# The Numerical Study for the Effect of Stiffness Matching on Wheel-Rail Curve Squeal Noise
## 3 Results
### Frequency Response Analysis of Wheel and Rail  
#### 3.2.1 Modal Analysis of Wheel  
The modal characteristics of the wheel encompass its natural frequency and modal shape, serving as direct indicators of the wheel's dynamic behavior. Wheel resonance may occur when the frequency of external excitation closely aligns with the wheel's natural frequency. Therefore, it is essential to extract the natural frequency of the wheel through modal analysis to gain insights into its dynamic response and identify potential resonance modes.  
On an international scale, curve squeal noise is typically defined as the sound radiation occurring above 1000 Hz due to wheel-rail vibration, specifically from the inner wheel of the front wheelset. The frequency range associated with curve squeal noise typically falls within the 1000 Hz to 5000 Hz range. Consequently, this paper primarily focuses on conducting complex eigenvalue analysis for modes within the wheel-rail system with frequencies below 5000 Hz. This analysis aims to predict the trends related to the occurrence of wheel-rail curve squeal noise.  
To determine the natural frequencies of the wheel within the frequency range of 0-5000 Hz, the Lanzos method within the modal analysis capabilities of software ABAQUS 2022 is employed. For circular axi-symmetric structures such as wheels, it is convenient to introduce the concepts of axial and radial modes for a more comprehensive understanding. Axial modes pertain to vibrations along the center axis of rotation of the wheel, representing out-of-plane modes. Radial modes, on the other hand, involve vibrations within the radial plane, oriented along the axial centerline direction, signifying in-plane modes. Typically, the notation (d,c) mode is employed to represent a specific--order-modal shape, where 'd' denotes the number of pitch diameters and 'c' represents the number of pitch circles. The term \"pitch diameter\" refers to the diameter at which the modal shape value is zero, while \"pitch circle\" refers to the circle where the modal shape value reaches zero. In this context, (d,c) mode signifies a particular axial modal shape. Figure 4 illustrates selected axial modal shapes of the wheel.  
Figure 3: Transverse frictional force.  
Analyzing the mode shapes, we can discern distinct vibration patterns associated with different axial and radial modes:  
1. Axial Modes with 0 Pitch Circles: These modes predominantly exhibit out-of-plane bending primarily focused on the rim of the wheel. The deformation primarily occurs along the rim, emphasizing its importance in these modes.
2. Axial Modes with 1 Pitch Circle: In these modes, the primary vibration pattern involves out-of-plane bending, similar to the 0-pitch-circle axial modes. However, there is an additional contribution from the spoke plate, resulting in a more complex vibration pattern that combines both rim and spoke plate deformations.
3. Radial Modes: Radial modes display a distinctive pattern. They entail not only the bending deformation observed on the spoke plate but also include bending deformations along the rim. These modes manifest more comprehensive deformations, incorporating both the rim and the spoke plate, setting them apart from the axial modes.  
These modal shapes provide critical insights into the dynamic behavior of the wheel under different axial and radial modes of vibration, aiding in the understanding of its response to various external forces and contributing to the broader comprehension of wheel-rail interactions and potential sources of curve squeal noise.