# 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
\tag{7}\]  
The wheel/rail friction coefficient \(\mu\) can be modeled by a constant [35], Fingberg law (Equation (8)) [13] or piecewise friction coefficient (Equation (9)) [21]. \[\mu(x,y)=\mu_{s}[\frac{50}{100+\left|\mathbf{v}(x,y)\right|^{2}}+\frac{0.1}{0 2+\left|\mathbf{v}(x,y)\right|}] \tag{8}\]  
where \(\mathbf{v}(x,y)\) is the slip velocity between rail and wheel, \(\mu_{s}\) is the static friction coefficient. \[\mu(x,y)=\left\{ \begin{array}{cc}\mu_{s}&\text{if }\left|\mathbf{v}(x,y) \right|=0\\ \mu_{s}-\frac{\mu_{s}-\mu_{d}}{v_{c}}\cdot\left|\mathbf{v}(x,y)\right|&\text{ if }0<\left|\mathbf{v}(x,y)\right|\leq v_{c}\\ \mu_{d}&\text{if }\left|\mathbf{v}(x,y)\right|>v_{c}\end{array} \right. \tag{9}\]  
where \(\mu_{d}\) is the sliding friction coefficient, \(v_{c}\) the critical sliding velocity, and \(\mathbf{v}(x,y)\) is the slip velocity in the local coordinate system. Figure 2 gives the relationship of Creep force and Creepage based on three kinds of friction coefficients. The Fingberg model and piecewise model both consider the falling down friction coefficient. The piecewise friction coefficient is simplified from Fingberg model. The falling friction coefficient needs to be taken into account in the process of curve squeal simulation. This study used the Fingberg' model. In this study, a special type of track, i.e., embedded track, is investigated. Figure 3 shows the dynamic model of filling material (including elastomer and rail pad) in the groove. The filling material is glued to the rail and slab, and it provides elastic support and damping. There is no relative motion between filling material and rail/slab. In the dynamic simulation of embedded track, the filling material of the embedded track is approximately simulated as mass-spring-damping system. Compared with the spring-damping simulation, the effects of the inertia of elastomer can be considered in the mass-spring-damping system in this paper. The slab of the embedded track is simulated by 3D finite element in the vehicle-track coupling dynamic model. Figure 4 shows the force diagram of elastomer of filling material (the mass in Figure 3). The lateral and vertical motions of elastomer of filling material on left side and right side of the one rail are written in Equations (10)-(13). The stiffness (\(K_{ph}\) and \(K_{fill\_v}\)) and damping (\(C_{ph}\) and \(C_{fill\_v}\)) are including in forces in Figure 4.  
Figure 4: Dynamic model of elastomer in groove. Figure 3: Dynamic model of filling material in groove. Figure 2: Creep force-Creepage relationship based on three kinds of friction coefficients. . Lateral motion equation of elastomer on the left side  
\[M_{fill}\ddot{y}_{lf(L,R)i}=F_{sfh(L,R)i}+F_{sfh(L,R)i} \tag{10}\]  
Vertical motion equation of elastomer on the left side  
\[M_{fill}\ddot{z}_{lf(L,R)i}=-F_{lfr(L,R)i}-F_{lfsw(L,R)i}+F_{lff(L,R)i}+F_{lfr(L,R)i} \tag{11}\]  
Lateral motion equation of elastomer on the right side  
\[M_{fill}\ddot{y}_{rf(L,R)i}=F_{sfh(L,R)i}+F_{rfh(L,R)i} \tag{12}\]  
Vertical motion equation of elastomer on the right side  
\[M_{fill}\ddot{z}_{rf(L,R)i}=-F_{rfr(L,R)i}-F_{rfsw(L,R)i}+F_{rff(L,R)i}+F_{rfr(L,R)i} \tag{13}\]  
where \(M_{fill}\) is the mass of the \(i\)-th elastomer on one side (left or right) of the rail. \(\ddot{z}_{lf(L,R)i}\) and \(\ddot{z}_{rf(L,R)i}\) are vertical accelerations of the \(i\)-th elastomer on the left and right sides of the rail. \(\ddot{y}_{lf(L,R)i}\) and \(\ddot{y}_{rf(L,R)i}\) are lateral accelerations of the \(i\)-th elastomer on the left and right sides of the rail. \(F_{sfh(L,R)i}\) (\(F_{sfh(L,R)i}\)) and \(F_{frh(L,R)i}\) (\(F_{rfh(L,R)i}\)) are lateral force between slab and elastomer, and lateral force between elastomer and rail. \(F_{lfr(L,R)i}\), \(F_{lfr(L,R)i}\), \(F_{lff(L,R)i}\) and \(F_{lfr(L,R)i}\) are vertical force between elastomer and rail, vertical force between elastomer and slab, vertical force between the \(i\)-th and the \((i-1)\)-th elastomers, and vertical force between the \(i\)-th and the \((i+1)\)-th elastomers on the left side.