# Analytical Determination of the Brake Temperature Mode during Repetitive Short-Term Braking

^{*}

## Abstract

**:**

## 1. Introduction

- initial value problem for vehicle motion;
- boundary-value problem of heat conduction, taking into account frictional heat generation (the so-called thermal problem of friction).

## 2. Statement of the Problem

- At the initial moment of each braking phase, the friction element is pressed against the primary element contact surface with uniform pressure $p$, which exponentially increases with time $t$, from zero to nominal value ${p}_{0}$, Equation (9):$$p(t)={p}_{0}{p}^{\ast}(t),{p}^{\ast}(t)=1-{e}^{-\frac{t}{{t}_{i}}},0\le t\le {t}_{s}^{(k)},k=1,2,\dots ,n,$$
- At the initial moment of the $k\mathrm{th}$ cycle of braking, the distribution of temperature in the tribosystem is homogeneous and equal to the averaged volumetric temperature of friction pair ${T}_{0}^{(k)}$;
- As a result of the friction forces acting on the contact area of friction pair elements, heat is generated and absorbed by these elements in the normal directions of their friction surfaces;
- The thermal contact of friction pair elements is perfect. In other words, the sum of heat flux intensities directed into friction elements, is equal to the specific friction power, and the temperatures of its contact areas are equal.
- During the subsequent braking phases, the free surfaces of the brake system are adiabatic and during the acceleration stages, unforced convection cooling takes place.

## 3. Solution to the Problem

#### 3.1. Heat Generation on the Nominal Contact Surface

- Based on the experimental data, by means of the approximation formulas (1)–(8), describe the thermal stability of friction $f(T)$ and temperature dependencies of thermal ${K}_{l}(T)$, ${c}_{l}(T)$ and mechanical ${\rho}_{l}(T)$, $H{B}_{l}(T)$ properties of friction materials $l=1,2$;
- Set the operation input parameters: ${p}_{0}$, ${V}_{0}$, ${T}_{0}$, ${W}_{0}$, $n$, ${A}_{a}$, $h$, $G$, ${t}_{i}$, ${t}_{c}$, ${d}_{l}$, ${K}_{l,0}$, ${c}_{l,0}$, $H{B}_{l,0}$, ${\rho}_{l,0}$, $l=1,2$;
- Begin the first ($k=1$) braking cycle;
- Establish the averaged volumetric temperature ${T}_{0}^{(k)}$ of the friction pair from Formulas (25)–(29);
- Taking into account dependencies (1)–(8) calculate the values of friction coefficient ${f}_{0}^{(k)}$ and materials properties ${K}_{l,0}^{(k)}$, ${\rho}_{l,0}^{(k)}$, $l=1,2$ (25) in temperature ${T}_{0}^{(k)}$;
- Determine braking time ${t}_{s}^{(k)}$ and temporal profile of velocity ${V}^{(k)}(t)$, $0\le t\le {t}_{s}^{(k)}$ from Equations (10)–(12) and (22);
- Calculate the evolution of mean temperature on nominal contact surface ${T}_{m}^{*(k)}(t)$, $0\le t\le {t}_{s}^{(k)}$ (33)–(38);
- Start the subsequent ($k+1$) cycle of braking and repeat the calculations, beginning from point 4). The calculation process ends when condition $k=n$ is met.

#### 3.2. Temperature of the Real Contact Region

- Asperities have the spherical shape and are located on the surface of the harder and stiffer primary element, while the friction lining surface is smooth.
- Plastic roughness deformation mechanism takes place. This means that the contact of a single asperity with the friction lining surface lasts until its material becomes plastic due to a rapid increase in temperature and the appearance of significant thermal stresses.
- Before coming into contact with the friction lining, the temperature of asperity does not change along its height and is equal to the mean temperature of nominal contact area ${T}_{m}^{(k)}$.

- On the basis of the friction surface profiles of the primary friction element, the average values of parameters ${b}_{0}^{}$, $\nu $, ${r}_{av}$, ${h}_{\mathrm{max}}$ characterizing the roughness shape and their distribution along the height were calculated in the longitudinal and transverse directions.
- Knowing the temporal profile of mean temperature on nominal contact surface ${T}_{m}^{(k)}$ (30)–(37), by means of approximation functions (1)–(8), the evolutions of $H{B}_{l,m}^{(k)}$ (39) and ${f}_{m}^{(k)}$, ${K}_{l,m}^{(k)}$, ${c}_{l,m}^{(k)}$, ${\rho}_{l,m}^{(k)}$, $l=1,2$ (44) were established.
- Variations of contour contact area ${A}_{c}^{(k)}$ (38), (39) and contour pressure ${p}_{c}^{(k)}$ (40) during braking were determined, taking into account pressure profile $p$ (9).
- Changes of diameter ${d}_{r}^{(k)}$ (41) and total area of real contact ${A}_{r}^{(k)}$ (42) in time were established.
- Evolution of flash temperature ${T}_{f}^{(k)}$ (43) was calculated, taking into account velocity temporal profile ${V}^{(k)}$ (10)–(12).

