio of elastic modulus between the rail and wheel is approximately 1.2, providing a guideline for effective noise control measures.
4. The rail support stiffness does have a discernible impact on curve squeal noise. Theoretically, maintaining an appropriate support stiffness level keeps the negative damping ratio of the system's unstable modes at a minimal level, thereby reducing the energy input into the system during self-excited vibration. Consequently, control of wheel-rail curve squeal noise can be achieved by carefully managing the rail support stiffness.  
Conceptualization, S.H.; data curation, Y.G. and G.Z.; investigation, J.J.; methodology, Y.G., G.Z., M.Y., N.C. and S.H.; supervision, Y.G., M.Y., J.J., N.C. and S.H.; writing--original draft, Y.G. and G.Z.; writing--review and editing, Y.G., G.Z., M.Y., J.J., N.C. and S.H. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the Fundamental Research Funds for the Central Universities (2023 ZYGXZR089), Shandong Provincial Natural Fund Project (ZR2020 QE248) and Science and Technology Planning Project of Guangdong Province (Foreign Experts Program of Department of Science and Technology of Guangdong Province, China).  
Not applicable.  
The authors declare no conflict of interest.