Matching Analysis of Carbon-Ceramic Brake Discs for High-Speed Trains

Abstract

Matching analysis is a key step in the process of verifying the adaptation of carbon-ceramic brake discs to high-speed trains’ braking system. Relevant research on matching analysis tends to be carried out only on a single parameter of the brake disc. Due to this lack of comprehensive analysis, a data-driven, parametric method is proposed to address the problem. We have summarised the matching parameters of carbon-ceramic brake discs in three dimensions: assembly interface, physical characteristics, and braking performance. The method is based on the feasibility of modelling the parameters, completing the analysis of non-modelled parameters through a comparative conformity check, and modelling parameters through a statistical analysis of the experimental data. Conformity comparison results show that the example carbon-ceramic brake disc is well suited to high-speed trains and is better matching than the example cast-steel brake discs in terms of mass and average frictional coefficient. Analysis of the simulated experimental data shows that under high-speed braking conditions, the maximum disc surface temperature and wear of the example carbon-ceramic disc is higher than that of the cast-steel disc, trains equipped with carbon-ceramic discs have shorter emergency braking distances and higher average braking deceleration, and the carbon-ceramic discs exhibit better matching performance.

1. Parameters Definition

Part of the parameters covered in the article, with their interpretations, values, and units, are shown in

Table 1. Description of some parameters.

Parameters	Interpretation	Value	Unit
AW_0	Train mass under no load	458.28	Tons
AW_1	Train mass under no load	481.14
AW_2	Train mass under fix crew	507.22
AW_3	Train mass under overload	516.44
FN	Brake pressure on brake pads	—	kN
FNi	Brake pressure of the i-th carriage	—	kN

.

2. Introduction

With the advantage of light weight, high structural strength, and stable friction performance, the carbon-ceramic brake disc can improve the braking capacity and operational safety of trains [1] and is expected to be equipped in high-speed trains.

However, the braking systems of high-speed trains are extremely demanding in terms of the matching performance of the brake discs, which leads to the necessity of a matching analysis of carbon-ceramic brake discs. The matching performance is usually analysed and verified in terms of reliability, smoothness, comfort, and service life, where the specific work is to investigate the characteristics and changing patterns of the parameters that represent these properties.

Research on matching performance by domestic and foreign scholars has been carried out mainly in a single direction in terms of the friction coefficient, temperature field, and abnormal wear of the brake disc. Wang Dongxing et al. [2] completed a comparative analysis of the average frictional coefficient of two brake discs by bench testing under three typical braking conditions and proved the matching of the discs. Wojciech et al. [3] also investigated the friction coefficient of brake discs; however, in his work the effect of disc wear was ignored, and the brake conditions were followed by the standard UIC conditions. Zhang et al. [4] investigated the coupling relationship between brake disc friction characteristics and brake pressure fluctuations and found that the thickness difference and end runout of the brake disc would cause fluctuations in the normal force, which in turn would cause fluctuations in the friction coefficient. These results help to incorporate the surface morphology of the brake disc into the analysis of matching performance.

Wu et al. [5] analysed the abnormal phenomena that tend to occur during braking of brake discs, including surface scratching, vibration, noise, and anomalous friction curves, and summarised the many factors affecting the temperature field, stress field, and noise of brake discs, providing a new research idea for analysing the matching performance of brake discs. Sai et al. [6] used the finite element method to analyse the evolution of the braking temperature of the brake disc and concluded that the maximum temperature of the disc surface is the key factor affecting matching performance. Zuo et al. [7] investigated the abnormal wear of brake discs under low temperatures together with rain and snow conditions, concluding that the train running speed, vertical pressure, and abrasive particle shape have obvious influence on the brake disc wear depth, and proposing a staged braking strategy to mitigate the wear. The study provided a research idea for analysis of matching performance in terms of reducing wear.

Although many studies on brake discs’ compatibility have been carried out, there is still insufficient research on the braking capacity of the discs, their installation problems, their dislodgement, and brake judder, which are exposed in actual operation and affect the safety of trains [8]. Mostly, there is a lack of a complete set of matching analysis methods.

