Study on railway curve squeal using a rigid-flexible coupling model of vehicle and track Qiuyong Tian 1 Institute of Land and Sea Transport Systems, Chair of Rail Vehicles, Technische Universität Berlin Sekr. SG 14, Salzufer 17-19, 10587 Berlin, Germany Yichang Zhou 2 Institute of Land and Sea Transport Systems, Chair of Rail Vehicles, Technische Universität Berlin Sekr. SG 14, Salzufer 17-19, 10587 Berlin, Germany Markus Hecht 3 Institute of Land and Sea Transport Systems, Chair of Rail Vehicles, Technische Universität Berlin Sekr. SG 14, Salzufer 17-19, 10587 Berlin, Germany ABSTRACT Squeal noise occurring in narrow curves is one of the most annoying noise issues for people living alongside the railway track. When vehicles pass through those curves, squeal noise is mainly generated due to the large, high-frequency, lateral sliding friction force. To investigate generation mechanism of curve squeal, a three-dimensional (3D) vehicle-track interaction model with a flexible wheelset and a flexible curved track is built in a multi-body simulation tool. Compared to conventional rigid models, this model allows to obtain the high frequency friction force between wheel and rail. Then, friction force in the frequency domain is analyzed to identify the major modes. Those modes are compared with the modal properties of the wheel- sets and rails in order to determine which modes have the main sensitivity to the friction force and the squeal noise. 1. INTRODUCTION

Curve squeal, one of the most annoying noises, occurs mainly on narrow curves, and it has a signifi- cant impact on the health of residents living near those curves. At present, there are two widely used generation mechanisms of curve squeal: the wheel-rail friction falling mechanism and the mode-cou- pling mechanism [1] . Rudd [2] first proposed that the unstable wheel behavior is due to negative damping caused by the friction falling characteristic at the large slip speed. For instance, when a vehicle passing by a small curve, the negative damping can provide energy to the system in each vibration cycle, which leads to a further system vibration and even instability. Since then, many scholars have carried out many studies based on the Rudd’s method [3] . The representative study on the curve squeal based on the mode-coupling characteristic is performed by Hoffmann et al. [4] . They

1 qiuyong.tian@tu-berlin.de

2 yichang.zhou@campus.tu-berlin.de

3 markus.hecht@tu-berlin.de

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used a two-degree-of-freedom model to explain the mode-coupling mechanism. Due to the non-sym- metric stiffness matrix and the friction coefficient, the system could be instable. Glock et al. [5] and Pieringer [6] also showed that curve squeal can be generated due to the coupling between normal and tangential dynamics when the friction coefficient is constant.

In the previous studies, a certain mode of the wheelset is mostly used and considered as a single degree of freedom (DOF) or two DOFs for steady-state analysis. The dynamic vehicle/track coupling influence on the stability of the vehicle are not considered in detail. In this paper, a rigid-flexible coupling model of vehicle and track is established in SIMPACK, and the effects of track flexibility and wheel/rail friction falling on the dominant modes of the wheelset vibration and the lateral contact friction force are investigated. 2. MODELING

SIMPACK provides a co-simulation platform between itself and finite element (FE) software such as ANSYS and ABAQUS. The first wheelset and rails are modelled as elastic bodies to consider the vibration of the modes in high frequency band. The sleepers are neglected and therefore, the rails are connected directly with ground with fasteners. The components of the vehicle except of the first wheelset are considered as rigid bodies. In this paper, ANSYS is chosen to perform the FE simulation. The modelling procedure is shown in Figure 1 .

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Figure 1: Modeling procedure of the flexible bodies in ANSYS and SIMPACK

2.1. FE modeling of the flexible bodies In this section, a common 3D FE model of the wheelset and a beam model of rail with a length of 30 m and a radius of 100 m are established in ANSYS. The nodes connected to other components are defined as master nodes, such as the nodes at the suspension positions and distributed on the rail/wheel contact surfaces on the wheel tire. Besides, the DOFs of master nodes will be retained. Based on the Craig-Chang method [7] , the modal mass and stiffness matrices of the flexible bodies are calculated by using the substructure analysis in ANSYS. The vibration equation of the wheelset or the rail is