## 4. Numerical Analysis

_{2}, 6% BaSO

_{4}, 3% asbestos and 9% graphite. Retinax FC-16L is a composite based on phenol-formaldehyde resins and reinforced with brass shavings [21]. The thermophysical and mechanical properties of materials and friction coefficients of selected pairs at the initial temperature are included in Table 1. The methodology of calculations adopted in the present study is shown in Figure 2.

#### 4.1. Disc Brake System

#### 4.2. Drum Brake System

## 5. Summary of the Results and Discussion

- Dependence of the friction coefficient on temperature (thermal stability curve) shows a significant influence on the time profiles of the velocity, specific friction power and maximum temperature. The coefficient of friction, which decreases with increasing temperature in the disc brake system, results in elongation of each subsequent braking stage and growth of the maximum values of the specific friction power. The effect of the friction coefficient increase, under temperature increase to about 300 °C in the drum brake system, is the reduction of the braking time and the increase of the maximum values of the specific friction power.
- In the disc brake system operating in heavy mode, the evolution of temperature and its maximum values ${T}_{\mathrm{max}}^{(k)}$ are determined by the mean temperature ${T}_{m}^{(k)}$ on the nominal contact area. The contribution of the flash temperature ${T}_{f}^{(k)}$ to the maximum temperature is negligible.
- In the drum brake operating under light conditions, at the beginning of each braking stage, maximum temperature is determined mainly by the flash temperature, while at the end of braking it depends mostly from the mean temperature ${T}_{m}^{(k)}$.
- The results obtained by means of the proposed analytical model show satisfactory compliance with the relevant data obtained with the use of numerical methods, published in the scientific literature. In particular, the highest values of the maximum temperature ${T}_{\mathrm{max}}^{(k)}$ at the subsequent stages of braking $k=1,2,3,4$, found as a result of our calculations, are 482 °C, 560 °C, 666 °C and 753 °C (Table 3), and the corresponding data obtained in the article [16] are equal to 491 °C, 615 °C, 720 °C, 847 °C, respectively. The greatest relative percentage difference of the results occurred in the fourth stage and is equal to $\approx 11\%$. In the drum brake the maximum temperatures ${T}_{\mathrm{max}}^{(k)}$ determined by means of the proposed model, are equal to 353 °C, 387 °C, 416 °C and 443 °C (Table 3), and corresponding results presented in monograph [9] are 295 °C, 330 °C, 400 °C and 440 °C. The highest relative difference in outcomes occurs in the stage one and is equal to $\approx 21\%$. It should be noted that the mean temperature ${T}_{m}^{(k)}$ of the selected friction pair drum-brake shoe with similar input parameters during single braking $(k=1)$ was analyzed with the use of the finite difference method in the article [32]. The highest value of the mean temperature on the nominal contact area of the drum-brake shoe obtained was equal to 210 °C [32], which is in good agreement with the value 280 °C presented in Table 3.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${A}_{a}$ | area of the nominal contact surface (${\mathrm{m}}^{2}$) |

${A}_{c}$ | area of the contour contact region (${\mathrm{m}}^{2}$) |

${A}_{r}$ | area of the real contact region (${\mathrm{m}}^{2}$) |

${A}_{\mathrm{vent}}$ | area of the ventilated surface of the disc or drum (${\mathrm{m}}^{2}$) |

${b}_{0}$ | parameter of the reference-surface curve (dimensionless) |

$c$ | specific heat ($\mathrm{J}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$) |

$d$ | thickness ($\mathrm{m}$) |

${d}_{r}$ | diameter of an average spot of the real contact region ($\mathrm{m}$) |

$f$ | coefficient of friction (dimensionless) |

$h$ | coefficient of heat transfer ($\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$) |

${h}_{\mathrm{max}}$ | maximum roughness height on the friction surface of the disc or drum (($\mathrm{m}$)) |

$HB$ | Brinell hardness ($\mathrm{Pa}$) |

$K$ | thermal conductivity ($\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$) |