To this end, this paper proposes a parametric- and data-driven approach to analyse the matching performance of carbon-ceramic brake discs and aims to fully assess the compatibility of the brake disc with the train braking system. Firstly, by consolidating the results of the prior research, the article proposes parameters that can characterise the matching performance of carbon-ceramic brake discs in three dimensions: assembly interface, physical characteristics, and braking performance. Then these parameters are divided into modelled and non-modelled parameters depending on whether they can be fitted by mathematical models. Statistical analysis of simulation data and comparative conformity check were using to analyse the modelled and non-modelled parameters, respectively. Relevant standards and well-matched cast-steel brake discs were used as non-modelled parameters for comparison, and modelled parameters were compared between cast-steel discs and carbon-ceramic discs. An innovative research idea was provided through the process of validating the analytical approach to the matching performance of the carbon-ceramic brake discs.

The virtual trials carried out in this paper do not aim to simulate realistic high-speed braking scenarios with carbon-ceramic brake discs, but to reproduce the whole process of matching analysis, and thus to verify the feasibility of the matching analysis method. The data on the non-modelled parameters of the example carbon-ceramic and cast-steel brake discs were taken from previous trials and research. In the future studies we will validate the matching analysis method against the data from the field trials.

3. Matching Parameters

3.1. Summary of Matching Parameters

This article argues that the matching parameters of brake discs for high-speed trains can be summarised from three aspects: the feasibility of installation between brake discs and the train, the characteristics of the brake discs during high-speed braking, and the integral braking efficiency of the train. Thus, they can be translated into three dimensions: the brake discs’ assembly interface, the brake discs’ physical characteristics, and the trains’ braking performance. Based on previous research, the parameters that characterise the matching performance of carbon-ceramic brake discs have been grouped according to these three dimensions as follows:

(a) The assembly interface dimension includes three categories of parameters: size, mass (or density), and structure of the brake discs. Analysis of these parameters helps to check whether the carbon-ceramic brake disc meets the changeover requirements of the train.

The size category includes the inner diameter, the outer diameter, and the thickness of the brake disc and the diameter of the blots located at the connection of the brake discs. The structural category includes the pattern of the heat sink distribution.

(b) The physical characteristics dimension includes five categories of parameters: mechanical effect, friction coefficient, temperature, wear, and the dust effect on the brake discs. Analysis of these parameters helps to investigate the braking capacity as well as the safety of carbon-ceramic brake discs during the braking process.

The mechanical effect category includes the elastic force parameters, including elastic modulus, yield strength, and hardness; the thermodynamic parameters include specific heat capacity, thermal conductivity, and the coefficient of linear thermal expansion of the carbon-ceramic material. The friction coefficient category includes the average frictional coefficient of the brake discs. The temperature category includes the peak temperature on the disc’s surface and thermal fatigue. The “dust effect” category includes the value of the adhesion coefficient of the rail due to carbon dust contamination.

(c) The braking performance dimension includes three categories of parameters: emergency braking distance, braking deceleration, and braking impulse, which can be used to evaluate the efficiency of high-speed trains’ braking system and thus reflect the reliability of the braking discs. The category of brake deceleration includes the average braking deceleration, and the braking impulse category includes the maximum common braking impulse.

3.2. Model-Based Classification

Within the matching parameters summarised in Section 3.1, the methodology including modelling and simulation were not appropriate for analysing certain parameters, such as size, mass, and the friction coefficient of the brake discs; field trials and analysis of tested data are preferred by the relevant researchers [9,10,11]. Meanwhile, parameters such as temperature, wear of brake discs, and average braking deceleration can be studied by using mathematical model fitting and simulation tests. Thus, based on whether the parameters were modellable or not, this article further divides the matching parameters into non-modelled and modelled parameters, as shown in

Table 2. Overview of non-modelled and modelled parameters.

	Category	Parameters
Non-modelled	Assembly interface	Size, mass (or density), structure
	Physical characteristics	Mechanical effect, friction coefficient, dust effect
Modelled	Physical characteristics	Temperature, wear
	Braking performance	Emergency braking distance, braking deceleration, braking impulse

.

