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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 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

- Crolla, D.A.; Lang, A.M. Brake noise and vibration: State of art. Veh. Tribol.
**1991**, 18, 165–174. [Google Scholar] - Kinkaid, N.M.; O’Reilly, O.M.; Papadopoulos, P. Automotive disc brake squeal. J. Sound Vib.
**2003**, 267, 105–166. [Google Scholar] [CrossRef] - Ouyang, H.; Nack, W.; Yuan, Y.; Chen, F. Numerical analysis of automotive disc brake squeal: A review. Int. J. Veh. Noise Vib.
**2005**, 1, 207–231. [Google Scholar] [CrossRef] - Signorini, A. Questioni di elasticita non linearizzata e semilinearizzata. Rendiconti di Matematica e delle sue Applicazioni
**1959**, 18, 95–139. [Google Scholar] - Tonazzi, D.; Massi, F.; Salipante, M.; Baillet, L.; Berthier, Y. Estimation of the normal contact stiffness for frictional interface in sticking and sliding conditions. Lubricants
**2019**, 7, 56. [Google Scholar] [CrossRef] - Lorang, X.; Foy-Margiocchi, F.; Nguyen, Q.S.; Gautier, P.E. Tgv disc brake squeal. J. Sound Vib.
**2006**, 293, 735–746. [Google Scholar] [CrossRef] - Lorang, X.; Chiello, O. Stability and transient analysis in the modeling of railway disc brake squeal. Notes Numer. Fluid Mech. Multidiscip. Des.
**2008**, 99, 447–453. [Google Scholar] - Brizard, D.; Chiello, O.; Sinou, J.-J.; Lorang, X. Performances of some reduced bases for the stability analysis of a disc/pads system in sliding contact. J. Sound Vib.
**2011**, 330, 703–720. [Google Scholar] [CrossRef] [Green Version] - Loyer, A.; Sinou, J.-J.; Chiello, O.; Lorang, X. Study of nonlinear behaviors and modal reductions for friction destabilized systems. application to an elastic layer. J. Sound Vib.
**2012**, 331, 1011–1041. [Google Scholar] [CrossRef] - Sinou, J.-J.; Loyer, A.; Chiello, O.; Mogenier, G.; Lorang, X.; Cocheteux, F.; Bellaj, A. A global strategy based on experiments and simulations for squeal prediction on industrial railway brakes. J. Sound Vib.
**2013**, 332, 5068–5085. [Google Scholar] [CrossRef] [Green Version] - Charroyer, L.; Chiello, O.; Sinou, J.-J. Parametric study of the mode coupling instability for a simple system with planar or rectilinear friction. J. Sound Vib.
**2016**, 384, 94–112. [Google Scholar] [CrossRef] [Green Version] - Charroyer, L.; Chiello, O.; Sinou, J.-J. Self-excited vibrations of a non-smooth contact dynamical system with planar friction based on the shooting method. Int. J. Mech. Sci.
**2018**, 144, 90–101. [Google Scholar] [CrossRef] [Green Version] - Lai, V.-V.; Chiello, O.; Brunel, J.-F.; Dufrenoy, P. Full finite element models and reduction strategies for the simulation of friction-induced vibrations of rolling contact systems. J. Sound Vib.
**2019**, 331, 197–215. [Google Scholar] [CrossRef] - Jean, M. The non-smooth contact dynamics method. Comput. Methods Appl. Mech. Eng.
**1999**, 177, 235–257. [Google Scholar] [CrossRef] [Green Version] - Alart, P.; Curnier, A. A mixed formulation for frictional contact problems prone to newton like solution methods. Comput. Methods Appl. Mech. Eng.
**1991**, 92, 353–375. [Google Scholar] [CrossRef] - Duvaut, G.; Lions, J.-L. Les Inéquations en Mécanique et en Physique Vol 1 dans Travaux et Recherches Mathématiques; Dunod: Malakoff, France, 1972. [Google Scholar]
- Khenous, H.B.; Pommier, J.; Renard, Y. Hybrid discretization of the signorini problem with coulomb friction. theoretical aspects and comparison of some numerical solvers. Appl. Numer. Math.
**2006**, 56, 163–192. [Google Scholar] [CrossRef] - Laborde, P.; Renard, Y. Fixed point strategies for elastostatic frictional contact problems. Math. Methods Appl. Sci.
**2008**, 31, 415–441. [Google Scholar] [CrossRef] - Khenous, H.B. Problèmes de Contact Unilatéral avec Frottement de Coulomb en Élastostatique et Élastodynamique. Etude Mathématique et Résolution Numérique. Ph.D. Thesis, INSA Toulouse, Toulouse, France, 2006. [Google Scholar]
- Kudawoo, A.D. Problèmes Industriels de Grande Dimension en Mécanique Numérique du Contact: Performance, Fiabilité et Robustesse. Ph.D. Thesis, Université Aix-Marseille, Marseille, France, 2012. [Google Scholar]
- Moirot, F. Étude de la Stabilité d’un Équilibre en Présence de Frottement de Coulomb. Ph.D. Thesis, École Polytechnique, Palaiseau, France, 1998. [Google Scholar]
- Bobillot, A. Méthodes de Réduction pour le Recalage Application au cas D’Ariane 5. Ph.D. Thesis, Ecole Centrale Paris, Paris, France, 2002. [Google Scholar]
- Vola, D.; Pratt, E.; Jean, M.; Raous, M. Consistent time discretization for a dynamical frictional contact problem and complementarity techniques. Revue Européenne des Éléments Finis
**1998**, 7, 149–162. [Google Scholar] [CrossRef] - Acary, V. Energy Conservation and Dissipation Properties of Time-Integration Methods for the Nonsmooth Elastodynamics with Contact; Research Report RR-8602: Project-Team Bipop; INRIA: Le Chesnay Cedex, France, 2014. [Google Scholar]
- Raous, M.; Barbadin, S.; Vola, D. Numerical Characterization and Computation of Dynamic Instabilities for Frictional Contact Problems in Friction and Instabilities. In Friction and Instabilities; International Centre for Machanical Sciences; Springer: Berlin, Germany, 2002; pp. 233–291. [Google Scholar]
- Lorang, X. Instabilité des Structures en Contact Frottant: Application au Crissement des Freins à Disque de TGV. Ph.D. Thesis, École Polytechnique, Palaiseau, France, 2009. [Google Scholar]
- Loyer, A. Etude Numérique et Expérimentale du Crissement des Systèmes de Freinage Ferroviaires. Ph.D. Thesis, École Centrale de Lyon, Lyon, France, 2012. [Google Scholar]
- Charroyer, A. Méthodes Numériques pour le Calcul des Vibrations Auto-Entretenues Liées au Frottement: Application au Bruit de Crissement Ferroviaire. Ph.D. Thesis, École centrale de Lyon, Lyon, France, 2017. [Google Scholar]
- Lai, V.V. Simulation Dynamique du Contact Roue/rail en Courbe—Application au Bruit de Crissement. Ph.D. Thesis, Université de Lille, Lille, France, 2018. [Google Scholar]

**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 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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