Modelling of wheel/rail squeal noise in curves from mono-harmonic vibratory limit cycles Olivier Chiello 1 UMRAE, Univ Gustave Eiffel, IFSTTAR, CEREMA, Univ Lyon, F-69675, Lyon, France Rita Tufano, Martin Rissmann 2 Vibratec, Railway Business Unit, 28 chemin du petit bois, 69131 Ecully Cedex, France

ABSTRACT Most of the works in the literature agree to attribute the generation of wheel/rail squeal noise in curves to the important lateral slip imposed in the curve and the resulting instabilities. In models, the occurrence of the phenomenon is thus generally studied through a stability analysis based on the linearization of the contact forces. Despite its undeniable interest, the stability analysis does not al- low the prediction of the amplitudes of the nonlinear self-sustained vibrations resulting from the in- stabilities. These nonlinear vibrations are most often calculated using a numerical integration of the dynamic equations of the system in the time domain. Some authors have proposed simplified methods allowing a direct calculation of stationary regimes, but limited to a reduced modal description of the system dynamics. In this presentation, an original method is proposed to determine approximate limit cycles from the wheel/rail contact mobilities expressed in the frequency domain. The contact conden- sation allows to be both more general and more functional to describe the dynamics of structures, in particular the one of the rail. Assuming a mono-harmonic vibratory cycle, the corresponding ampli- tude is determined from a power balance and the squeal level at the considered pulsation is obtained from analytical radiation factors.

1. INTRODUCTION

Since Rudd [1], most of the works in the literature agree to attribute the generation of wheel/rail squeal noise in curves to the high lateral slip imposed in the curve and to the resulting instabilities [2]. In squeal models, the occurrence of the phenomenon is thus generally studied through a stability analysis based on the linearization of the contact forces. Two families of methods are used: those leading to a generalized eigenvalue system via a modal description of the system [3] and those de- scribing the wheel/rail interaction at the contact's degrees of freedom using their respective mobilities (Nyquist criterion) [4].

Despite its undeniable interest, stability analysis does not allow the prediction of the amplitudes of the nonlinear self-sustained vibrations resulting from instabilities, which is a necessary step for the determination of the squeal noise levels and thus for the comprehension of the severity of squeal instabilities. These nonlinear vibrations are most often calculated using a numerical integration of the dynamic equations of the system in the time domain. As in the case of stability analysis, these equa- tions are based either on a modal description [3] or on contact condensation, but this time from the wheel/rail impulse responses [5]. The possible stationary regimes or "limit cycles" obtained are then

1 olivier.chiello@univ-eiffel.fr

2 martin.rissmann@vibratec.fr

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re-expressed in the frequency domain for the calculation of the acoustic radiation and the comparison with experimental data. A disadvantage is that the integration has to be carried out over a sufficiently long period of time for the transient regime to stabilize.

Some authors have proposed simplified methods allowing a direct computation of stationary re- gimes [1,6,7]. These methods are mainly based on the assumption of mono-harmonic limit cycles. Unfortunately, they are limited to a reduced modal description of the system dynamics: one wheel mode [1,6] or one wheel mode and one rail mode [7]. In this paper, a more general method using wheel/rail mobilities is proposed to determine approximate limit cycles.

2. DESCRIPTION OF THE MODEL

2.1. Main assumptions and contact mobilities The studied configuration is shown in Figure 1. The pass-by of the guided vehicle at speed 𝑉 in a curve of small radius is characterized by a high angle of attack of some wheels, in particular the front inner wheel of the vehicle bogies. The angle of attack 𝛼 designates the misalignment of the wheel axis with the rolling direction tangent to the rail and leads to the lateral slip of the wheel on the rail head at a speed 𝑉 𝑡 ≈𝛼𝑉 .

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Figure 1: Position of the problem

In the model, the interaction of a single wheel with the rail is considered. The flange contact that often occurs, especially on the outer wheels, is not considered. The wheel/rail interaction is point-like and reduced to the vertical and lateral directions. Two degrees of freedom at the contact, normal and tangential, are thus considered. The wheel and the rail are described by their point and cross mobilities 𝑌 𝑊𝑡𝑡 , 𝑌 𝑊𝑛𝑛 , 𝑌 𝑤𝑡𝑛 , 𝑌 𝑅𝑡𝑡 , 𝑌 𝑅𝑛𝑛 and 𝑌 𝑅𝑡𝑛 at the contact’s degrees of freedom in the frequency domain where subscripts 𝑊 , 𝑅 , 𝑛 and 𝑡 stand for wheel, rail, normal and tangential contact’s degree of free- dom. These mobilities can be determined using various types of models.

