# Non Smooth Contact Dynamics Approach for Mechanical Systems Subjected to Friction-Induced Vibration

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## Abstract

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## 1. Introduction

## 2. Physical Modeling and Mathematical Formulation

#### 2.1. Unilateral Contact Laws

#### 2.2. Description of the Nonlinear Problem

#### 2.3. Quasi-Static Solution and Stability Analysis of the Mechanical System with a Frictional Interface

## 3. Finite Element Formulation and Solving

#### 3.1. Finite Element Discretization of the Nonlinear Dynamics Problem

#### 3.2. Stability Analysis

#### 3.3. Transient Non-Linear Temporal Response and Integration Scheme

## 4. Numerical Example

#### 4.1. Finite Element Model of the Brake System under Study

- 27,090 degrees-of-freedom for the bell/disc system;
- 4884 degrees-of-freedom for all the lining;
- 1368 degrees of freedom at the frictional contact zone between the disc and the 12 small cylindrical pins.

#### 4.2. Stability Analysis

#### 4.3. Nonlinear Simulation

#### 4.3.1. Friction-Induced Steady State Vibrations

#### 4.3.2. Evolution of the Mechanical Energy

#### 4.3.3. Effective Braking Force over Time

#### 4.3.4. Node Status at Frictional Interfaces over Time

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Description of the general contact problem on an elastic body [9].

**Figure 3.**Picture of a TGV brake system [10].

**Figure 4.**Finite element model of the brake system (

**a**) the bell-disc system (

**b**) the 12 small cylindrical pins.

**Figure 5.**Stability chart of the brake system for $\mu =0.2$ (red: with Rayleigh damping, blue: without Rayleigh damping).

**Figure 7.**Evolution of the coupling modes versus the friction coefficient (

**a**) frequencies of the first coupling modes (

**b**) frequencies of the second coupling modes (

**c**) real parts.

**Figure 9.**Temporal evolution of the contact and no-contact states at the upper frictional interface between the disc and the 6 cylindrical pins during the steady state vibrations (for 8 specific times denoted from 1 to 8).

**Figure 10.**Displacement evolutions and phase diagrams of node 1864 in the (

**a**,

**b**) tangential (

**c**,

**d**) radial and (

**e**,

**f**) normal directions (

**a**,

**c**,

**e**) displacements (

**b**,

**d**,

**f**) limit cycles.

**Figure 11.**Evolution of the mechanical energy (

**a**) mechanical energy (blue: $E\left(t\right)$; red circles: $\overline{E}\left(t\right)$) (

**b**) spectrogram of the mechanical energy in dB (

**c**) variation of the mechanical energy (blue: $\frac{\partial E}{\partial t}$; red circles: $\frac{\partial \overline{E}}{\partial t}$) (

**d**) variation rate of mechanical energy ${\overline{\tau}}_{E}$.

**Figure 12.**Braking force over time (blue: instantaneous effective braking force ${N}_{b}$; red circles: mean $\overline{{N}_{b}}$).

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

Rayleigh damping for hub and disc—mass contribution | ${\alpha}_{disc}$ | $7.50$ s${}^{-1}$ |

Rayleigh damping for hub and disc—stiffness contribution | ${\beta}_{disc}$ | $1.0\times {10}^{-7}$ s |

Rayleigh damping for pads and backplate—mass contribution | ${\alpha}_{gar}$ | 135 s${}^{-1}$ |

Rayleigh damping for pads and backplate—stiffness contribution | ${\beta}_{gar}$ | $1.80\times {10}^{-6}$ s |

Normal force applied on the caliper support | ${F}_{ext}$ | 2000 N |

Young modulus for hub and disc | ${E}_{\mathrm{disc}}$ | 210 GPa |

Mass density for hub and disc | ${\rho}_{\mathrm{disc}}$ | 7850 kg/m${}^{3}$ |

Poisson coefficient for hub and disc | ${\nu}_{\mathrm{disc}}$ | $0.28$ |

Young modulus for pads | ${E}_{\mathrm{pad}}$ | $2.0$ GPa |

Mass density for pads | ${\rho}_{\mathrm{pad}}$ | 5000 kg/m${}^{3}$ |

Poisson coefficient for pads | ${\nu}_{\mathrm{pad}}$ | $0.3$ |

Normal displacement imposed on the outer face of the pins | ${U}_{pin}^{imposed}$ | $1.60\times {10}^{-6}$ m |

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**MDPI and ACS Style**

Sinou, J.-J.; Chiello, O.; Charroyer, L. Non Smooth Contact Dynamics Approach for Mechanical Systems Subjected to Friction-Induced Vibration. *Lubricants* **2019**, *7*, 59.
https://doi.org/10.3390/lubricants7070059

**AMA Style**

Sinou J-J, Chiello O, Charroyer L. Non Smooth Contact Dynamics Approach for Mechanical Systems Subjected to Friction-Induced Vibration. *Lubricants*. 2019; 7(7):59.
https://doi.org/10.3390/lubricants7070059

**Chicago/Turabian Style**

Sinou, Jean-Jacques, Olivier Chiello, and Lucien Charroyer. 2019. "Non Smooth Contact Dynamics Approach for Mechanical Systems Subjected to Friction-Induced Vibration" *Lubricants* 7, no. 7: 59.
https://doi.org/10.3390/lubricants7070059