# The Numerical Study for the Effect of Stiffness Matching on Wheel-Rail Curve Squeal Noise
## 4 Discussion
### The Effect of Elastic Modulus of Rail on The Propensity of Squeal Noise  
Finite element software ABAQUS 2022 has the capability to extract eigenvalues and damping ratios for specific modes, enabling the assessment of mode stability based on the real part of the eigenvalues and their symbols. However, solely evaluating the stability of individual modes does not provide a direct means of predicting the likelihood of squeal noise occurrence. Moreover, it presents challenges when attempting to assess and compare system stability across various operational conditions. To address these limitations, the concept of \"TOI\" (tendency of instability) is introduced as an evaluation index for gauging the overall stability of the wheel-rail system. Its definition is outlined as follows [31]:  
\[\mathrm{TOI}= \sum_{\mathrm{i}}\,(\frac{1000\mathrm{Re}(\lambda_{\mathrm{i}})}{ \mathrm{Im}(\lambda_{\mathrm{i}})}),\,\mathrm{Re}(\lambda_{\mathrm{i}})\ >\ 0 \tag{21}\]  
where \(\,\mathrm{Re}(\lambda_{\mathrm{i}})\) is the real part of the eigenvalue of the i-th-order mode and \(\,\mathrm{Im}(\lambda_{\mathrm{i}})\) is the imaginary part of the eigenvalue of the i-th-order mode. \[\lambda_{\mathrm{i}}=-\,\xi_{\mathrm{mi}}\omega_{\mathrm{mi}}\pm\mathrm{j} \omega_{\mathrm{mi}}\sqrt{1-\,\xi_{\mathrm{mi}}^{2}} \tag{22}\]where: \(\omega_{\rm mi}\) is the natural frequency of the order mode without damping, and \(\xi_{\rm mi}\) denotes the complex mode damping ratio, so that \(\xi_{\rm mi}\)\(<0\) when \(\rm{Re}(\lambda_{\rm i})~{}>~{}0\). The absolute value of the ratio of the real part to the imaginary part is set to \(\mu_{\rm i}\):  
\[\mu_{\rm i}=\left|\frac{\rm{Re}(\lambda_{\rm i})}{\rm{Im}(\lambda_{\rm i})} \right|=-\frac{\xi_{\rm mi}}{\sqrt{1-\xi_{\rm mi}^{2}}},\rm{Re}(\lambda_{\rm i })~{}>~{}0 \tag{23}\]  
Usually, for the unstable modes of the wheel-rail system, the damping ratio satisfies \(\xi_{\rm mi}\)\(<0\) and \(\xi_{\rm mi}~{}>~{}-0.01\), so that Equation (12) can be expressed as:  
\[\mu_{\rm i}=\left|\frac{\rm{Re}(\lambda_{\rm i})}{\rm{Im}(\lambda_{\rm i})} \right|=|\xi_{\rm mi}|,\rm{Re}(\lambda_{\rm i})~{}>~{}0 \tag{24}\]  
Therefore, TOI represents 1000 times the sum of the absolute values of the complex damping ratios \(\xi_{\rm mi}\) for each order of unstable modes of the system, so the bigger the value of TOI, the stronger the tendency of the system to be unstable and the more obvious the tendency for squeal noise to occur. The complex eigenvalues of the wheel-rail system are extracted by using the finite element software ABAQUS 2022. The frequency domain distribution of the negative damping ratio of the above four operating conditions is shown in Figure 11, and the number of modes and TOI values under each working condition are summarized as shown in Table 3.  
Figure 11: Frequency domain distribution of negative damping ratio under different elastic modulus of the rail: (**a**) 1.5 \(\times\) 10\({}^{5}\) MPa; (**b**) 1.8 \(\times\) 10\({}^{5}\) MPa; (**c**) 2.1 \(\times\) 10\({}^{5}\) MPa; (**d**) 2.4 \(\times\) 10\({}^{5}\) MPa. Figure 11 reveals that when the elastic modulus is set at 1.5 \(\times\) 10\({}^{5}\) Mpa, the negative damping ratio is minimal, with the highest magnitude of negative damping occurring within the frequency domain below 1000 Hz. From a system stability perspective, this configuration appears relatively stable. However, in this scenario, the rail exhibits a low elastic modulus, implying weak resistance to deformation. This situation is not conducive to both the safe operation of trains and the long-term durability of the rail. On the other hand, when the rail's elastic modulus is either equivalent to or significantly greater than that of the wheel, the system's instability tendencies are accentuated, especially when the rail's elastic modulus is excessively high. Under such conditions, the number of unstable modes notably increases.