An eulerian referential in the axle moving frame is considered. Rotation (for the wheel) and mov- ing (for the rail) dynamic effects are neglected in this paper but they could be advantageously con- sidered in the computation of contact mobilities.

2.2. Contact and friction laws The contact is modelled by Hertz’s normal stiffness 𝑘 𝐻 and Mindlin’s tangential stiffness 𝑘 𝑀 lead- ing to normal and tangential point mobilities 𝑌 𝐶𝑛𝑛 = 𝑘 𝐻

−1 and 𝑌 𝐶𝑡𝑡 = 𝑘 𝑀

−1 (see for instance rolling noise models [8]) while a nonlinear friction/creep law is considered for friction. With such a law, the total friction force 𝑓 𝑡 acting at the wheel/rail interface can be related to the wheel/rail creep 𝑠 and the normal contact force 𝑓 𝑛 by the following equation:

𝑓 𝑡 ሺ𝑠, 𝑓 𝑛 ሻ= 𝜇ሺ𝑠, 𝑓 𝑛 ሻ𝑓 𝑛 with 𝑠= 𝛼+ ∆𝑣 𝑡 /𝑉 (1)

where 𝜇 denotes the non-linear dynamic friction coefficient and ∆𝑣 𝑡 = 𝑣 𝑊𝑡 −𝑣 𝑅𝑡 −𝑣 𝐶𝑡 stands for the relative instantaneous tangential velocity at the wheel/rail interface. In this paper, the choice was made for a nonlinear creep law of Shen-Hedrick-Elkins type, combined with a velocity-weakening friction coefficient (as shown in Figure 2) but this choice does not restrict the generality of Equa- tion (1).

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Figure 2: Friction/creep law

2.3. Stability analysis A stability analysis is first performed using the Nyquist criterion as proposed by De Beer et al [4]. For small oscillations around the quasi-static equilibrium characterized by normal load 𝑁 , creep 𝛼 and quasi-static friction force 𝑇= 𝜇ሺ𝛼, 𝑁ሻ𝑁 , the dynamic part of the friction force can be linearized:

𝑁

𝜕𝜇

𝜕𝜇

𝛼,𝑁 ∆𝑣 𝑡 + ൬𝜇ሺ𝛼, 𝑁ሻ+ 𝑁

൰ሺ𝑓 𝑛 −𝑁ሻ (2)

𝑓 𝑡 −𝑇=

𝜕𝑠 ቚ

𝜕𝑓 𝑛 ቚ

𝑉 0

𝛼,𝑁

Considering furthermore harmonic variations of the oscillations such that ∆𝑣 𝑡 = ∆𝑣 𝑡 ෝ𝑒 𝑖𝜔𝑡 , 𝑓 𝑡 − 𝑇= 𝑓 𝑡 ෡ 𝑒 𝑖𝜔𝑡 and 𝑓 𝑛 −𝑁= 𝑓 𝑛 ෡ 𝑒 𝑖𝜔𝑡 , the wheel, rail and contact mobilities can be used to express the normal and tangential coupling between the components in the frequency domain [4]:

𝑌 𝑛𝑛 𝑓 𝑛 ෡ + 𝑌 𝑛𝑡 𝑓 𝑡 ෡ = 0 𝑌 𝑡𝑛 𝑓 𝑛 ෡ + 𝑌 𝑡𝑡 𝑓 𝑡 ෡ = ∆𝑣 𝑡 ෝ (3)

Fton cnet 6}

where 𝑌 𝑛𝑛 = 𝑌 𝑊𝑛𝑛 + 𝑌 𝑅𝑛𝑛 + 𝑌 𝐶𝑛𝑛 , 𝑌 𝑡𝑡 = 𝑌 𝑊𝑡𝑡 + 𝑌 𝑅𝑡𝑡 + 𝑌 𝐶𝑡𝑡 and 𝑌 𝑛𝑡 = 𝑌 𝑡𝑛 = 𝑌 𝑊𝑛𝑡 + 𝑌 𝑅𝑛𝑡 are the to- tal contact mobilities. By using Equations (2) and (3) together with harmonic variations, a complex equation is obtained for the friction force:

𝑓 𝑡 ෡ = 𝐻ሺ𝜔ሻ𝑓 𝑡 ෡ (4)

where closed-loop transfer function 𝐻ሺ𝜔ሻ is given by:

𝐻ሺ𝜔ሻ= 𝑁

𝜕𝜇

−1 𝑌 𝑛𝑡 ሻ−ቆ𝜇ሺ𝛼, 𝑁ሻ+ 𝑁 𝜕𝜇

−1 𝑌 𝑛𝑡 (5)

ሺ𝑌 𝑡𝑡 −𝑌 𝑡𝑛 𝑌 𝑛𝑛

𝜕𝑠 ฬ

ฬ

ቇ𝑌 𝑛𝑛

𝑉 0

𝜕𝑓 𝑛

𝛼,𝑁

𝛼,𝑁

According to the Nyquist criterion, oscillations are unstable for pulsations 𝜔 such that ℑሺ𝐻ሺ𝜔ሻሻ= 0 and ℜሺ𝐻ሺ𝜔ሻሻ> 1 . This technique is useful to find the pulsations where instabilities may occur and initiate self-sustained vibrations but does not give information on the amplitude of these vibra- tions.