4. Analysis of Matching Parameters

4.1. Non-Modelled Parameters

The non-modelled matching parameters summarised by this article are inherent properties of carbon-ceramic brake discs, which maintain a static value during train braking. There is no better fitting model for such parameters; it is appropriate to use a method of comparative conformity check for analysis.

The comparative conformity check process is divided into two phases: Firstly, values of the carbon-ceramic brake discs’ parameters were checked with the range of values offered by standard documents; secondly, numerical comparison between carbon-ceramic discs and well-matched cast-steel discs’ parameters were carried out after the preceding checking work was finished. Parameters with different values were filtered as the second phase, and a matching analysis was carried out based on the objective of improving the overall braking efficiency of the high-speed train.

The study took the axle-mounted carbon-ceramic and cast-steel brake discs for examples to reproduce the process of the comparative conformity check, and the parameters’ data comes from Hunan Shixin New Materials Co., Ltd, Zhuzhou, China, and the literature [12], respectively. Non-modelled matching parameters for these two brake discs meet the requirements of “Brakes-Disc brakes and their application—General conditions for the approval of brake pads” [13] and “Brake disc rolling stock” [14]. The specific comparative analysis of the non-modelled parameters is shown in

Table 3. Comparative conformity check on non-modelled matching parameters.

Category	Parameters		Value
			Carbon-Ceramic Brake Disc	Cast-Steel Brake Disc
Assembly interface	Size	Outer diameter/mm	670	640
		Inner diameter/mm	350	350
		Thickness/mm	80	80
		Bolts’ diameter/mm	18	18
	Density/(g/cm3)		1.95–2.10	7.8
	Construction
Physical characteristics	Elastodynamic parameters	Modulus of elasticity/Gpa	151	215
		Yield strength/Mpa	348.5	1054
		Hardness/HB	172	220
	Thermodynamic parameters	Specific heat capacity/(J/kg/K)	900	457
		Thermal conductivity/(W/m/K)	56.4	19.5
		Linear expansion coefficient/($\mathsf{\mu }$/K)	1.2 × 10−6	1.04 × 10−5
	Friction coefficient	Average-friction coefficient	0.36	0.338

.

Comparing the parameters with the different values in Table 1, we can find that:

(a) The outer diameter of the carbon-ceramic brake disc is 30 mm larger than that of the cast-steel brake disc; therefore, the train bogies need to be equipped with more space during the assembly process of the carbon-ceramic brake discs. In other words, the clamp-axle distance must be greater than the outer diameter of the brake discs. In Figure 1, the axle of high-speed trains (type CR400AF) is shown in the top diagram and the clamp-axle distance is shown in the bottom diagram; it has a value of 400 mm, which is larger than the outer radius of the carbon-ceramic brake disc, indicating that carbon-ceramic brake disks are well matched in the parameters of size.

(b) The density of carbon-ceramic brake discs is one-fourth of that cast-steel discs, and the weight of carbon-ceramic is about two-sevenths that of cast-steel discs. The use of carbon-ceramic brake discs significantly reduces the burden on the axle and improves the safety of the bogie structure.

(c) The structure of the brake discs affects the maximum load pressure that the disc can withstand. It also determines the thermal Li et al. [15,16] found that, compared with the random-distribution-type heat-dissipation structure used in the example cast-steel brake discs, the radial-vane-type heat-dissipation structure used in the example carbon-ceramic brake discs has increased air flow pumped by the brake disc during rotation, reduced the air return area inside the disc body. Clearly, the air circulates more quickly and evenly through the carbon-ceramic discs, which improves the thermal efficiency, also indicating that carbon-ceramic brake disks are well matched in the parameters of structure.

(d) The values of the elastodynamic parameters of the carbon-ceramic brake disc are smaller than those of the cast-steel brake disc to varying degrees. Under the same load pressure, the microstructure of the carbon-ceramic brake disc is more susceptible to damage and thermal cracking, as well as increased wear and fracture rate, compared with the cast-steel brake disc. The values of specific heat capacity and thermal conductivity in thermodynamic parameters are larger than those of cast-steel brake discs, while the coefficient of linear expansion is smaller, indicating that more heat is absorbed with lower volume expansion during the braking process of carbon-ceramic discs, improving the heat resistance of braking devices for high-speed trains.

