# A Comparison of 3D and 2D FE Frictional Heating Models for Long and Variable Applications of Railway Tread Brake

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}were studied in the article [23]. The measurements were carried out on a pin-on-ring tribometer.

## 2. Experimental Research on a Dynamometer Test Stand for Railway Brakes

## 3. Statement of the Problem

- kinetic energy of rotating masses is entirely converted into heat;
- deformation of the system components due to mechanical forces and temperature is neglected—only thermal problem is considered;
- materials of the friction elements are isotropic and their properties depend on temperature;
- geometrical model of the wheel does not account for holes drilled for thermocouples due to very small diameter of the tip of the thermocouple; such dimension would imply very fine distribution of the finite element mesh in the vicinity of the holes;
- the outer regions of the brake shoes were not modelled since the temperature in that location does not change during the entire analyzed process;
- the coefficient of friction changes during braking, and these changes are known a priori on the basis of measurements on a full-scale dynamometer;
- on the free surfaces, convective cooling takes place with the constant heat transfer coefficient, defined separately for specific areas on the surface of the wheel and brake shoes;
- thermal radiation was not accounted for.

## 4. Description of the 3D and 2D Finite Element Models of the Wheel-Brake Shoe System

## 5. Numerical Analysis

_{1–3}= 652 s. During the subsequent braking sequence (brakings no. 4–6 from 80 km/h, 120 km/h and 160 km/h, at ${F}_{c}=30\hspace{0.17em}\mathrm{kN}$ and $m=2.5\hspace{0.17em}\mathrm{t}$) average temperature reached maximum values of ${T}_{1-3E}=89.5\xb0\mathrm{C}$, $120\xb0\mathrm{C}$, $158.4\xb0\mathrm{C}$, respectively, where t

_{4–6}= 766 s. Throughout the final part of the program (brake applications no. 7–9 from 80 km/h, 120 km/h, and 160 km/h, at ${F}_{c}=10\hspace{0.17em}\mathrm{kN}$ and $m=10\hspace{0.17em}\mathrm{t}$), maximum values of the average temperature are ${T}_{1-3E}=126\xb0\mathrm{C}$, $173\xb0\mathrm{C}$, $213.8\xb0\mathrm{C}$, respectively, for t

_{7–9}= 1879 s.

^{®}Xeon

^{®}E5-2698 v4 @ 2.20GHz; RAM 64 GB (DDR4). The calculation times for the Material A/ER7 pair performed were equal to 1h 35 min for the 2D model and 114 h 50 min for the 3D model. However, in the case of the Material B/ER7 friction pair, calculation times were 1 h 13 min and 51 h 32 min, respectively.

^{−1}s) occurring e.g., in motor vehicles. The long duration of the processes, in addition to the large size of the objects such as the wheel of a rail vehicle, with an outer diameter of 870 mm (assumed rolling radius was 437 mm), require a significant computing power and lead to many longer calculations. Therefore, the results obtained in this study, apart from their correctness verified by experimental research, are also intended to justify the previously formulated statement that the 2D axisymmetric model is sufficient to provide results at a similar level of accuracy. In order to perform such analysis, temperature points 1–6 are marked on the time courses of temperature obtained from calculations made with the 2D and 3D models presented in Figure 9a and d. For the abovementioned time points the temperature distributions are shown in Figure 9b and c (Material A/ER7 pair) and Figure 9e and f (Material B/ER7 pair)—for the curve lying on the circumference of the circle in the middle of the brake shoe thickness (Figure 9b,e), and in the axial direction (Figure 9c,f)—for a profile marked in the schematic drawing with a thick line. As can be seen from the time courses of temperature in Figure 9b,e, during braking no. 7 for both considered friction pairs at time points no. 2–5, when the temperature rises and reaches maximum values, this distribution is non-uniform. The maximum temperature fluctuations around the circumference at these points are at a level of 5.5% (Material A) and 13% (Material B), which confirms the assumption that the axisymmetric model will be sufficient to obtain similar results. By contrast, the temperature distribution in the axial direction shown in Figure 9c,f, for both friction pairs, differ by a maximum of 171% for Material A and 115% for Material B. The temperature along the analyzed curve for two friction pairs is higher on the right-hand side of the line z = 91 mm due to the lower width of the area absorbing heat through conduction.

