# Influence of Material-Dependent Damping on Brake Squeal in a Specific Disc Brake System

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## Abstract

**:**

## 1. Introduction

- Can material-dependent (nonproportional) damping have a significant influence on the stability of specific disc brake structures?
- Can material-dependent damping be directly applied to computational finite element models (FEMs) to streamline the instability prediction process?

## 2. Mathematical Description of the Disc Brake Minimal Model

#### 2.1. Two Degrees of Freedom (DoF) Disc Brake Model

#### 2.2. Results

## 3. Pad-on-Disc System Description

#### 3.1. Analyses of Brake Squeal on Simplified Experimental Brake Model

#### 3.2. Mathematical Description of Pad-on-Disc System

#### 3.3. Numerical Verification of the Proposed FE Model

#### 3.4. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CEVA | Complex Eigenvalue Analysis |

DoF | Degree of Freedom |

EMA | Experimental Modal Analysis |

FE | Finite Element |

FEM | Finite Element Method |

FM | Friction Material |

FRF | Frequency Response Function |

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**Figure 1.**Minimal model of rigid disc brake including friction force between the pads and a wobbling disc.

**Figure 2.**Influence of the damping matrix structure on the stability of the break disc model. The structure of damping matrix is changed by two dimensionless parameters. Red zone—unstable systems, green zone—stable systems, black dashed line—the proportionality line.

**Figure 3.**(

**a**) Pad-on-disc experimental test bench, (

**b**) working principle of pad-on-disc; 6—pad support, 7—thin plate, 8—friction material, 9—disc.

**Figure 4.**Schematics of frequency response function (FRF) measurement on the (

**a**) free and (

**b**) attached discs.

**Figure 5.**(

**a**) Averaged FRF of the free disc with highlighted mode shapes and (

**b**) comparison of FRF of the attached disc with different pad loading variations in the 5800–6200 Hz region.

**Figure 6.**(

**a**) Sensor positions for brake squeal measurement and (

**b**) single-side amplitude spectrum of the measured squeal signal [27].

**Figure 7.**(

**a**) Geometry of the simplified pad-on-disc brake model, (

**b**) FEM representation of the model counting 75,520 elements and 184,225 nodes.

**Figure 9.**(

**a**) Numerically obtained and (

**b**) experimentally obtained mode shape corresponding to the (4,0) bending mode of the disc.

**Figure 10.**Calculated unstable mode shape of the system at (

**a**) 5966 Hz (experimentally verified) and (

**b**) 8079 Hz (over-predicted).

**Figure 11.**(

**a**) Stability map with respect to ${\zeta}_{d}$, ${\zeta}_{p}$ damping ratios and (

**b**) stability map in the ${\zeta}_{d}$–${\zeta}_{p}$ plane. Bold lines—stability boundary, dashed line—proportional damping line, and red circle—default damping ratio values.

${\mathit{k}}_{\mathit{t}}$$\left(\frac{\mathbf{N}}{\mathbf{m}}\right)$ | k$\left(\frac{\mathbf{N}}{\mathbf{m}}\right)$ | I (kg m${}^{2}$) | $\mathit{\mu}$ (-) | ${\mathit{N}}_{0}$ (N) | h (m) | r (m) | $\mathit{\beta}$ (-) |
---|---|---|---|---|---|---|---|

5.4 × 10${}^{3}$ | 9.3 × 10${}^{3}$ | 4 × 10${}^{-6}$ | 0.58 | 4000 | 0.037 | 0.11 | 1.5 × 10${}^{-7}$ |

**Table 2.**Basic dimensions of pad-on-disc system components: $\varphi {D}_{D}$—disc diameter, ${t}_{D}$—disc thickness, l—yellow friction material (FM) length, w—FM width, h—FM height, $\varphi {D}_{H}$—hub diameter, ${t}_{t}$—thin plate thickness.

