# The Numerical Study for the Effect of Stiffness Matching on Wheel-Rail Curve Squeal Noise
## 4 Discussion  
### Complex Modal Analysis of Wheel-Rail System  
The outcomes of the complex modes analysis for the wheel-rail system within the 5000 Hz range have been extracted using ABAQUS software, and these results are depicted in Figure 9. Furthermore, a scatter plot representing the distribution of the negative damping ratio of the unstable modes across the frequency domain is generated for this operational scenario. Upon examination of the figure, it becomes apparent that three unstable modes are present, specifically the 52 nd-, 147 th-, and 272 nd-order modes. These modes correspond to frequencies of 1265.1 Hz, 2378.4 Hz, and 3799.8 Hz, respectively, as derived from the output file. The modal shapes obtained through complex eigenvalue analysis are intricate, resulting in relative phase differences among the degrees of freedom within each mode. In essence, these points do not simultaneously reach their maximum values (some lag behind others) and do not cross the equilibrium position at the same moment. Consequently, the modal shape of complex modes lacks certainty. To facilitate comprehension, this paper briefly introduces the concepts of real part mode and imaginary part mode:  
1. Real Part Mode: When the phase aligns with the real phase, the modal shape is referred to as a real part mode;
2. Imaginary Part Mode: Imaginary part modes are characterized by a 90\({}^{\circ}\) phase difference from the real phase. By combining the real part mode and imaginary part mode, it becomes possible to analyze the overall amplitude of the wheel-rail system model and ascertain the phase differences of vibration at each node within the model. The vibration amplitude diagram provides insights into the vibration amplitudes at specific nodes within a given order mode, offering valuable information for understanding the underlying causes of curve squeal noise. The subsequent analysis focuses on the third-order unstable modes of the wheel-rail system, examining the amplitudes of the modal shapes. Figure 10 illustrates these amplitudes, with a deformation amplification factor of 100 applied for clarity. Figure 9 reveals that the modal shapes of the three modes of the wheel take the form of out-of-plane modes, specifically the (3,0) mode, (2,1) mode, and (6,0) mode. These modes primarily exhibit out-of-plane bending, occurring predominantly on the rim and spoke plate. Conversely, the rail experiences more pronounced vibrations at the rail's foot, while the vibrations at the top of the rail comprise both transverse and vertical components. In overall terms, the rail can be characterized as undergoing longitudinal torsion with a predominant vertical bending component. Figure 9: Complex mode analysis results of the wheel–rail system. Upon comparing the results of the frequency response analysis for the wheel and rail, it becomes apparent that the frequencies of the unstable modes closely align with the resonant frequencies of the wheel and rail. Consequently, friction at these frequencies readily excites the resonant modal shapes of both the wheel and rail, resulting in amplitude levels that closely resemble those of the resonant mode shapes of the wheel and rail. Furthermore, through an examination of the positions where maximum amplitudes are observed, it becomes evident that noise primarily arises from vibrations occurring at three locations: the rim, railhead, and rail foot. The results of the unstable modal shape analysis for the wheel-rail system are summarized in Table 2.  
Upon conducting a comprehensive analysis of the amplitudes associated with the aforementioned three modes, several noteworthy observations emerge. In the wheel-rail system's unstable modes, the wheel consistently exhibits axial vibrations, while the rail primarily undergoes vertical vibrations. When the wheel's amplitude is relatively small, transverse vibrations become more pronounced at the top of the rail. Conversely, when the