$n$ | number of braking in RST brake mode |

$p$ | contact pressure ($\mathrm{Pa}$) |

${p}_{0}$ | nominal value of the contact pressure ($\mathrm{Pa}$) |

$q$ | specific power of friction ($\mathrm{W}{\mathrm{m}}^{-2}$) |

${q}_{0}$ | nominal value of the specific power of friction ($\mathrm{W}{\mathrm{m}}^{-2}$) |

${r}_{av}$ | average rounding radius of roughness on the friction surface (${r}_{av}$) |

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

${t}_{b}$ | time of performance of all RST mode of braking (s) |

${t}_{c}$ | cooling time at acceleration (s) |

${t}_{i}$ | time of pressure increase (s) |

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

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

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

${T}_{f}$ | flash temperature (${}^{\circ}\mathrm{C}$) |

${T}_{m}$ | mean temperature (${}^{\circ}\mathrm{C}$) |

${T}_{\mathrm{max}}$ | maximum temperature (${}^{\circ}\mathrm{C}$) |

$V$ | velocity ($\mathrm{m}{\mathrm{s}}^{-1}$) |

${V}_{0}$ | initial velocity ($\mathrm{m}{\mathrm{s}}^{-1}$) |

${W}_{0}$ | initial kinetic energy ($\mathrm{J}$) |

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

Greek Symbols | |

$\nu $ | parameter of the reference-surface curve (dimensionless) |

$\rho $ | density ($\mathrm{kg}{\mathrm{m}}^{-3}$) |

Index | |

upper $k$ | number of a stage of braking |

lower $l$ | number of the main ($l=1$) and frictional ($l=2$) elements of the friction couple |

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**Figure 3.**Graphs of functions approximating the experimental data of the dependence on temperature: (

**a**) coefficient of friction $f$; (

**b**) thermal conductivity ${K}_{l}$; (

**c**) specific heat capacity ${c}_{l}$; (

**d**) Brinell hardness $H{B}_{l}$, $l=1,2$.

**Figure 4.**Changes in: (

**a**) velocity ${V}^{(k)}$ and (

**b**) specific friction power ${q}^{(k)}$ during $k=1,2,3,4$ braking applications of a disc brake system.

**Figure 5.**The values of: (

**a**) friction coefficient ${f}_{0}^{(k)}$; (

**b**) stop time ${t}_{s}^{(k)}$ for disc brake in each braking cycle $k=1,2,3,4$.

**Figure 6.**Evolutions of the mean ${T}_{m}^{(k)}$, flash ${T}_{f}^{(k)}$ and maximum ${T}_{\mathrm{max}}^{(k)}$ temperatures for disc brake during each cycle: (

**a**) k = 1; (

**b**) k = 2; (

**c**) k = 3; (

**d**) k = 4.

**Figure 7.**Variations of the maximum ${T}_{\mathrm{max}}^{(k)}$ (solid lines), the mean ${T}_{m}^{(k)}$ (dashed lines) and the flash ${T}_{f}^{(k)}$ (dotted lines) temperatures during disc brake RST mode.

**Figure 8.**Changes in time during RST mode of the drum brake: (

**a**) velocity ${V}^{(k)}$; (

**b**) specific friction power ${q}^{(k)}$, $k=1,2,3,4$.

**Figure 9.**The values of: (

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

**b**) stopping time ${t}_{s}^{(k)}$ for the drum brake in each braking cycle $k=1,2,3,4$.

**Figure 10.**Evolutions of the mean ${T}_{m}^{(k)}$, flash ${T}_{f}^{(k)}$ and maximum ${T}_{\mathrm{max}}^{(k)}$ temperatures for drum brake during each cycle: (

**a**) k = 1; (

**b**) k = 2; (

**c**) k = 3; (

**d**) k = 4.

**Figure 11.**Evolutions of the maximum ${T}_{\mathrm{max}}^{(k)}$ (solid lines), mean ${T}_{m}^{(k)}$ (dashed lines) and flash ${T}_{f}^{(k)}$ (dotted lines) temperatures during drum brake RST mode.

