# Temperature during Repetitive Short-Term Operation of a Brake with Functionally Graded Friction Element

^{*}

## Abstract

**:**

_{2}–Ti-6Al-4V) in combination with gray cast iron (ChNMKh). It was found that for selected friction pair materials, consideration of their thermal sensitivity reduces the time of braking and the value of temperature achieved on the friction surfaces. At the same time, the whole process was characterized by a good stability of braking with a slight decrease in efficiency in each subsequent cycle.

## 1. Introduction

## 2. Scheme of Braking and Model Assumptions

- Initial temperature of considered a friction pair at the beginning of the subsequent braking is equal to the volume-averaged temperature of the system;
- As a result of the friction forces interaction, the heat is generated on the contact area of the elements and absorbed by them along the normal direction to the friction surface. The friction thermal contact of the elements during heating is perfect;
- Unforced convection cooling of the system during braking stages is omitted.

## 3. Analytical Model

## 4. Numerical Analysis

- Based on experimental data, finding the dependences of material properties and the friction coefficient on temperature in forms (6)–(10). Determining the value of material properties ${K}_{1,m}^{(0)}$, ${c}_{1,m}^{(0)}$, ${\rho}_{1,m}^{(0)}$, $m=1,2$, (7), ${K}_{2}^{(0)}$, ${c}_{2}^{(0)}$, ${\rho}_{2}^{(0)}$ (9) and the coefficient of friction ${f}_{0}$ (10) at the initial temperature ${T}_{0}$;
- Introduction of the input operational parameters: ${p}_{0}$, ${V}_{0}$, ${T}_{0}$, ${W}_{0}$, $n$, ${A}_{a}$, ${A}_{\mathrm{vent}}$, ${d}_{1}$, ${d}_{2}$, $h$, ${t}_{c}$,$\mathrm{v}$;
- Start of the first braking: $k=1$;
- Determination of the volume temperature ${\widehat{T}}_{}^{(k)}$ (24)–(32);
- Using the dependencies (6)–(10), establishment of the material properties values ${K}_{1,m}^{(k)}$, ${c}_{1,m}^{(k)}$, ${\rho}_{1,m}^{(k)}$, $m=1,2$ (19), ${K}_{2}^{(k)}$, ${c}_{2}^{(k)}$, ${\rho}_{2}^{(k)}$ (20), the friction coefficient ${f}^{(k)}$, and specific friction power ${q}_{0}^{(k)}$ (21) at the volume temperature ${\widehat{T}}_{}^{(k)}$;
- Determination of the stop time ${t}_{s}^{(k)}$ (2) and temporal profile of velocity ${V}^{(k)}(t)$, $0\le t\le {t}_{s}^{(k)}$ (1);
- Calculation of the temperature evolution ${T}^{(k)}(t)$, $0\le t\le {t}_{s}^{(k)}$ (11)–(23);
- Starting the next $k+1$ braking cycle and repeating starting from point (5) or ending the calculation process after reaching the equality $k=n$.

## 5. Conclusions

_{2}, core Ti–6Al–4V) and ChNMKh gray cast iron, for five braking actions. It was found that the braking time, the volume, and maximum temperature values increased almost linearly with the number of braking cycles. Involving the thermal sensitivity of materials into the calculation model causes a decrease in the maximum temperature value in relation to the results obtained for materials with invariant properties under temperature changes. This effect becomes more noticeable with each subsequent braking cycle. The coefficient of friction decreases rapidly at the beginning of each braking to a minimum value, then begins to increase slightly until standstill. The considered friction pair is characterized by good braking stability with sufficient efficiency, slightly decreasing with each successive braking.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$a$ | Effective depth of heat penetration of the friction element ($\mathrm{m}$) |

${A}_{a}$ | Area of the nominal contact region (m^{2}) |

${A}_{vent}$ | Area of the ventilated surface of the brake disc (m^{2}) |

$c$ | Specific heat ($\mathrm{J}\hspace{0.17em}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$) |

d | Thickness of friction elements ($\mathrm{m}$) |

$f$ | Coefficient of friction (dimensionless) |

h | Coefficient of heat transfer ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$) |

${J}_{k}(\cdot )$ | Bessel functions of the first kind of the kth order |

$k$ | Thermal diffusivity (${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$) |

$K$ | Thermal conductivity ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$) |

${K}_{\epsilon}$ | Dimensionless coefficient of thermal activity of friction couple |

n | Number of brakings in RST brake mode |

$p$ | Pressure on the contact surface (Pa) |

${p}_{0}$ | Nominal value of the contact pressure (Pa) |

$q$ | Specific friction power ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-2}$) |

${q}_{0}$ | Nominal value of specific friction power ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-2}$) |

