## 4 Results and Discussion
A momentary difference between the rolling radii of the right and left wheels is a function of lateral displacement \(y\) of the wheel shaft in reference to the center of the wheel shaft and RRD(\(\Delta r\)) can be defined as follows:  
\[\Delta r(y)\equiv r_{1}(y)-r_{2}(y) \tag{12}\]  
As the RRD function is determined by the wheel and rail profiles, it is affected by the changes in the wheel shape caused by wear, and is a very important parameter in the wheel shaft dynamics. As an RRD is the difference between the radii of the outer and inner wheels at the contact point of a wheel and a rail, it is determined by the wheel and rail profiles, and affects sliding between the rail and wheel, wheel shaft safety, critical speed, and wheel shaft movement. As shown in Figure 7, in the case of the optimized wheel profile, a bigger lateral displacement has occurred in curved sections than in the case of the existing profile, and the cases where the contact point between the wheel and the rail has more uniformly changed than in the case of the existing profile, as the RRD has changed softly. On the other hand, as the wheel tread profile has not been changed, the optimized profile is the same as the existing profile in straight sections. The contact point between the existing wheel and rail profile is shown in Figure 8.  
Figure 7: Comparison of RRD. In general, the contact point between a wheel and a rail and an increase in the angle of attack affect the increase in the wear. An angle of attack is generated by the angle between the movement direction of the railway vehicle and the rail direction. An angle of attack affects the lateral pressure and vertical force, and an increase in the angle of attack leads to an increase in the wear. The location of the contact point between the wheel and rail profiles with optimized wear is shown in Figure 9. Lateral displacement occurs between 3 mm and 8 mm in a curved section, and the wheel flange and rail gauge contact each other, as shown in Figure 9, as a result of the lateral displacement. Figure 8: Position of contact points on wheel and rail depending on lateral displacement of wheelset (initial wheel profile). Figure 9: Position of contact points on wheel and rail depending on lateral displacement of wheelset (wear-optimized wheel profile). In this case, the contact point distribution of the profile with optimized wear is more uniform (that is to say, the jump at the contact point is smaller) than that of the existing profile. The location of the contact point between the wheel and rail profiles with optimized fatigue is shown in Figure 10.  
As shown in Figure 10, the wheel flange and the rail gauge contact each other in a curved section at the point where the lateral displacement of the wheel is between approximately 2 mm and 8 mm. In the case of the profile with optimized fatigue, the contact points between the wheel and the rail are relatively concentrated on one point. The contact point distribution of the profile with optimized fatigue is not uniform when compared with that of the existing wheel profile. Among the total 100 Pareto-optimal Solutions, the wheel design with optimized wear and fatigue was selected from the values between the wear number of the existing wheel and the wear number of the profile with optimized wear, and between the surface fatigue index of the existing wheel and the surface fatigue index of the profile with optimized fatigue. As shown in Figure 11, in the case of the wheel design with optimized wear and fatigue, the flange contact occurs at the point where the lateral displacement of the wheel shaft is between 3 mm and 8 mm, and the jump between contact points is not great, and the contact points are quite uniformly distributed when compared to the existing profile. As shown in Figure 11, the jump at a contact point is relatively great when compared with the design with optimized wear. Accordingly, the wear number of the wheel profile with optimized wear and fatigue is between the wear numbers of the existing wheel and the wheel with optimized wear.