# The Numerical Study for the Effect of Stiffness Matching on Wheel-Rail Curve Squeal Noise
## 4 Discussion
### The Effect of Rail Support Stiffness on The Propensity of Squeal Noise  
To research the influence of different support stiffness on the stability of the wheel-rail system, the complex modal analysis of the wheel-rail system with rail support stiffness of 0.5 Keq, 0.75 Keq, 1.25 Keq, and 1.5 Keq was carried out, respectively. The distribution of the negative damping ratio of unstable modes under different overall support stiffnesses in the frequency domain is shown in Figure 12, and the number of modes and TOI values under each working condition are summarized as shown in Table 4.  
The relationship between the overall support stiffness and system stability is not linear, as indicated by Figure 13. Both excessively small and excessively large bracing stiffness levels have an exacerbating effect on the system's instability trend. Therefore, when considering rail bracing stiffness as a means to control wheel-rail curve howling noise, it is imperative to identify a suitable rail stiffness level. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Support Stiffness (Keq)** & **Number of Unstable Modes** & **TOI** \\ \hline
0.5 & 8 & 14.198 \\
0.75 & 4 & 59.383 \\
1 & 3 & 54.293 \\
1.25 & 7 & 68.101 \\
1.5 & 11 & 78.444 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: TOI values under different support stiffness conditions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Elastic Modulus (MPa)** & **Number of Unstable Modes** & **TOI** \\ \hline \(1.5\times 10^{5}\) & 4 & 14.398 \\ \(1.8\times 10^{5}\) & 5 & 54.293 \\ \(2.1\times 10^{5}\) & 3 & 46.870 \\ \(2.4\times 10^{5}\) & 11 & 64.067 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: TOI values under different elastic modulus conditions. Comparing the distribution of unstable modes under standard support stiffness conditions (as shown in Figure 11c) with conditions of varying support stiffness, it becomes evident that the unstable modes exhibit a noticeable increasing trend. However, the negative damping associated with these unstable modes does not display a linear relationship with rail support stiffness, and the modal frequencies of the unstable modes do not deviate significantly from their original values. Figure 12: Frequency domain distribution of negative damping ratio under different support stiffness: (a) 0.5 \(\mathrm{K_{eq}}\); (b) 0.75 \(\mathrm{K_{eq}}\); (c) 1.25 \(\mathrm{K_{eq}}\); (d) 1.5 \(\mathrm{K_{eq}}\). Figure 13: Negative damping ratio variation curve with the support stiffness of rail. To further examine the variation in the negative damping ratio with changes in rail support stiffness, five unstable modes are extracted for comparative analysis. The resulting sample curve depicting the negative damping ratio in relation to the variation in overall support stiffness is presented in Figure 13.  
Among the five modes selected, four modes began to become unstable between 0.5 K\({}_{\text{eq}}\) and 0.75 K\({}_{\text{eq}}\), and then tended to be stable and decreased slightly. The 39 th-order mode is in a stable state under 0.75 K\({}_{\text{eq}}\), and instability occurs at lower stiffness and higher stiffness. Upon comparing the various curves in Figure 13, it becomes evident that the negative damping ratio of unstable modes responds differently to changes in rail support stiffness. These responses include decreasing, increasing, and maintaining stability, signifying that rail support stiffness exerts varying degrees of influence on different modes. Given that the frequency characteristics of actual wheel-rail curve squeal noise may be determined by one of these unstable modes, it is essential to minimize the negative damping of the actual self-excited vibration input system. This can be achieved by incorporating as many unstable modes as possible within a low damping ratio range during the optimization of support stiffness. As exemplified by the observation in Figure 11, it is noticeable that the negative damping ratio for each mode remains at a low level.