(e) The friction sub combines the brake disc and pad and is the most critical parameter controlling the braking force of trains. A set of standard procedures for measuring the friction coefficient of the friction sub was proposed by “Provisional technical conditions for locomotive gates” [17], and the process of obtaining the values of the average frictional coefficient by means of field trials and averaging the tested data was described by the literature [18] in detail. The friction coefficient data in Table 3 were obtained from standard test procedures on friction sub, which combined brake discs with copper-based alloy brake pads, and the material parameters of the pad are shown in

Table 4. Material parameters of copper-based alloy pad.

Materials	Density (kg/m3)	Modulus of Elasticity (Gpa)	Poisson’s Ratio	Thermal Conductivity (W/(m × °C))	Coefficient of Thermal Expansion (10−6/°C)	Specific Heat Capacity (J/kg × °C)
Copper-based alloy	5500	180	0.3	436	11.1	74

. The higher value of the average coefficient of friction indicates a better match performance for the carbon-ceramic brake disc.

4.2. 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).

$$\begin{array}{c}q\left(t\right)=\mu F_{N}v\left(t\right)=\mu F_{N}w\left(t\right)r\end{array}$$

(1)

Here, $q\left(t\right)$ 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].

(b)

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].

$$\begin{array}{c}W=Ks\frac{F_N}{H}\end{array}$$

(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.

(c)

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).

$$\begin{array}{c}{a}_{AW\_j}\left(v\right)=\frac{W\left(v\right)+F_b\left(v\right)}{M_{AW\_j}+M_{AW\_0} · J_z}\end{array}$$

(3)

$$F_b\left(v\right)=\sum \mu F_{Ni}\left(v\right)+F_{ele}\left(v\right)$$

(4)

$$W\left(v\right)=A+Bv+C{v}^2$$

(5)

Here, $M_{AW\_j}$ refers to the mass of the train under the load condition $AW\_j$; $J_z$ refers to the inertia conversion factor, whose value was taken as 0.065 for motor carriages and 0.045 for the trailers. The denominator part of Equation (3) together constitutes the train’s conversion mass. $F_b$ refers to the available braking force and includes the electric brake force $F_{ele}$, whose value was set to zero in order to focus on the braking performance of the brake disc; the frictional force, whose value is related to the friction coefficient of the friction sub; and the braking pressure of the braking unit $F_{Ni}$. $W$ refers to the resistance of the train, whose value was related to factors such as the train’s speed and the slope of the railway lines. In Equation (5), A, B, and C are the resistance coefficients related to the train type [23], such as 0.35, 2.4 × 10⁻³, and 1.03 × 10⁻⁴ for the CR400AF.

The braking impulse of the train is derived, by definition, from the derivation of the deceleration function with respect to the time variable. For high-speed trains, I_max ≤ 0.75 must be satisfied.

(d)

Braking distance and average brake deceleration

The calculating model of braking distance is based on that of braking deceleration; because the deceleration obtained from Equation (3) is not a constant, it is necessary to use a discrete method to calculate the train braking distance.

Assuming a functional relationship with speed as the independent variable and deceleration as the dependent variable, $a\left(v\right)$, the deceleration was divided into segments according to an equivalent interval of speed (3 km/h in this article). Then, the deceleration within the same segment was set as a constant, which is the average of the deceleration corresponding with the two boundary speeds, and the overall braking distance was calculated by summing up the distances of each segment. The calculating model is shown in Equation (6).

$$\begin{array}{c}{S}_{AW\_j}\left(v_0\right)={\displaystyle {\displaystyle \sum }_{k=0}^{k=\left(Ns-1\right)}}\left\{\frac{\left[{\left(V_k/3.6\right)}^2-{\left(V_{k+1}/3.6\right)}^2\right]}{a_{AW\_j_k}+a_{AW\_j\_k+1}}\right\}+\frac{V_0}{3.6}·(t_{empty}+\frac{t_{U_{Max}}}{2})\end{array}$$

(6)

Here, k is the mark number of the deceleration segment; $V_k$ refers to the speed corresponding with the segment number k; $N_S$ refers to the total number segments; $V_0$ refers to the initial braking speed; $t_{empty}$ is the empty travel time; and $t_{U_{Max}}$ refers to the maximum of the brake cylinder ramp-up time and the ramp-up time limited by the braking impulse.

