## 4 Effect of Vibration on Emergency Braking Tribological Behaviors of Brake Interfaces
### Emergency Braking Model of Brake Shoe under Vibration  
An emergency braking finite element model of a brake shoe under vibration (Figure 8) was established, employing the finite element software ABAQUS, according to Figure 3. The dimensions of the brake disc and the brake shoe, and the material parameters of the brake shoe can be found in Sections 1 and 2, respectively. The brake disc made of 16 Mn material has a density of 7850 kg.m\({}^{-3}\), elastic modulus of 2.2 \(\times\) 10\({}^{5}\) MPa, specific heat capacity of 460 J.(kg.K)\({}^{-1}\), Poisson's ratio of 0.3, thermal expansion coefficient of 1.2 \(\times\) 10\({}^{-5}\) K and thermal conductivity of 58 W.(m\(\cdot\)K)\({}^{-1}\)[19]. The eight-node hexahedral element (C3D8 RT) was employed to mesh the finite element model. The meshes at the contact regions of the brake disc and brake shoe were refined. The finite element model has 75,800 elements and 93,656 nodes, a full Newton solver, and uses a linear interpolation method; the brake disc and brake shoe have the smallest mesh sizes of 18 mm \(\times\) 48 mm \(\times\) 4 mm and 11 mm \(\times\) 19 mm \(\times\) 6 mm, respectively. The master-slave surface-to-surface contact method and the finite sliding contact state tracking method were employed between the brake disc and brake shoe. The bilinear Coulomb friction model was adopted between the brake disc and the brake shoe in the finite element model. The average coefficient of friction between the brake disc and the brake shoe in the stabilized stage during emergency braking was chosen to be 0.4, according to the results presented in Section 1. All degrees of freedom of the brake shoe except the Z direction were constrained, which indicated that the brake shoe was allowed to move along the direction of normal pressure between the brake disc and the brake shoe. Reference nodes were set at the centers of the brake discs in order to couple the nodes of the corresponding brake disc surfaces. Therefore, the degree of freedom of the reference node was constrained to constrain the degrees of freedom of the surface nodes of the brake disc. The degrees of freedom were constrained except for the degree of freedom of rotation in the Z direction (\(RZ\)), which allowed for the rotation of the brake disc along its axis. The initial temperature of the brake disc and all brake shoes was set to be 20 \({}^{\circ}\)C [3; 13]. The convective heat transfer coefficient, obtained via Equation (7) [20], was applied to brake disc surface due to the convective heat transfer between the brake disc surface and the air during emergency braking. The speed and emergency braking deceleration of the brake disc were set through reference nodes. According to the vibration acceleration of the brake shoe during emergency braking in Section 3.3 and Equation (8), the obtained dynamic evolution of the braking force of the brake shoe was used as the vibration boundary condition to the brake shoe. Figure 9 shows the evolution curve of the specific braking pressure and the application of braking force.  
\[h=\left\{\begin{array}{l}0.7\left(\frac{3\alpha}{d_{a}}\right)R_{e}^{0.55},R _{e}\leq 2.4\times 10^{5}\\ 0.04\left(\frac{3\alpha}{d_{a}}\right)R_{e}^{0.8},R_{e}\geq 2.4\times 10^{5} \end{array}\right. \tag{7}\]  
where \(R_{e}\) is the Reynold number, \(R_{e}=\frac{v_{a}\rho_{a}d_{a}}{3.6\mu_{a}}\), \(\lambda_{a}\) is the thermal conductivity and \(d_{a}\) is the outer diameter of the brake disc. In the equation of \(R_{e}\), \(v_{a}\) is the rotational speed, \(\rho_{a}\) is the fluid density and \(\mu_{a}\) is the fluid viscosity.  
\[P(t)=\frac{\overset{\rightarrow}{F}+m\cdot\overset{\rightarrow}{a}(t)}{S} \tag{8}\]  
where \(P(t)\) is the specific braking pressure, \(\overset{\rightarrow}{F}\) is the braking force provided by the braking system, \(m\) is the mass of the brake shoe, \(\overset{\rightarrow}{a}(t)\) is the vibration acceleration and \(S\) is the frictional area of the brake shoe.