2.4. Power balance in the linear frequency domain Actually, the Nyquist criterion may be interpreted as a power balance in the linear frequency do- main. On the one hand, the mean dissipated power in the wheel/rail system during a period 𝑇= 2𝜋𝜔 −1 is obtained by using mobilities involved in Equations (3):

∗ ሻ= ȁ ∆𝑣 𝑡 ෝ ȁ 2

𝑊 dis തതതതതത= 1

−1 𝑌 𝑛𝑡 ሻ −1 ሻ (6)

2 ℜሺ∆𝑣 𝑡 ෝ𝑓 𝑡

2 ℜሺሺ𝑌 𝑡𝑡 −𝑌 𝑡𝑛 𝑌 𝑛𝑛

On the other hand, the mean injected power in the system is obtained by using the linearized fric- tion force given by Equation (2) together with the expression of 𝑓 𝑛 ෡ as a function of ∆𝑣 𝑡 ෝ resulting from Equations (3):

𝑊 inj തതതതത= 1

∗ ሻ

2 ℜሺ∆𝑣 𝑡 ෝ𝑓 𝑡

(7)

= ȁ ∆𝑣 𝑡 ෝ ȁ 2

2 ሺ 𝑁

𝜕𝜇

+ ቆ𝜇ሺ𝛼, 𝑁ሻ+ 𝑁 𝜕𝜇

−1 𝑌 𝑛𝑛 ሻ −1 ሻሻ

ቇℜሺሺ𝑌 𝑡𝑛 −𝑌 𝑡𝑡 𝑌 𝑛𝑡

𝜕𝑠 ฬ

ฬ

𝑉 0

𝜕𝑓 𝑛

𝛼,𝑁

𝛼,𝑁

It is easy to verify that at pulsations 𝜔 when ℑሺ𝐻ሺ𝜔ሻሻ= 0 , the Nyquist criterion ℜሺ𝐻ሺ𝜔ሻሻ> 1 is equivalent to 𝑊 inj തതതതത> 𝑊 dis തതതതതത which means that the system cannot dissipate all the power injected through friction at the wheel/rail interface. Consequently, the amplitude ∆𝑣 𝑡 ෝ of the oscillations in- creases until the non-linearities appear and allow to balance the injected and dissipated powers.

2.5. Power balance for non-linear harmonic cycles Considering larger oscillations around the quasi-static equilibrium, the friction force can no longer be linearized and in most cases the frequency domain is not appropriate to describe the response of the structure. Nevertheless, it is assumed a mono-harmonic response of the structure at a pulsation 𝜔 𝑖 for which the system is unstable.

With this assumption the power dissipated in the system can be evaluated with the same method (e.g. Equation (4)) since the behavior in the structure remains linear. However, for the evaluation of the injected power a time integration over the cycle has to be performed since the friction force is non-linear:

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𝑇 0

1

𝑊 inj∗ തതതതതത=

𝑇 ׬ ∆𝑣 𝑡 ሺ𝑡ሻ𝜇൫𝑠ሺ𝑡ሻ, 𝑓 𝑛 ሺ𝑡ሻ൯𝑓 𝑛 ሺ𝑡ሻ𝑑𝑡

1

(8)

with ∆𝑣 𝑡 ሺ𝑡ሻ= ℜ൫∆𝑣 𝑡 ෝ𝑒 𝑖𝜔 𝑖 𝑡 ൯ , 𝑠ሺ𝑡ሻ= 𝛼+

𝑉 ℜ൫∆𝑣 𝑡 ෝ𝑒 𝑖𝜔 𝑖 𝑡 ൯

−1 𝑌 𝑛𝑛 ሻ −1 ∆𝑣 𝑡 ෝ𝑒 𝑖𝜔 𝑖 𝑡 ൯

and 𝑓 𝑛 ሺ𝑡ሻ= 𝑁+ ℜ൫ሺ𝑌 𝑡𝑛 −𝑌 𝑡𝑡 𝑌 𝑛𝑡

This integral may be computed numerically for a given value of amplitude ∆𝑣 𝑡 ෝ . It is important to note that, unlike dissipated power 𝑊 dis തതതതതത , injected power 𝑊 inj∗ തതതതതത is not proportional to ȁ∆𝑣 𝑡 ෝȁ 2 so that the power balance may vary with ∆𝑣 𝑡 ෝ and differ from the linear case. The search for stationary self-sus- tained vibrations thus amounts to find ∆𝑣 𝑡 ෝ such that:

𝑊 inj∗ തതതതതതሺ∆𝑣 𝑡 ෝሻ= 𝑊 dis തതതതതതሺ∆𝑣 𝑡 ෝሻ (9)

which consists in solving a nonlinear algebraic equation on ∆𝑣 𝑡 ෝ .

