# The Numerical Study for the Effect of Stiffness Matching on Wheel-Rail Curve Squeal Noise
## 3 Results  
### Establishment and Verification of Complex Modal Analysis Model for Wheel-Rail System  
#### 3.1.1 Establishment of Complex Modal Analysis of Wheel-Rail System  
This paper primarily focuses on the analysis of wheel sliding friction under conditions of creep-slip force saturation. To facilitate this analysis, certain assumptions are made regarding the friction at the contact surface. These assumptions include the following: (1) The wheel's sliding process is considered to be pure sliding, with a constant contact area and a fixed coefficient of sliding friction. (2) The transverse creep-slip force, when it reaches saturation, is modeled as Coulomb friction. In other words, the lateral creep force (essentially friction) is assumed to be equal to the product of the contact surface's friction coefficient and the normal force.  
1. Finite element model of a wheel-rail system. Software Solidworks 2020 is used to geometrically model the wheel, rail, sleeper, and track plate, and is geometrically cleaned in software Hypermesh 2020 and meshed by 8-node hexahedral incompatible element (C3 DSI); the finite element model of each part is imported into software ABAQUS 2022, and each part is assembled according to the actual action relationship. The finite element model of the wheel-rail system and each part is shown in Figure 2. The sliding process of the wheel is pure sliding, and the sliding friction is Coulomb friction with a constant friction coefficient.
2. Defining material properties. Define the material properties of each structure specifically as shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structure Name** & **Elastic Modulus (MPa)** & **Density (kg/m\({}^{3}\))** & **Poisson Ratio** \\ \hline Wheel & \(1.8\times 10^{5}\) & 7800 & 0.28 \\ Rail & \(2.1\times 10^{5}\) & 7800 & 0.3 \\ Sleeper & \(3.45\times 10^{4}\) & 2500 & 0.2 \\ Track plate & \(3.45\times 10^{4}\) & 2500 & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of the wheel–rail system.  
Figure 1: Theoretical analysis framework.  
Analysis step settings. Four analysis steps are defined: the first one applies the normal load, the second one introduces the transverse sliding between the wheel and rail, and the last two analysis steps extract the real modes and complex modes.
4. Connection relationship settings. The contact relationship between wheel and rail is simulated by \"Surf-to-Surf Contact\", the rigid connection between wheel and axle is simulated by \"MPC-Beam\", and the fastener system between rail and sleeper is simulated by \"Spring\".
5. Determining the solution condition setting. Considering that this model is a system stability analysis of a unilateral wheel-rail, for the track plate, six degrees of freedom (three translational degrees of freedom and three rotational degrees of freedom) are restricted in the symmetry plane and the bottom plane; for the rail, three translational degrees of freedom are restricted in both end faces; for the wheel, the longitudinal and lateral translational degrees of freedom are initially restricted and the vertical load of 90 kN is applied to it, but the wheel can slide, and the sliding effect of the wheel in the second analysis step is added in Keywords.