## 6. Summary and Conclusions

^{®}Xeon

^{®}E5-2698 v4 @ 2.20GHz; RAM 64 GB (DDR4)) performed with the 2D model compared to the more general 3D model is reduced by approximately 85 times for the braking cycle lasting 5032 s, and approximately 45 times for the braking cycle lasting 3297 s.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${A}_{a}$ | nominal area of the contact region between brake shoe and wheel (m^{2}) |

$c$ | specific heat capacity (J/(kg K)) |

${d}_{eq}$ | nominal wheel outer diameter ${d}_{eq}=2{r}_{eq}$ (m) |

$f$ | coefficient of friction (dimensionless) |

${F}_{c}$ | contact force (N) |

$h$ | heat transfer coefficient (W/(m^{2} K)) |

$k$ | thermal diffusivity (m^{2}/s) |

$K$ | thermal conductivity (W/(m K)) |

$m$ | braking mass per one wheel (kg) |

$M$ | braking torque (N m) |

$p$ | contact pressure (MPa) |

$q$ | specific friction power (W/m^{2}) |

$r$ | radial coordinate (m) |

${r}_{eq}$ | equivalent radius of the contact region (m) |

${R}_{w}$ | equivalent radius of the wheel ${R}_{w}={r}_{eq}$ (m) |

$t$ | time (s) |

${t}_{s}$ | total braking time (s) |

$T$ | temperature (°C) |

${T}_{a}$ | ambient temperature (°C) |

${T}_{0}$ | initial temperature (°C) |

${T}_{1-3}$ | average temperature from ${T}_{1}$, ${T}_{2}$ and ${T}_{3}$ (°C) |

${T}_{4-6}$ | average temperature from ${T}_{4}$, ${T}_{5}$ and ${T}_{6}$ (°C) |

${T}_{1},\hspace{0.17em}{T}_{2},\hspace{0.17em}{T}_{3}$ | temperature at specific location inside the wheel (°C) |

${T}_{4},\hspace{0.17em}{T}_{5},\hspace{0.17em}{T}_{6}$ | temperature at specific location on the contact surface of the wheel (°C) |

$V$ | velocity on the equivalent radius of the wheel (vehicle velocity) (m/s) |

${V}_{0}$ | initial vehicle velocity (m/s) |

$z$ | axial coordinate (m) |

Greek symbols | |

$\gamma $ | heat partition coefficient (dimensionless) |

$\mathrm{\Gamma}$ | contact region of the brake shoe and wheel (m^{2}) |

$\eta $ | coverage factor (dimensionless) |

$\theta $ | circumferential coordinate (rad) |

$\rho $ | mass density (kg/m^{3}) |

$\omega $ | angular velocity of the wheel (rad/s) |

${\omega}_{0}$ | initial angular velocity of the wheel (rad/s) |

${\mathrm{\Omega}}_{s}$ | region within the volume of the brake shoe (m^{3}) |

${\mathrm{\Omega}}_{w}$ | region within the volume of the wheel (m^{3}) |

Subscripts | |

$s,\hspace{0.17em}w$ | brake shoe, wheel |

$E$ | experiment |

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**Figure 1.**A full-scale inertia dynamometer for testing of brake friction pairs at the Railway Research Institute in Warsaw (

**a**–

**c**) (pictures courtesy of the Railway Research Institute in Warsaw) and (

**d**) 1xBg railway tread brake configuration, (

**e**) scheme of the test rig with main components: 1—electric motor; 2—drive shaft; 3—flywheels; 4—railway wheel; 5—brake block; 6—brake cylinder; 7—load cell.

**Figure 2.**View of the brake shoes made of: (

**a**) Material A; (

**b**) Material B after tests on a full-scale dynamometer.

**Figure 3.**Changes in contact force ${F}_{c}$ and braking torque $M$ during braking for: (

**a**) Material A; (

**b**) Material B.

**Figure 4.**Changes in simulated velocity $V$ of the vehicle during braking for: (

**a**) Material A; (

**b**) Material B.

**Figure 5.**Finite element meshes of the tread brake used in braking simulation: (

**a**) spatial; (

**b**) two-dimensional (axisymmetric) with measurement points; $r$—radial coordinate, ${T}_{1},\hspace{0.17em}{T}_{2},\hspace{0.17em}{T}_{3}$ —temperature at specific location inside the wheel, $\theta $—circumferential coordinate, ${\mathrm{\Omega}}_{s}$ —region within the volume of the brake shoe, ${\mathrm{\Omega}}_{w}$ —region within the volume of the wheel, $\omega $ —angular velocity of the wheel.

**Figure 6.**Changes in frictional heat flux density generated in the wheel-brake shoe contact area during braking: (

**a**) Material A—ER7; (

**b**) Material B—ER7.

**Figure 8.**Mean temperature changes: (

**a**) Material A; (

**b**) Material B; average values from three thermocouples (dashed lines) ${T}_{1-3E}$, three points from 3D and 2D numerical calculations under the surface of the wheel ${T}_{1-3FEM}$ and three points on the surface of the wheel 3D ${T}_{4-6FEM}$.

