# Effect of Control Measures on Wheel/Rail Noise When the Vehicle Curves
## 2 The Time Domain Model for Predicting Wheel/Rail Noise under Lateral Excitation  
### Vehicle/Track Interaction Model with Falling Friction Coefficients  
Figure 1 shows a three-dimensional dynamic model of a metro vehicle which coupled with a conventional ballastless track which can predict the curve squeal when a vehicle negotiates narrow curves. The dynamic model is based on the classical dynamic theory, which was also successfully applied to many research topics in rail transit field [28; 29; 30; 31; 32; 33]. In this paper, the car-body and two bogies are modelled as rigid bodies. The primary and secondary suspensions are modelled as spring-damp elements. The track system is a triple-layers model as demonstrated in [30], in which the couple of rails are modelled as Timoshenko beams. The explicit integration method in [34] is used to solve dynamic equations. Wheel-rail normal force is calculated based on non-linear Hertzian contact theory. Non-linear Hertzian contact force is simulated by Equation (1). The normal force depends on the relationship between normal loading and wheel, rail deformation if the normal compression at the wheel/rail contact point is larger than zero when wheel and rail are in contact. The normal force is zero if the normal compression at the wheel/rail contact point is not larger than zero when the wheel and rail separates. \[N(t)=\left\{ \begin{array}{ll}\left[\frac{1}{\alpha_{\text{ Hertz}}}Z_{\text{wrmc}}(t)\right]^{3/2},&Z_{\text{wrmc}}(t)>0\\ 0,&Z_{\text{wrmc}}(t)\leq 0\end{array} \right. \tag{1}\]  
where \(G_{\text{Hertz}}\) is the contact constant of wheel and rail (m/N\({}^{2/3}\)). For LM wheel tread profile and CN60 rail, \(G_{\text{Hertz}}\) = 3.86 \(R^{-0.115}\)\(\times\)\(10^{-8}\). \(R\) is the wheel rolling radius. \(Z_{\text{wrmc}}\) (\(t\)) is the normal compression at the wheel/rail contact point. The tangential creep forces of wheel and rail are calculated based on [PERSON]'s simplified creep theory, in which the falling down friction law [13] is introduced. The contact area is discretized and the tangential wheel-rail creep force is solved through iteration from the leading part of the contact area to its trailing part. Thus, the tangential wheel-rail creep forces can be written as follows:  
\[\left\{ \begin{array}{ll}p_{xi}=p_{x(i-1)}-\left(\frac{\xi_{x}}{L_{3}}-\frac{ \xi_{z}y_{i}}{L_{3}}\right)\Delta x\\ p_{yi}=p_{y(i-1)}-\left(\frac{\xi_{y}}{L_{3}}-\frac{\xi_{z}(x_{i}+0.5\Delta x) }{L_{3}}\right)\Delta x\end{array} \right. \tag{2}\]  
Figure 1: Coupling dynamic model of a metro vehicle and a conventional ballastless track: (**a**) Elevation; (**b**) End view. where \(p_{xi}\) and \(p_{yi}\) are longitudinal and lateral tangential wheel-rail creep force. \(\xi_{x}\), \(\xi_{y}\), \(\xi_{z}\) are longitudinal, lateral and spin creepage, respectively. \(\Delta x\) is the longitudinal width of discrete grid. \(L_{1}\), \(L_{2}\), \(L_{3}\) are flexibility coefficients and can be written as follows:  
\[L_{1}=\frac{8a}{3 GC_{11}},\ L_{2}=\frac{8a}{3 GC_{22}},\ L_{3}=\frac{\pi a\sqrt{ a/b}}{4 GC_{23}} \tag{3}\]  
where \(a\) and \(b\) are the lengths of longitudinal and lateral semi-axis, respectively. \(G\) is the shear modulus of wheel/rail. \(C_{ij}\) is the coefficient calculated by [PERSON] in 1967 [20]. The tangential wheel-rail creep forces at any point of contact area can be written as follows:  
\[p_{ti}=\sqrt{p_{xi}+p_{yi}} \tag{4}\]  
If  
\[p_{ti}\leq\mu p_{n}(x,y) \tag{5}\]  
where \(\mu\) is the wheel/rail friction coefficient, and \(p\) is the distribution of wheel/rail normal force, the particle in contact area is in an adhesion state. The tangential wheel-rail creep forces can be calculated by Equation (2). If  
\[p_{ti}>\mu p_{n}(x,y) \tag{6}\]  
the particle in contact area is in a slide state. The tangential wheel-rail creep forces can be reduced to Equation (7):  
\[\left\{ \begin{array}{c}p_{xi}=p_{xi}\left(\frac{up_{n}(x,y)}{p_{xi}}\right) \\ p_{yi}=p_{yi}\left(\frac{up_{n}(x,y)}{p_{xi}}\right)\end{array} \right.