$\mathit{\varphi}{\mathit{D}}_{\mathit{D}}$ (mm) | ${\mathit{t}}_{\mathit{D}}$ (mm) | l (mm) | w (mm) | h (mm) | $\mathit{\varphi}{\mathit{D}}_{\mathit{H}}$ (mm) | ${\mathit{t}}_{\mathit{t}}$ (mm) |
---|---|---|---|---|---|---|

215 | 14 | 40 | 20 | 8 | 108 | 2 |

Friction Material | ||||||
---|---|---|---|---|---|---|

T (${}^{\circ}$C) | $\rho $ (kg/m${}^{3}$) | ${E}_{x}$ (Pa) | ${E}_{y}$ (Pa) | ${E}_{z}$ (Pa) | ${\nu}_{xy}\phantom{\rule{0.166667em}{0ex}}(-)$ | ${\nu}_{yz}\phantom{\rule{0.166667em}{0ex}}(-)$ |

23 | 2930 | 8.7 × 10${}^{9}$ | 8.7 × 10${}^{9}$ | 2.5 × 10${}^{9}$ | 0.24 | 0.1 |

100 | 2930 | 5.9 × 10${}^{9}$ | 5.9 × 10${}^{9}$ | 1.7 × 10${}^{9}$ | 0.24 | 0.1 |

${\nu}_{xz}\phantom{\rule{0.166667em}{0ex}}(-)$ | ${G}_{xy}$ (Pa) | ${G}_{yz}$ (Pa) | ${G}_{xz}$ (Pa) | ${\zeta}_{p}\phantom{\rule{0.166667em}{0ex}}(\%)$ | ||

23 | 0.35 | 3.5 × 10${}^{9}$ | 1.6 × 10${}^{9}$ | 1.6 × 10${}^{9}$ | 0–10 (5) * | |

100 | 0.35 | 2.4 × 10${}^{9}$ | 1.1 × 10${}^{9}$ | 1.1 × 10${}^{9}$ | 0–10 (5) * | |

Disc | ||||||

T (${}^{\circ}$C) | $\rho $ (kg/m${}^{3}$) | E (Pa) | $\nu \phantom{\rule{0.166667em}{0ex}}(-)$ | ${\zeta}_{d}\phantom{\rule{0.166667em}{0ex}}(\%)$ | ||

23 | 7683 | 1.9 × 10${}^{11}$ | 0.25 | 0–0.8 (0.4) * | ||

Thin Plate, Backing Plate, Hub | ||||||

T (${}^{\circ}$C) | $\rho $ (kg/m${}^{3}$) | E (Pa) | $\nu \phantom{\rule{0.166667em}{0ex}}(-)$ | $\zeta \phantom{\rule{0.166667em}{0ex}}(\%)$ | ||

23 | 7800 | 2.1 × 10${}^{11}$ | 0.3 | 0.4 | ||

* default value. |

**Table 4.**Comparison of experimentally and numerically obtained eigenfrequencies ${f}_{n1}$ and ${f}_{n2}$ and modal damping ratios ${\zeta}_{1}$ and ${\zeta}_{2}$ for couples of (4,0) disc-bending modes.

Experiment | FEM | Error | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

No. | ${f}_{n1}$(Hz) | ${\zeta}_{1}$(%) | ${f}_{n2}$(Hz) | ${\zeta}_{2}$(%) | ${f}_{n1}$(Hz) | ${\zeta}_{1}$(%) | ${f}_{n2}$(Hz) | ${\zeta}_{2}$(%) | $\mathsf{\Delta}{f}_{n1}$(%) | $\mathsf{\Delta}{f}_{n2}$(%) |

1 | 6000 | 0.4 | * | * | 5937 | 0.4 | 5960 | 0.4 | 1.1 | * |

2 | 6140 | 0.4 | 6154 | 0.4 | 6122 | 0.4 | 6159 | 0.4 | 0.3 | 0.1 |

3 | 6146 | * | 6180 | * | 6166 | 0.4 | 6540 | 1.2 | 0.3 | 5.8 |

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**MDPI and ACS Style**

Úradníček, J.; Musil, M.; Gašparovič, L.; Bachratý, M. Influence of Material-Dependent Damping on Brake Squeal in a Specific Disc Brake System. *Appl. Sci.* **2021**, *11*, 2625.
https://doi.org/10.3390/app11062625

**AMA Style**

Úradníček J, Musil M, Gašparovič L, Bachratý M. Influence of Material-Dependent Damping on Brake Squeal in a Specific Disc Brake System. *Applied Sciences*. 2021; 11(6):2625.
https://doi.org/10.3390/app11062625

**Chicago/Turabian Style**

Úradníček, Juraj, Miloš Musil, L’uboš Gašparovič, and Michal Bachratý. 2021. "Influence of Material-Dependent Damping on Brake Squeal in a Specific Disc Brake System" *Applied Sciences* 11, no. 6: 2625.
https://doi.org/10.3390/app11062625