2.6. From the amplitude of the cycle to the radiated sound power In cases where a solution is found to Equation (9), the complex magnitudes 𝑓 𝑡 ෡ et 𝑓 𝑛 ෡ of the tangent and normal forces acting at the wheel/rail interface at unstable pulsation 𝜔 𝑖 are calculated using Equa- tion 3. The sound power radiated by the wheel is finally estimated from contact forces by a “rolling noise” type method based on analytical radiation factors [8].

3. RESULTS

3.1. Components models and input data The method is tested in a realistic case of a metro wheel rolling in a curve at speed 𝑉= 30 km/h and angle of attack 𝛼= 11 mrad. As shown in Figure 2, at this angle of attack and for the chosen friction parameters, the friction/creep curve is decreasing which reflects a potential source of insta- bility. A normal load 𝑁= 51 kN is considered.

The wheel is a steel monobloc wheel with a diameter of 86 cm. It is first modelled by the Finite Element Method considering clamped boundary conditions at the axle axis. Contact mobilities are computed by modal superposition from 37 normal modes obtained with the Finite Element model in the frequency range 0 – 6250 Hz. Damping factors of 0.01 % are chosen for most of modes above 400 Hz. These low values can be related to the high resonances in wheel mobility shown in Figure 3.

A classical “Rodel” model (infinite Timoshenko beam with uniform elastic support [8]) is chosen for the rail lying on monobloc concrete sleepers through elastic rail pads of medium stiffness. Contact mobilities are computed analytically from rail and support parameters.

3.2. Stability analysis The results of the stability analysis are given in Figure 3. On the left, the contour of the closed- loop transfer function 𝐻ሺ𝜔ሻ is plotted, highlighting 9 frequencies (red crosses) where instabilities may occur. On the right, these particular frequencies are reported on the axial wheel mobility. They correspond for the most part to the natural frequencies of axial wheel modes without nodal circles. These modes are known to play an important role in the generation of curve squeal.

3.3. Non-linear harmonic cycles Non-linear Equation (9) is solved for each frequency where instability may occur. For each of them an amplitude ∆𝑣 𝑡 ෝ is found allowing a balance between the injected and dissipated powers. As an example, the variations of these quantities (normalized by ȁ∆𝑣 𝑡 ෝȁ 2 ) are plotted on Figure 4 as a

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function of ∆𝑣 𝑡 ෝ for the frequency 1856 Hz, i.e. the natural frequency of the axial mode without nodal circle and 4 nodal diameters (0L4). The figure highlights the difference between the linear domain (amplitudes ∆𝑣 𝑡 ෝ< 0.05 m/s) where the normalized powers are both constant and the non-linear do- main (higher amplitudes ∆𝑣 𝑡 ෝ> 0.05 m/s) where the normalized injected power decreases with the amplitude of the cycle until it reaches the normalized dissipated power at ∆𝑣 𝑡 ෝ≈0.06 m/s.

The corresponding radiated sound powers are given in Figure 5 for each unstable mode provided by the stability analysis. It can be observed that the amplitudes are quite similar for all modes. Com- puted sound powers are realistic and provide a rapid estimation of the squeal noise potentially emitted by the system.

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Figure 3: Stability - Contour of the closed-loop transfer function and wheel axial mobility

Figure 4: Mode 0L4 (1856 Hz) – Power balance as a function of the cycle amplitude

Figure 5: Radiated sound power for each unstable mode

4. CONCLUSIONS

An original method for the approximate estimation of curve squeal levels is proposed, directly targeting the non-linear stationary regime and using wheel/rail mobilities at contact instead of modal characteristics of the structures. As for the stability analysis or the time integration, the condensation at contact allows a more general and more functional description of the dynamics of the structures, in particular the behaviour of the rail for which a modal representation is not very adapted. The method is tested in a realistic case for which the wheel and rail mobilities are obtained from elaborate models. It proves to be very efficient and adapted to parametric studies.

5. ACKNOWLEDGEMENTS

This work was carried out within the framework of the LabEx CeLyA (ANR-10-LABX-0060) of the University of Lyon, as part of the "Investissements d'Avenir" program managed by the Agence Nationale de la Recherche (ANR).

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