## 5 Relationship between Contact Forces and Wear as well as New Insights
### Preferred Frequency of Contact Forces  
The dynamic normal and longitudinal contact forces are determined by the combined effects of the parameters of the structure, the excitation by the corrugation and the transient rolling contact. To investigate the contact force sensitivity to different corrugation wavelengths, the wavelength is varied at 28, 30, 31, 32 and 35 mm. Figure 12 shows the relationships between the PSD and frequencies of the contact forces and the corrugation wavelength. The following observations can be made. 1. The frequencies of the normal and longitudinal forces are different. The frequencies of the contact forces are different from that of the excitation, i.e., the passing frequency of the corrugation, and they are sensitive to and change with the corrugation wavelengths. The dynamic forces are stronger at certain wavelengths than others. With the current track parameters, both the longitudinal and vertical contact forces are strongest at \(\lambda\) = 30 mm. This again is in agreement with the observed corrugation wavelength of 30 mm. 2. The bandwidth of the frequency change of the longitudinal force is between 1116 Hz and 1288 Hz (Figure 12a), which is narrower than that of the normal force (between 1073 and 1361 Hz). The frequency band of the vertical force is broader because it follows the change in the corrugation wavelength, as shown in Figure 12b for the relatively constant ratio of the corrugation passing frequency to the contact force frequency. The frequency of the longitudinal contact force is lower than that of the normal contact force when the corrugation wavelength is short, i.e., between \(\lambda\) = 28 mm and 31 mm (Figure 12b), and vice versa when the corrugation wavelength is longer than 31 mm. 3. The frequency of the normal contact force is always lower than that of the excitation (Figure 12b). The presence of corrugation is an excitation mainly in the normal direction; thus, the response always follows the excitation. However, the frequency of the longitudinal contact force can be lower or higher than that of the excitation, which likely depends on the nearest natural frequency, as well as the complex relationship between the tangential and normal contact forces. This pattern reveals a strong dependence of the normal contact force on the excitation and a relatively weaker dependence of the longitudinal contact force on the excitation. These dependencies are in line with the narrower band of the frequency change of the longitudinal force compared with the normal force. 4. In Figure 12b, the frequency of the longitudinal contact force has the largest deviation from the corrugation passing frequency when the wavelength is shorter, i.e., at \(\lambda\) = 28 mm. With increasing \(\lambda\), the deviation decreases. At approximately \(\lambda\) = 31.5 mm, the frequency of the longitudinal contact force equals the passing frequency. The longitudinal contact force subsequently follows the excitation closely. 5. The frequency curves (Figure 12b) of the normal and longitudinal contact forces cross each other at a wavelength of 31 mm, where the frequencies of the normal and longitudinal forces are equal. Is an equal frequency of the two forces a condition for the corrugation to initiate, grow and become a wavelength-fixing mechanism? As shown in Sections 4.2 and 5.1, the wear is strongest when the normal and longitudinal forces \(F_{N}\) and \(F_{L}\) are in phase at 1.23 m. Because an equal frequency is a necessary condition for \(F_{N}\) and \(F_{L}\) to be in phase over many wavelengths, it is indeed a favorable condition for corrugation development. These observations indicate that the system appears to have a preference for certain frequencies, which is further confirmed by Figure 12a, in which the frequency of the longitudinal contact force is 1226 Hz for \(\lambda\) = 30 mm, 31 mm and 32 mm. This apparent preference is also in agreement with the results of [3], which show that most models in the literature suggest a fixed frequency related to vertical resonances in the system.