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T.G.V disk brake squeal : understanding and modeling

1X. Lorang, 2F. Margiocchi and 3O. Chiello SNCF, Innovative & Research Department,

PSF, 45 rue de Londres, 75379, Paris, France (1) Tel: +33 (0)1 53 42 92 28, E-mail: xavier.lorang@sncf.fr

(2) Tel: +33 (0)1 53 31 31 72, E-mail: florence.margiocchi@sncf.fr (3) INRETS, 25 av. F. Mitterrand, 69675 Bron cedex, France

Tel: +33 (0)4 72 14 24 05, Fax: +33 (0)4 72 37 68 37, E-mail: olivier.chiello@inrets.fr Introduction Brakes are one of the most important safety components in train operating condition but the brake squeal generated by disk brakes is an everyday source of discomfort for the passengers both inside and outside the train in stations. This is the case in particular for TGV. The development of a refined mechanical modeling of the phenomenon was carried out in order to understand the mechanism of squeal generation. Principles of solutions and noise reduction assessment through the developed models taking into account the braking noise and the safety will be now possible. The paper deals with friction induced vibrations and especially with TGV disk brake squeal. The paper is divided in three parts. It begins with the strategy to model such friction induced vibrations and its application to a 3D test case. In the second part, an experimental investigation of the phenomenon is presented both in real conditions and on a fixed testing plant. The third part is devoted to the model of the brake mechanism. A finite element approach is used to model the phenomenon via the presented strategy and is compared with the experimental results. The first part of the paper is devoted to the strategy used to model the general problem of self-excited vibrations of two elastic bodies in frictional contact. Unilateral contact conditions with Coulomb friction and constant friction coefficient are considered. In order to predict the occurrence of self-excited vibrations, a classical stability analysis is performed, which consists on computing the complex modes associated to the linearized problem. A common interpretation of the stability analysis is that frequencies of unstable complex modes correspond to squeal frequencies. To check this assumption, the behavior of the solution far from the sliding equilibrium is determined by using a non linear transient analysis. Moreover, an expansion of the transient solution on the complex modes provided by the stability analysis helps us to highlight the role of the unstable modes and the vibratory field in general. The influence of initial conditions of the transient analysis is analyzed. In the second part, a complete experimental investigation is presented. In order to have a good representativity of the phenomenon, acoustic measurements have been done in the station on various kind of TGV trains. To go ahead into deeper understanding, one brake system was equipped when the train is running. The vibration spectrum in one point of the disk was explored via a single point laser doppler vibrometer. To get more information than a vibration spectrum at a point, a fixed testing plant was used to continue the investigation. The motivation to get the entire vibration field is the identification of the “exited modes” of the disk during the braking phase. The scanning laser doppler vibrometry technique helped us to highlight the wave propagation along the disk which are predicted numerically. For a given frequency, when a rotating wave is detected, it means that the field is a consequence of an unstable mode at the same frequency. The third part of the paper focuses on the application of the approach to a Finite Element model of the TGV disk brake system. The numerical model is first validated with a classical modal analysis when the disk is in free condition. The modal damping factors are measured. The complex eigenvalue problem is computed (stability analysis) and shows that for a given friction coefficient (f = 0.4) the sliding equilibrium is unstable even taking into account the material damping.

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A. Mechanical strategy to model brake squeal

1. Formulation of the problem

The mechanism of the simplified disc brake system represented on figure 1 is considered. The rotation speed Ω of the disc is assumed to be constant and sufficiently small so that the gyroscopic terms and the volume forces induced by rotation may be neglected. An “eulerian” description is adopted. Unilateral contact with Coulomb friction conditions are taken into account. By using a finite element method, the nonlinear dynamics problem may be written in a discrete form as follows (see details in reference [3]):