Material | $\mathit{f}$ | ${\mathit{K}}_{\mathit{l}},{\mathbf{Wm}}^{-1}{\mathbf{K}}^{-1}$ | ${\mathit{c}}_{\mathit{l}},{\mathbf{Jkg}}^{-1}{\mathbf{K}}^{-1}$ | ${\mathit{\rho}}_{\mathit{l}},\mathbf{kg}{\mathbf{m}}^{-3}$ | $\mathit{H}{\mathit{B}}_{\mathit{l}},\mathbf{MPa}$ |
---|---|---|---|---|---|

ChNMKh | 0.45 | 52.17 | 444.6 | 7100 | 2100 |

FMC-11 | 35 | 479 | 4700 | 137 | |

30KhHSA | 0.39 | 38 | 490 | 7800 | 2050 |

FC-16L | 0.79 | 961 | 2500 | 392 |

**Table 2.**Coefficients in approximation functions (4)–(6) and (8) for considered materials [31].

Coefficients | Material | i = 1 | i = 2 | i = 3, ${}^{\circ}{\mathbf{C}}^{-1}$$,\times {10}^{3}$ | i = 4, ${}^{\circ}\mathbf{C}$ | i = 5 | i = 6, ${}^{\circ}{\mathbf{C}}^{-1}$$,\times {10}^{3}$ | i = 7, ${}^{\circ}\mathbf{C}$ |
---|---|---|---|---|---|---|---|---|

${f}_{i}$ | ChNMKh/FMC-11 | 0.01 | 1.07 | 1.5 | –250 | 0 | 0 | 0 |

30KhHSA/FC-16L | 0 | 1.1 | 0.0014 | 300 | 0 | 0 | 0 | |

${K}_{l,i}$ | ChNMKh | –2.37 | 4.22 | 0.196 | –2543 | 0 | 0 | 0 |

FMC-11 | 1.125 | –0.64 | 2.3 | 900 | 0 | 0 | 0 | |

30KhHSA | 2.455 | –1.58 | 0.86 | 847 | –1.05 | 6.3 | –163 | |

FC-16L | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |

${c}_{l,i}$ | ChNMKh | –0.85 | 6.6 | 0.57 | 4903 | 1.37 | 1.2 | 443 |

FMC-11 | 0.78 | 0.74 | 3.5 | 1059 | 0.5 | 2.6 | 573 | |

30KhHSA | 2.99 | −1.4 | $2\cdot {10}^{-6}$ | 859 | –0.59 | 1.36 | 20 | |

FC-16L | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |

$H{B}_{l,i}$ | ChNMKh | –0.54 | 1 | 2 | –50 | 1 | 1.7 | 500 |

FMC-11 | –0.93 | 0.83 | 2.34 | 546 | 2.02 | 2 | –233 | |

30KhHSA | –0.55 | 1 | 3.3 | 0 | 1 | 2.5 | 400 | |

FC-16L | 0.43 | 1.05 | 3.5 | –250 | 0 | 0 | 0 |

Characteristic | Brake System | k = 1 | k = 2 | k = 3 | k = 4 |
---|---|---|---|---|---|

${f}^{(k)}$ | Disc | 0.45 | 0.38 | 0.32 | 0.28 |

Drum | 0.39 | 0.40 | 0.41 | 0.42 | |

${t}_{s}{}^{(k)},\mathrm{s}$ | Disc | 1.54 | 1.73 | 1.96 | 2.22 |

Drum | 6.17 | 6.00 | 5.87 | 5.77 | |

${T}_{0}{}^{(k)},{}^{\circ}\mathrm{C}$ | Disc | 20 | 168 | 296 | 418 |

Drum | 20 | 53 | 83 | 109 | |

$\mathrm{max}{T}_{m}{}^{(k)},{}^{\circ}\mathrm{C}$ | Disc | 434 | 542 | 641 | 741 |

Drum | 208 | 243 | 272 | 298 | |

$\mathrm{max}{T}_{f}{}^{(k)},{}^{\circ}\mathrm{C}$ | Disc | 103 | 70 | 53 | 42 |

Drum | 237 | 226 | 220 | 216 | |

$\mathrm{max}{T}_{\mathrm{max}}{}^{(k)},{}^{\circ}\mathrm{C}$ | Disc | 482 | 560 | 666 | 753 |

Drum | 353 | 387 | 416 | 443 |

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**MDPI and ACS Style**

Yevtushenko, A.; Topczewska, K.; Kuciej, M.
Analytical Determination of the Brake Temperature Mode during Repetitive Short-Term Braking. *Materials* **2021**, *14*, 1912.
https://doi.org/10.3390/ma14081912

**AMA Style**

Yevtushenko A, Topczewska K, Kuciej M.
Analytical Determination of the Brake Temperature Mode during Repetitive Short-Term Braking. *Materials*. 2021; 14(8):1912.
https://doi.org/10.3390/ma14081912

**Chicago/Turabian Style**

Yevtushenko, Aleksander, Katarzyna Topczewska, and Michal Kuciej.
2021. "Analytical Determination of the Brake Temperature Mode during Repetitive Short-Term Braking" *Materials* 14, no. 8: 1912.
https://doi.org/10.3390/ma14081912