# The Impact of Hollow Wear on the Stability of High-Speed Railway Vehicles
## 3 Impact of Hollow Worn Wheel on Vehicle Stability
### Evaluation of Vehicle Stability Using Bifurcation Configuration Method  
#### 3.2.1 Critical Speed and Bifurcation Type of Vehicles  
Due to the nonlinear contact relationship between wheels and rails, bifurcation phenomena may occur during the movement of wheelsets, commonly represented by bifurcation diagrams. According to the degrees of freedom listed in Table 2 and the motion differential equations of the vehicle system [22], let  
\[\left\{ \begin{array}{l}y=\left[y_{u1},y_{u2},y_{u3},y_{u4},\dot{y}_{u1},\dot{y }_{u2},\dot{y}_{u3},\dot{y}_{u4},y_{u1},y_{u2},\dot{y}_{u1},y_{l2},\dot{y}_{l1}, \dot{y}_{l2},\dot{y}_{l1},\dot{y}_{l2},y_{c},\dot{y}_{c},\dot{y}_{c}]^{T}\\ \dot{y}=\left[\dot{y}_{u1},\dot{y}_{u2},\dot{y}_{u3},\dot{y}_{u4},\dot{y}_{u1}, \dot{y}_{u2},\dot{y}_{u3},\dot{y}_{u4},\dot{y}_{u1},\dot{y}_{l2},\dot{y}_{l1}, \dot{y}_{l2},\dot{y}_{l1},\dot{y}_{l2},\dot{y}_{c},\dot{y}_{c},\dot{y}_{c}]^{T} \\ z=\left[\beta_{u1},\beta_{u2},\beta_{u3},\beta_{u4},y_{syl1},y_{syl1},y_{syl2}, y_{syl2},y_{syl1},x_{ssl1},x_{ssl2},x_{ssl2}]^{T}\right.\end{array} \right. \tag{5}\]  
Then, the nonlinear ordinary differential equation can be described the vehicle's dynamic behavior by reducing the order. \[\frac{d\mathbf{X}}{dt}=f(\mathbf{X},a) \tag{6}\]  
Figure 8: Nonlinear critical speed corresponding to different worn wheels. where \(a\) is called the control parameter or bifurcation parameter [23], and \(\mathbf{X}=[y;y;z]\). In this article, the vehicle operating speed \(v\) was used as the bifurcation parameter of the system, and the equilibrium solution of the nonlinear differential equation \(\mathbf{X}_{0}=(\overline{x}_{1},\overline{x}_{2},\ldots,\overline{x}_{n})^{T}\) satisfied the following equation:  
\[f(\overline{x}_{1},\overline{x}_{2},\ldots,\overline{x}_{n},v)=0\qquad(i=1,2, \ldots,n) \tag{7}\]  
As the velocity of the vehicle rises, the complex conjugate eigenvalues of the Jacobi matrix \(\mathbf{A}(v)=\left[\partial f_{i}/\partial x_{j}\right](i,j=1,2,\ldots,n)\) cross the imaginary axis, and the system exhibits Hopf bifurcation [24]. Meanwhile, the equilibrium solution for the vehicle system transitions from a stable state to an unstable state. Therefore, in the stability analysis of the vehicle system, the bifurcation diagram illustrates the amplitude of the limit cycle (often represented by the lateral displacement of the wheelset) as a function of the vehicle's speed. The commonly used numerical calculation methods for solving limit cycles include the direct integration method, shooting method, and continuation algorithm. In this study, the shooting method is applied to solve the bifurcation configuration diagram of vehicles. The bifurcation configuration of vehicles traveling in a straight line usually has three main forms, as shown in Figure 9. Figure 9a shows subcritical bifurcation, while Figure 9b,c show supercritical bifurcation. In the figure, the solid line denotes the stable limit cycle, while the dashed line denotes the unstable limit cycle; Point A is the Hopf bifurcation point of the vehicle system, defined as the linear critical speed, which is denoted as V_lin; Point B is the minimum speed at which vehicle hunting occurs, which is the nonlinear critical speed of the vehicle system, which is denoted as V_nl. In addition, Point D represents the saddle point of stable and unstable limit cycles. In Figure 9a, when the vehicle's speed is below the nonlinear critical speed (Point B), the limit cycle approaches zero irrespective of the initial excitation amplitude. However, once the vehicle's speed surpasses this threshold, the amplitude of the limit cycle becomes contingent upon the initial excitation amplitude. If the initial excitation amplitude is within the AB line, the limit cycle converges to zero. If the initial excitation amplitude exceeds the amplitude corresponding to the AB line, the limit cycle will converge to the amplitude corresponding to the BC line.