[ ] [ ] [ ] [ ] [ ] [ ] ( )( ) [ ] ( )( )VUPrr

gUPrrrPrPFUKUCUM

tttCt

nnnn

tT

tnT

n

+−=

−−=++=++

−ℜ

&

&&&

α

α

Proj

Proj 0 (1)

where [M], [C] and [K] denote the mass, Rayleigh damping and stiffness matrices whereas U and F represent the vectors of nodal displacements and external nodal forces. In addition, rn and rt denote the vectors of normal and tangential reactions forces at the contact nodes whereas [Pn] and [Pt] are projection matrices on the normal and tangential relative displacements between the disc and the pads at the contact nodes. C is the Coulomb cone and coefficients αn and αt must be positive. Finally, g0 is the vector of initial gaps whereas V is the vectors of imposed sliding velocities due to the disc rotation speed Ω at the contact nodes.

2. Stability and transient analysis

Linear stability analysis

System (1) is a set of non-linear differential equations characterised by a sliding equilibrium. Considering small regular perturbations that does not break the contact (bilateral contact), the frictional forces may be linearised and the evolution of the perturbations U* verifies (see [2]):

[ ] [ ]( ) [ ] [ ] [ ]( ) [ ] [ ]( ) [ ]

[ ] 0*

****

=

=++++++

UP

rPUKfKUCfCfCUMfM

n

nT

nfeff&&&

(2)

where [Mf], [Cf], [Kf] and [Ce] are non symmetrical matrices provided by the linearisation of the frictional forces, f is the friction coefficient and rn* denote the vector of perturbed normal reactions forces at the contact nodes. By eliminating the bilateral contact constraints, a non symmetrical linear system of equations is obtained and the stability of the equilibrium may then be deduced from a complex eigenvalue analysis of the system, providing complex modes and complex eigenvalue λi . A complex mode i is unstable if Re(λi)>0, which may happen since the system is non symmetric. A modal growth rate ζi=Re(λi)/Im(λi) may also be defined (physically equivalent to a negative modal damping). Non linear transient analysis In addition to the stability analysis, an implicit numerical resolution of the system of equations (1) may be performed. A time discretisation method is used here, resting on a former work of M. Jean and J.J. Moreau (see [3]). The θ-method allows one to avoid numerical problems at the time of an impact. Indeed, the instability of the sliding equilibrium may lead to strongly non-linear events like a separation followed by a shock, as well as stick/slip transitions. In order to introduce an inelastic shock law, one uses the modified version of the θ-method (see [4]).

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Relations between stability and transient analyses In order to understand the role of the unstable modes in the non linear part of the transient behaviour, it is proposed to expand the numerical transient solution on the basis of the complex modes. Taking into account the non symmetrical character of the matrices involved in the problem (2), it is necessary to introduce the complex modes provided by transposed linearised problem. Indeed, the generalisation of the usual orthogonality conditions in the case of symmetrical matrices leads to bi-orthogonality conditions, and given by :

[ ] 0=Φ≠∀ jT

i ALji (3) where φj are the complex modes of the direct problem (2) and Li the complex modes of the transposed problem of (2). [A] is the mass matrix in state space variable (see [5]). This bi-orthogonality property allows one to compute the evolution of the complex amplitude of the mode j, βj(t), from the transient perturbation written in state-space variables α*(t):

[ ] [ ] j

Tj

Tj

j AL

tALt

Φ=

)()(

*αβ (4)

with α*(t) provided by numerical resolution of system (1). The contribution of the mode j to the perturbed displacement and velocity fields can then computed from βj(t). Finally, the variation of the total perturbed energy of a mode j can be computed from these contributions.

3. Application to a simplified disc brake system In this part, the simplified disc brake system of figure 1 is considered. The geometric and physical characteristics of the structures are given in table 1. The other parameters are f=0.35, Ω=2.5 rad/s and δ=3.33×10-6 m. A stability analysis is performed in the [0 15kHz] frequency range. Among the 100 computed complex modes, three modes are found unstable, called M1 (8583 Hz, ζ=0.05 %), M2 (9288 Hz, ζ=0.13 %) and M3 (10130 Hz, ζ=0.17 %). The transient solution is also calculated with θ = 0.5 and time step Δt = 5×10-6 s. In order to study the influence of the initial conditions on the transient solution, 4 cases are considered (A to D) for which the initial contributions of the unstable modes are different. The contributions of the unstable modes to the total perturbed energy ETM1(t), ETM2(t) and ETM3(t) are represented on the figure 2 for the 4 cases. The different figures show that the stabilised solution is not dependent on the initial conditions. It may be observed that this stabilised solution is made up of 2 of the 3 unstable modes (M1 and M2). These computations show that the stability analysis is able to predict the prone-squeal modes but that a transient analysis is necessary to predict which modes remain in the stabilised solution. It also highlights the possible coexistence of several unstable modes in the self-sustained vibrations.