The calculating model of the average braking deceleration is shown in Equation (7).

$$\begin{array}{c}{\overline{a}}_{A{W}_j}=\frac{{(V_0/3.6)}^2}{2 · S_{AW\_j}}\end{array}$$

(7)

5. Matching Analysis of Modelled Parameters

5.1. Selection of Braking Conditions

From Section 4.2 it is concluded that the interactive coupling of braking condition parameters are involved in the modelling process of modelled parameters, such as the braking speed v, train load $M_{AW\_j}$, braking mode, and line’s slope. The modelled parameters of carbon-ceramic brake discs have different patterns of variation under different braking conditions, and the research on carbon-ceramic brake disc parameters under typical braking conditions allows for an efficient matching performance analysis.

This paper provides a gradient of values for the above braking condition parameters shown in

Table 5. Braking condition parameters of high-speed trains.

Braking Conditions Parameters	Values	Unit
Initial speed	160, 200, 250, 300, 350, 360, 380	km/h
Braking mode	Emergency braking, Segmented emergency braking, Maximum common braking	/
Train load	AW_0, AW_1, AW_2, AW_3	/
Line’s slope	8~17	‰

, which can reflect most of the braking scenarios of high-speed trains in China.

In the table, different braking modes correspond to different braking pressures applied by the train braking system to form different target decelerations.

5.2. Parameters’ Calculation and Analysis

The virtual trial was divided into two groups: the high-speed train CR400AF equipped with friction sub, which combined the example carbon-ceramic brake disc with a copper-based brake pad, was set as the test group, briefly recorded as “carbon-ceramic group”; and the friction sub combined with the example cast-steel brake disc with a copper-based brake pad, was set as the comparison group, briefly recorded as “cast-steel group”.

Under the selected typical braking conditions, modelled parameters such as braking deceleration, braking distance, average braking deceleration of high-speed trains, maximum disc surface temperature, and the wear volume of the brake disc were calculated using a means of data-driven analysis in the process.

5.2.1. Braking Deceleration

In the process of the virtual trial, the target deceleration data for the cast-steel group was set as initial conditions, and the curve is shown as the yellow line in Figure 2. Under the condition that the train’s braking system remains unchanged, the braking pressure $F_{Ni}$ applied to the friction sub by the train braking devices at all brake models remains unchanged, and the target deceleration curve for the carbon-ceramic group can be calculated by combining Equations (3) and (4), as shown by the blue line in Figure 2.

Because the average frictional coefficient of carbon-ceramic brake discs is higher than that of cast-steel brake discs, the trend of target braking deceleration is the same for both groups; however, the carbon-ceramic brake disc provides a higher value of braking deceleration than cast-steel brake disc for the train under the same speed conditions.

5.2.2. Braking Distance and Average Braking Deceleration

Based on the target braking deceleration data shown in Figure 2, the braking distances for high-speed trains, at all levels of braking initial speed and under typical braking conditions, consisting of no line slope, train loads of AW_2, and the three braking modes, could be calculated as Equation (6). Results are shown in

Table 6. Braking distance data for braking simulation.

		Value of Distance							Unit
Initial Speed		160	200	250	300	350	380	Average Value	km/h
Carbon-ceramic group	Emergency braking	894	1366	2085	3017	4300	6068	3217	m
	Segmented emergency braking	844	1329	2072	3051	4368	6151
	Maximum common braking	1013	1546	2360	3504	5457	8483
Cast-steel group	Emergency braking	1092	1668	2547	3718	5445	8569	4335
	Segmented emergency braking	1046	1603	2452	3732	6145	9186
	Maximum common braking	1272	2020	3499	5591	8271	10,173

.

