## 4 Analysis of Matching Parameters
### Mathematical Modelling of Modelled Parameters  
The values of the modelled parameters, such as braking temperature, wear, braking deceleration, distance, and impulse of the carbon-ceramic brake disc change continuously during the braking process, and there are interactive relationships within these parameters. There exist computational models for fitting these parameters, which means that simulation trials could be carried out based on the model. The calculated data of the modelled parameters were analysed by means of statistical methods including hypothesis testing, test of normality, and regression analysis. This paper proposes a mathematical modelling approach for each modelled parameter, respectively, as follows. (a) Maximum temperature of the brake disc surface  
The study of the brake disc temperature is a huge project with multiple input and output parameters, and the parameter of peak temperature of the brake disc surface was focused in the process of matching analysis. The variation of the brake disc's temperature is affected by a combination of heat input, conduction, radiation, and heat convection, where the rate of frictional heat generation during heat input is one of the key parameters affecting the braking temperature [19; 20]. The calculating model is shown in Equation (1). \[q(t)=\mu F_{N}v(t)=\mu F_{N}w(t)r \tag{1}\]  
Here, \(q(t)\) refers to the frictional heat generation rate; \(\mu\) refers to the instantaneous friction coefficient of the friction sub; and \(v\) refers to the relative sliding speed of the pad, which equals the multiplication of the angular speed of axle \(w\) and friction radius \(r\). The values of \(r\) as well as \(F_{N}\) remain static once the braking system of trains does not change; thus, under the same speed conditions, the rate is positively correlated with the friction coefficient of the friction sub, concluded from the Equation (1) [21]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Materials** & **Density** & **Modulus of** & **Poisson’s** & **Thermal** & **Coefficient of** & **Specific Heat** \\
**(kg/m3)** & **Elasticity (Gpa)** & **Ratio** & **Conductivity** & **Thermal Expansion** & **Capacity (J/kg** \\
**(W/(m \(\times\) \({}^{\circ}\)C))** & **(10\({}^{-6}\)/C)** & **(10\({}^{-6}\)/C)** & \(\times\ \ {}^{\circ}\)C** \\ \hline Copper-based alloy & 5500 & 180 & 0.3 & 436 & 11.1 & 74 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material parameters of copper-based alloy pad. 2. The wear of the brake disc  
The physical wear of brake discs exists in every braking process, and the wear limitations are often used in engineering to determine the service life of the brake discs. The average thickness of a brake disc is reduced by brake pad friction during a braking session. This can be calculated by dividing the wear volume by the friction area, and the wear volume can be calculated basing on the generalised Archard wear model, as shown in Equation (2) [22]. \[W=Ks\frac{F_{N}}{H} \tag{2}\]  
Here, \(W\) refers to the wear volume; \(K\) is a dimensionless wear coefficient, and the value of this parameter is obtained by generalization from previous experiments; \(s\) refers to sliding distance; and \(H\) refers to the hardness of the brake disc, whose data has been given in Table 3.  
2. Braking deceleration and impulse  
High-speed trains are controlled mainly by resistance and braking force during the braking process, and the braking deceleration comes from the combined effect of both. The calculating models are shown in Equations (3)-(5). \[a_{AW\_j}(v)=\frac{W(v)+F_{b}(v)}{M_{AW\_j}+M_{AW\_0}\cdot I_{z}} \tag{3}\]  
\[F_{b}(v)=\sum\mu F_{Ni}(v)+F_{ele}(v) \tag{4}\]  
\[W(v)=A+Bv+Cv^{2} \tag{5}\]  
Here, \(M_{AW\_j}\) refers to the mass of the train under the load condition \(AW\_j\); \(I_{z}\) refers to the inertia conversion factor, whose value was taken as 0.065 for motor carpiages and 0.045 for the trailers. The denominator part of Equation (3) together constitutes the train's conversion mass.