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Pads

Disc

- t

zPad

Disc

P

D

δ

δ

Disc Pads Young’s modulus E 2.02×1011 Pa 4.5×109 Pa Poisson’s ratio ν 0.29 0.3 Density ρ 7850 kg.m−3 5250 kg.m−3 Damping param. α 0 s−1 108 s−1 Damping param. β 7.510−9 s 1.4410−6 s Ext. diameter 0.6 m 0.16 m Int. diameter 0.2 m Thickness 0.04 m 0.04 m

Tab 1. Physical and geometric characteristics

Fig 1. Simplified disc brake system

Case A Case B

Case C Case D Fig 2. Evolution of modal contribution to total perturbed energy ET(t) [J] for the different cases as a function of time [s]. ETM1(t), ◊ ETM2(t), O ETM3(t) .

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B. Experimental investigation on a TGV brake system This part presents the analysis of experimental data coming from a statistical study in stations, on-board measurements and tests on bench in laboratory.

1. Measurements in stations

In the first experimental step of the project, the noise of 41 TGV coming into Saint-Pierre des Corps and Avignon-TGV stations has been performed with a microphone located on the platform, at 1m from the train. From these recordings, noise levels L

Aeq, 125 ms and maximum

spectra have been calculated for 173 squealing bogies in braking operation. The measured levels are contained between 85 and 106 dB(A), with a rate of 89%, superior at 90 dB(A). The main squealing frequencies are represented on figure 3. Despite the dispersion of these measured frequencies, 3 groups may be clearly distinguished. The first group corresponds to frequencies contained between 1000 and 5000 Hz. These frequencies vary a lot and their contributions are often secondary. The second group is represented by only one frequency around 6600 Hz. It has a significant contribution to the global noise level. The third group is composed of the frequencies contained between 8000 and 16000 Hz, which dominate the spectrum.

Figure 3.-Squeal frequencies measured in stations

2. In-board measurements

In a second step, more detailed measurements have been carried out on rolling stock in controlled conditions. Different linings and discs have been instrumented with a microphone in the close field, a laser vibrometer aiming at the disc surface and some thermocouples and accelerometers on the linings. The main characteristics of the squeal noise measured in station have been found again. The effect of several parameters has been studied as the running direction, the braking pressure or the vehicle speed. One of the main results is the relationship between the noise level and the vibration level at the disc surface. Indeed, it appears that above 5000 Hz (groups 2 and 3), most of the squealing frequencies correspond to the vibration frequencies (see figure 4). For these groups, the squeal noise is mainly radiated by the disc in axial vibrations (out-of-plane).

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Fig 4. Experimental power Spectrum of the normal velocity [dB] (below, ref 1 [m/s]) and acoustic

pressure [dB] (above, ref 20 [μPa])

3. Laboratory measurements

From these measurements, considered like references, the representativeness of some braking benches has been tested. It appears that no bench is able to reproduce exactly the phenomenon observed on the train. It strongly depends on the lining and the frequency of interest. Generally speaking, only the squealing frequencies of the third group (above 8000 Hz) are well reproduced. The 6600 Hz frequency never appears and the frequencies less than 5000 Hz are too variables. Some experiments are in progress to explain these differences. However, the best test bench has been used to characterize the vibrations field on the disc by laser measurements during braking (see figure 5). At frequencies above 7000 Hz, the measurements have shown that the disc vibrates mainly axially. The measured vibrations fields do not correspond to stationary flexion modes but rather flexion waves, which may be interpreted as flexion modes rotating along the circumference of the disc (see figure 6). From 7000 to 18000 Hz, the identified modes have no nodal circles and nodal diameters varying from 9 to 15.