The data in Table 6 were tested separately for normality, and the braking distance data of the carbon-ceramic group significantly obeyed the normal distribution with N₁ (3217, 2185²) at the confidence level of 0.05; that of the cast-steel group significantly obeyed the normal distribution with N₂ (4335, 3028²) l at the confidence level of 0.05.

A statistical hypothesis test was performed on both groups of data to verify that at the confidence level of 0.05, trains equipped with a carbon-ceramic friction sub could reach a 20% reduction in braking distance compared with trains equipped with a cast-steel friction sub. The carbon-ceramic group braking distance data was set as sample 1, and a new group of data assumed to obey the normal distribution with the N₃ (3468, 3028²) was set as sample 3, whose average value is 80% of the cast-steel group braking distance data set as sample 2. Then, to verify the hypothesis, the data from sample 1 and sample 3 were compared to prove that: $H_0: {\mu }_1\le {\mu }_3 (H_1: {\mu }_1>{\mu }_3)$.

Such questions are categorised in statistics as mean tests for multiple-sample functional data [24]. The rejection domain of the mean test is constructed as shown in Equation (8), since both groups of samples obey different normal distribution and each variance is known.

$$\begin{array}{c}{W}_1=\left\{\left(x_1,\dots ,x_m,y_1,\dots ,y_n\right):U=\frac{\overline{x}-\overline{y}}{\sqrt{\frac{{\sigma }_1^2}{m}+\frac{{\sigma }_2^2}{n}}}>{\mu }_{1-\alpha }\right\}\end{array}$$

(8)

Here, $\overline{x}$, $\overline{y}$ are the means of the two groups of samples; ${\sigma }_1^2,{\sigma }_2^2$ are the variances; $m,n$ are the data amounts of the two groups of samples; ${\mu }_{1-\alpha }$ denotes the standard normal distribution; and the confidence level α is 0.05.

Taking the relevant data of sample 1 and sample 3 into Equation (8), the calculated value of U equals 0.285, which is smaller than ${\mu }_{0.95}$, which equals 1.645. The rejection domain is not satisfied. This means that the $H_0$ hypothesis is accepted, which proves that the mean value of sample 1 is smaller than that of sample 3. Then, the conclusion that the trains equipped with a carbon-ceramic friction sub can shorten the braking distance by 20% compared with those equipped with a cast-steel friction sub can be accepted.

The average braking deceleration can be calculated from Equation (7) and the data in Table 6, and the results are shown in

Table 7. Average braking deceleration data for braking simulation.

		The Value of Average Brake Deceleration							Unit
Initial Speed		160	200	250	300	350	380	Average Value
Carbon-ceramic group	Emergency braking	1.104	1.129	1.156	1.151	1.099	0.918	1.038	m/s2
	Segmented emergency braking	1.17	1.161	1.164	1.138	1.082	0.906
	Maximum common braking	0.975	0.998	1.022	0.991	0.866	0.657
Cast-steel group	Emergency braking	0.904	0.925	0.946	0.933	0.867	0.650	0.799
	Segmented emergency braking	0.944	0.962	0.983	0.930	0.769	0.606
	Maximum common braking	0.776	0.764	0.689	0.621	0.571	0.548

and Figure 3.

A considerable reduction of the average braking deceleration can be captured in the progress of high-speed braking ($v_0\ge 300 \mathrm{km}/\mathrm{h}$) from Figure 3; however, the value of the carbon-ceramic group is still significantly higher than that of the cast-steel group in each braking mode.

Figure 4 reflects the changing patterns of the average deceleration difference between the carbon-ceramic group and the cast-steel group under the same braking mode. The graph shows that the difference in all three modes peaks in the process of high-speed braking, and under the initial braking speed, the average braking deceleration of the carbon-ceramic group compared with the cast-steel group is increased by 0.225~0.325 m/s², which greatly improves the braking efficiency and safety of trains.

5.2.3. Maximum Temperature of Disc Surface

The modelling and meshing of the brake disc in the temperature calculation process were carried out using ANSYS Workbench software. The data of the frictional heat generation rate for the two groups were calculated with Equation (2) and applied to the model of the brake disc as input parameters, whose braking conditions were set as a train load of AW_2, emergency braking mode, no line slope, and +15‰. The two sets of maximum temperature data calculated from the virtual trial are shown in

Table 8. Maximum disc surface temperature under no-line-slope operation.