**Figure 10.**Temperature distribution in the wheel and brake shoe for selected time points marked in Figure 9a,d, for braking no. 7: (

**a**) Material A/ER7; (

**b**) Material B/ER7.

Parameter | Value |
---|---|

nominal diameter of the wheel ${d}_{eq}$, m | 0.87 |

nominal surface area ${A}_{a}$, mm^{2} | 25,337.91 |

thickness of the brake shoe, m | 0.08 |

width of the brake shoe, m | 0.32 |

Base Formulation | Glass Fibre | Steel Fibre | |
---|---|---|---|

Material A | 65–75% | 25–35% | 0% |

Material B | 0% | 25–35% |

Number of Braking | $\mathbf{Initial}\mathbf{Velocity}\mathbf{of}\mathbf{the}\mathbf{Vehicle}{\mathit{V}}_{0},\mathbf{km}/\mathbf{h}$ | $\mathbf{Contact}\mathbf{Force}{\mathit{F}}_{\mathit{c}},\mathbf{kN}$ | $\mathbf{Initial}\mathbf{Temperature}{\mathit{T}}_{\mathit{a}},\xb0\mathbf{C}$ | Mass per Wheel m, t | Comments |
---|---|---|---|---|---|

1 | 160 | ambient | 2.5 | ||

2 | 120 | 10 | 50–60 | ||

3 | 80 | ||||

4 | 160 | ||||

5 | 120 | 30 | Material A/ER7 | ||

6 | 80 | ||||

7 | 160 | ||||

8 | 120 | 10 | 10 | ||

9 | 80 | ||||

1 | 80 | ambient | 2.5 | ||

2 | 120 | 10 | 50–60 | ||

3 | 160 | ||||

4 | 80 | ||||

5 | 120 | 30 | Material B/ER7 | ||

6 | 160 | ||||

7 | 80 | ||||

8 | 120 | 10 | 10 | ||

9 | 160 |

**Table 4.**Properties of the brake shoe materials [31].

Specific Heat Capacity ${\mathit{c}}_{\mathit{s}}$$,\mathbf{J}/(\mathbf{kg}\hspace{0.17em}\mathbf{K})$ | Thermal Conductivity K_{s}$,\mathbf{W}/(\mathbf{m}\mathbf{K})$ | Mass Density ${\mathit{\rho}}_{\mathit{s}}$$,\mathbf{kg}/{\mathbf{m}}^{3}$ | Thermal Diffusivity ${\mathit{k}}_{\mathit{s}}$$,{\mathbf{m}}^{2}/\mathbf{s}$ | ||||
---|---|---|---|---|---|---|---|

at temperature | 30 °C | 100 °C | 30 °C | 100 °C | 20 °C | ||

brake shoe | material A | 870 | 1040 | 1.18 | 1.47 | 1930 | 7.013·10^{−7} |

material B | 730 | 860 | 1.41 | 1.74 | 2350 | 8.594·10^{−7} |

Temperature, °C | Thermal Conductivity ${\mathit{K}}_{\mathit{w}}$$,\mathbf{W}/(\mathbf{m}\mathbf{K})$ | Specific Heat Capacity ${\mathit{c}}_{\mathit{w}}$$,\mathbf{J}/(\mathbf{kg}\hspace{0.17em}\mathbf{K})$ |
---|---|---|

0 | 47.3 | 440 |

20 | 44.1 | 510 |

400 | 39.3 | 570 |

^{1}Density of the wheel ${\rho}_{w}=7850\hspace{0.17em}\mathrm{kg}{\mathrm{m}}^{-3}.$

**Table 6.**Convection coefficient values in the various wheel surface zones [34].

Zone | Heat Transfer Coefficient $\mathit{h},{\mathbf{W}/(\mathbf{m}}^{2}\hspace{0.17em}\mathbf{K})$ |
---|---|

1 | 32.6 |

2 | 55.9 |

3 | 55.9 |

4, 5 | 65.3 |

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**MDPI and ACS Style**

Kuciej, M.; Grzes, P.; Wasilewski, P.
A Comparison of 3D and 2D FE Frictional Heating Models for Long and Variable Applications of Railway Tread Brake. *Materials* **2020**, *13*, 4846.
https://doi.org/10.3390/ma13214846

**AMA Style**

Kuciej M, Grzes P, Wasilewski P.
A Comparison of 3D and 2D FE Frictional Heating Models for Long and Variable Applications of Railway Tread Brake. *Materials*. 2020; 13(21):4846.
https://doi.org/10.3390/ma13214846

**Chicago/Turabian Style**

Kuciej, Michal, Piotr Grzes, and Piotr Wasilewski.
2020. "A Comparison of 3D and 2D FE Frictional Heating Models for Long and Variable Applications of Railway Tread Brake" *Materials* 13, no. 21: 4846.
https://doi.org/10.3390/ma13214846