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Figure 5 : Vibration field investigations with laser measurements

Figure 6 : Vibration field at different frequencies

The main components of the brake system (disc and linings) have also been tested in “free” conditions (e.g. without frictional contact). The results of these experimental modal analyses have been used to improve the finite element modelling of the components. C. Stability analysis of a TGV brake system In this part, a finite element model of the TGV brake system (cf. fig 3) is considered. First, the modes of the disc in free conditions have been calculated and have been classified according to the direction of the dominant deformations. In particular, in-plane circumferential modes Cn-m, in plane radial modes Rn-m and out-of-plane axial modes An-m may be distinguished where n and m denote respectively the number of nodal circles and nodal diameters. A stability analysis has been performed. More than 800 complex modes have been calculated to reach an upper limit frequency of about 14 kHz. The corresponding growth factors are

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represented on figure 5. Two kinds of modes may be distinguished: the pad modes, for which the disc vibrations are very small, and the disc modes for which the vibrations of the disc are dominating. The corresponding mode shapes and frequencies of the disc modes are close to the modes of the disc in free conditions but rotate along the disc. Most of these modes are axial modes without nodal circles and with one nodal circle (see fig 3). Another mode is rather an in-plane mode (C0-2) but with some axial components. Some experimental results have been obtained from a TGV in braking operation at about 10 km/h. The acoustic pressure at one centimetre from the disc and the axial vibratory velocity at a point on the disc surface have been measured during braking. Figure 4 shows that the disc is responsible for the emitted noise and that the vibrations are composed of 8 high frequencies from 5000 to 15000 Hz. Except the axial modes with one nodal circle, all the unstable modes are close to the experimental squeal frequencies.

Mode C0-2 : 6684 Hz - ζ=0.02% Mode A1-6 : 9494 Hz - ζ=0.12% Mode A0-10 : 10040 Hz -

ζ=0.23%

Fig 3. Some unstable modes of the TGV brake F.E. model

Fig 5. Growth rates ζ [%] of complex modes as a function of frequency [Hz] (O : disc mode × : Pad modes) Conclusion In this paper, the modelling strategy of disc brake squeal has been studied. In order to investigate the relations between stability and transient classical analyses, a new method has been proposed, which consists on expanding the transient vibratory field on the complex modes provided by the stability analysis. This method has been tested on a simplified disc brake model

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for various initial conditions. Results have shown that the stabilised solution is made up of two coexisting unstable modes and that this solution is the same for different initial conditions. A stability study has also been performed on a finite element TGV brake system and compared with vibration measurements. It has been found that the frequencies of the unstable disc modes correspond to most of the vibration frequencies. The experimental investigations performed on TGV disc brakes have been presented. The experimental data coming from a prior statistical study in stations, on-board measurements and tests on bench in laboratory has been analyzed. References [1] F. Moirot and Q.S. Nguyen, Brake squeal: a problem of flutter instability of the steady

sliding solution ? Arch. Mech. 52, 2000, pp. 645-661. [2] F. Moirot, Etude de la stabilité d'un équilibre en présence de frottement de coulomb. PhD

Thesis, Ecole polytechnique, Palaiseau, France, 1998. [3] M. Jean, The non-smooth contact dynamics method, Comput. Methods Appl. Mech. Eng.,

177, 1999, pp. 235-257. [4] D. Vola, E. Pratt, M. Jean and M. Raous, Consistent time discretization for a dynamical

frictional contact problem and complementarity techniques. REEF Volume 7, 1998. [5] E. Balmès, Structural Dynamics Toolbox, 2006, www.sdtools.com [6] X. Lorang, Instabilité des structures en contact frottant : application au crissement des

freins à disque de TGV. PhD Thesis, Ecole polytechnique, Palaiseau, France, 2007.