Initial Speed (km/h)		160	200	250	300	350	380
Maximum Temperature of the Plate (°C)	Carbon-ceramic group	545.4	567.5	685.6	773	852.9	912.3
	Cast-steel group	319.5	346.1	417.6	461.3	539.9	608.5

and

Table 9. Maximum temperature of the disk surface under 15‰ line slope condition.

Initial Speed (km/h)		160	200	250	300	350	380
Maximum Temperature of the Plate (°C)	Carbon-ceramic group	456.5	484.6	583.5	711.9	818.8	895.9
	Cast-steel group	274.8	299	376.7	429.5	518.4	597.5

and in Figure 5.

The same statistical method as described in Section 5.2.2 was used to analyse the maximum temperature, and it was found that the maximum temperature data of both carbon-ceramic and cast-steel groups obeyed normal distributions. The hypothesis that under the confidence level of 0.05, the maximum temperature of the carbon-ceramic brake disc increased by 200 °C compared with the cast-steel brake disc, could be accepted. Where the calculated value U equals 0.883, it is less than ${\mu }_{0.95}$, which equals 1.645.

Figure 5 shows that the peak temperature of the disc surface is approximately linear with the initial braking speed for both carbon-ceramic discs and cast-steel discs, and the temperature changes more slowly in the process of low-speed braking ($v_0\le 200 \mathrm{km}/\mathrm{h}$) than that in high-speed braking. The effect of the maximum temperature on the disc surface, compared between the non-slope and +15‰ slope, is more pronounced in low-speed braking than that in high-speed braking, and there is almost no difference at the initial speed of 380 km/h.

5.2.4. The Wear of Brake Discs

Setting the braking condition as train load of AW_2 in emergency braking mode with no line slope, the wear volume of brake discs was calculated with Equation (2); the calculated results are shown in

Table 10. Brake discs’ wear data.

Initial Speed (km/h)		160	200	250	300	350	380
Brake Disc Wear (cm3)	Carbon-ceramic group	0.008	0.016	0.115	0.199	0.274	0.322
	Cast-steel group	0.006	0.012	0.085	0.148	0.203	0.238

, and Figure 6 shows the wear curve and the difference between the two groups.

From curves shown in Figure 6, we can find that the wear volume of carbon-ceramic brake discs is larger than that of cast-steel brake discs through the virtual trial, and the difference between the two groups soars as the initial braking speed increases. In the process of low-speed braking, the difference is not apparent and the average value is at a low level; once the initial braking speed is greater than 200 km/h, the wear of both groups increases linearly.

6. Conclusions

(a) The parameters characterizing the matching performance of the carbon-ceramic brake disc were summarised from three categories and classified into non-modelled and modelled parameters. A comparative conformity check of the non-modelled parameters revealed that the parameters such as size, density, structure, and frictional coefficient of carbon-ceramic brake discs were obviously better matched than those of the cast-steel brake discs. However, the mechanical effect parameters should be analysed in conjunction with the temperature and wear of the brake disc. (b) A mathematical modelling approach was proposed for each of the modelled parameters, and a means of combining different parameters’ values of braking conditions, which could reflect the full braking scenario of high-speed trains, was also proposed. (c) Computational data of the modelled parameters were obtained through virtual trial, and the analysis of these data by using statistical methods found that there was a greater value of the target and average braking deceleration of trains in the carbon-ceramic group process than those in the cast-steel group, as well as a 20% reduction in the braking distance, which indicates a better matching performance of the carbon-ceramic brake disc. However, the maximum temperature of the carbon-ceramic disc’s surface is increased by 200 °C compared with the cast-steel disc, leading to negative effects on the safety of the brake disc and its peripheral equipment. The wear of the brake discs is approximately linearly related to the initial braking speed of the train, and the difference in wear between carbon-ceramic discs and cast-steel discs changes linearly as initial speed increases, with poor matching performance regarding the service life of the carbon-ceramic brake disc.