$t$ | Time ($\mathrm{s}$) |

t_{b} | Time of performance of all RST mode of braking (s) |

t_{c} | Cooling time during acceleration stage (s) |

${t}_{s}$ | Stop time ($\mathrm{s}$) |

$T$ | Temperature (${}^{\circ}\mathrm{C}$) |

${T}^{\ast}$ | Dimensionless temperature |

$\widehat{T}$ | Volume temperature (${}^{\circ}\mathrm{C}$) |

${T}_{0}$ | Initial temperature (${}^{\circ}\mathrm{C}$) |

v | Volume fraction of the FGM components (dimensionless) |

$V$ | Velocity ($\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$) |

${V}_{0}$ | Initial velocity ($\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$) |

W_{0} | Initial kinetic energy (J) |

$z$ | Spatial coordinate in axial direction ($\mathrm{m}$) |

$\alpha $ | Heat partition ratio (dimensionless) |

$\gamma $ | Parameter of material gradient (${\mathrm{m}}^{-1}$) |

$\Lambda $ | Scaling factor of temperature (${}^{\circ}\mathrm{C}$) |

$\rho $ | Density ($\mathrm{kg}\hspace{0.17em}{\mathrm{m}}^{-3}$) |

$\tau $ | Dimensionless time |

${\tau}_{s}$ | Dimensionless time of braking |

superscript k | Number of a braking cycle |

subscript l | Number of the friction pair element |

subscript m | Number of the component material of functionally graded element |

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**Figure 2.**Temperature dependencies of the dimensionless material properties: (

**a**) coefficient of thermal conductivity; (

**b**) specific heat; (

**c**) density; and (

**d**) friction coefficient of the considered friction pair.

**Figure 3.**Evolutions of the: (

**a**) velocity; (

**b**) specific friction power, during each of the five braking applications.

**Figure 4.**Evolutions of temperature on the friction surface during each of the five braking actions with (solid lines) and without (dotted lines) taking into account the thermal sensitivity of materials.

**Figure 5.**Dependencies on the number of braking applications $k$ of: (

**a**) friction coefficient ${f}^{(k)}$ (21); (

**b**) braking time ${t}_{s}^{(k)}$ (2); (

**c**) volume temperature ${\widehat{T}}^{(k)}$ (24); (

**d**) maximum temperature on the friction surface ${T}_{\mathrm{max}}^{(k)}$.

k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

${f}_{}^{(k)}$ | 0.27 | 0.25 | 0.23 | 0.21 | 0.20 |

${t}_{s}^{(k)},\hspace{0.17em}\mathrm{s}$ | 1.77 | 1.94 | 2.10 | 2.26 | 2.41 |

${\widehat{T}}_{}^{(k)},\hspace{0.17em}\xb0\mathrm{C}$ | 20 | 86.85 | 143.21 | 193.48 | 239.74 |

${T}_{\mathrm{max}}^{(k)},\hspace{0.17em}\xb0\mathrm{C}$ | 530.27 | 561.29 | 591.46 | 621.11 | 650.49 |

k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

${f}_{}^{(k)}$ | 0.27 | 0.24 | 0.22 | 0.20 | 0.18 |

${t}_{s}^{(k)},\hspace{0.17em}\mathrm{s}$ | 1.77 | 1.95 | 2.15 | 2.36 | 2.60 |

${\widehat{T}}_{}^{(k)},\hspace{0.17em}\xb0\mathrm{C}$ | 20 | 89.83 | 158.79 | 226.89 | 294.12 |

${T}_{\mathrm{max}}^{(k)},\hspace{0.17em}\xb0\mathrm{C}$ | 530.27 | 575.76 | 621.75 | 668.19 | 715.04 |

k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

${f}_{m}^{(k)}$ | 0.149 | 0.142 | 0.135 | 0.129 | 0.123 |

${f}_{s}^{(k)}$ | 0.553 | 0.576 | 0.593 | 0.608 | 0.620 |

${f}_{f}^{(k)}$ | 0.489 | 0.515 | 0.534 | 0.550 | 0.563 |

${f}_{eff}^{(k)}$, ${\mathrm{s}}^{-2}$ | 0.177 | 0.153 | 0.134 | 0.119 | 0.107 |

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**MDPI and ACS Style**

Yevtushenko, A.; Topczewska, K.; Zamojski, P.
Temperature during Repetitive Short-Term Operation of a Brake with Functionally Graded Friction Element. *Materials* **2023**, *16*, 881.
https://doi.org/10.3390/ma16020881

**AMA Style**

Yevtushenko A, Topczewska K, Zamojski P.
Temperature during Repetitive Short-Term Operation of a Brake with Functionally Graded Friction Element. *Materials*. 2023; 16(2):881.
https://doi.org/10.3390/ma16020881

**Chicago/Turabian Style**

Yevtushenko, Aleksander, Katarzyna Topczewska, and Przemysław Zamojski.
2023. "Temperature during Repetitive Short-Term Operation of a Brake with Functionally Graded Friction Element" *Materials* 16, no. 2: 881.
https://doi.org/10.3390/ma16020881