𝑀𝑀𝑥𝑥̈ + 𝐾𝐾𝐾𝐾= 𝐹𝐹 (1)

where 𝑀𝑀 and 𝐾𝐾 are the mass and stiffness matrices of wheelset, respectively, the damping is ne- glected due to the modal analysis. 𝑥𝑥 and 𝐹𝐹 are the state coordinate and force vector, respectively. The

modal coordinate vector 𝑞𝑞 can be transformed from the state coordinate vector by using a transfor- mation matrix Ψ . The modal mass and stiffness matrices as well as the modal force vector can be calculated using modal superposition, and it described as

Ψ 𝑇𝑇 𝑀𝑀Ψ ᇣᇧᇤᇧᇥ

𝑞𝑞̈ + Ψ 𝑇𝑇 𝐾𝐾Ψ ᇣᇤᇥ

𝑞𝑞= Ψ 𝑇𝑇 𝐹𝐹 ถ

𝑀𝑀 𝑀𝑀

𝐾𝐾 𝑀𝑀

𝐹𝐹 𝑀𝑀

(2)

The highest investigated frequency for the wheelset and the rail is 5000 Hz. In this step, two files are created. One is the the *.sub file including the modal mass and stiffness matrices as well as the mode information. The other is *.cdb file which saves the geometry information. These two files generated in ANSYS are used to create a *.fbi file, and it will be imported into SIMPACK. In addi- tion, a *.ftr file must be generated for the flexible track. In the *.ftr file, the *.fbi file of the rail and the exact coordinate positions of the master nodes in the rails are included so as to connect the rail and the wheel.

2.2. Vehicle-track coupled model A vehicle-track coupled model is established in the multi-body simulation (MBS) software SIMPACK as shown in Figure 2 . It includes a passenger vehicle model and a flexible track with 30 m long. The vehicle model, which based on the Manchester benchmark model [8] , consists of a car body, two bogies each with two wheelsets. As noted, the first wheelset is modelled as a flexible body and the track flexibility is also considered in the vehicle-track coupled model. The normal force be- tween wheel/rail point is calculated by an equivalent elastic contact model [9] based on Hertz contact theory, while the tangential force is calculated by the FASTSIM algorithm [10] . The main parameters of this vehicle-track model are summarized in Table 1 .

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Figure 2: (a) Vehicle-track coupled model;(b) Bogie

(a) Flexible whéelset (b) Primary suspension: 1. Spring 2. Vertical damper Secondary suspension: 3. Vertical damper 4. Spring 5. Lateral damper 6, Anti-roll bar 7. Traction rod 8, Lateral bump stop

Table 1. Main parameters of the vehicle-track system

Element Value Unit Wheelset mass 1116 kg Moments of inertia I xx (wheelset) 602 kgm 2 Moments of inertia I yy (wheelset) 101 kgm 2 Moments of inertia I zz (wheelset) 602 kgm 2 Pivot distance 12 m Wheelset distance 1.8 m Wheel radius 0.46 m Track gauge 1435 mm Wheel/rail friction coefficient 0.35 – Wheel/rail profile S1002/UIC60 – Rail inclination 1:40 – Longitudinal stiffness (a single rail and ground) 66 MN/m Lateral stiffness (a single rail and ground) 66 MN/m Vertical stiffness (a single rail and ground) 330 MN/m Longitudinal damping (a single rail and ground) 81 kNs/m Lateral damping (a single rail and ground) 81 kNs/m Vertical damping (a single rail and ground) 55 kNs/m 3. RESULTS

The objective of this section is to investigate the influence of track flexibility and wheel/rail friction falling on the wheelset vibration during curves. Different to the conventional rigid MBS simulation, the natural frequency of the system is significantly increased due to the consideration of flexible bodies. The maximum step size is set to be 1.0 × 10 -5 s in order to ensure the system convergence. The highest frequency of interest in this paper is 5000 Hz, and therefore the sample rate is set as 10 kHz. The deformations of the flexible wheelset and rails are illustrated in Figure 3 with a scaling of 100 . The results show that the deformation shape of the wheelset is mainly the first bending mode shape due to the vehicle weight. Besides, the deformation (red line) of the outer rail is larger than that of the inner rail.

Deformation of the rail

Figure 3: Deformations of the flexible wheelset and rails

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3.1. Influence of the rail flexibility In this section, the influence of the rail flexibility is studied. Figure 4 (a,b) show the lateral wheel/rail contact force of the inner wheel of the first flexible wheelset in the time and frequency domain re- spectively when the vehicle negotiates the curve with a radius of 100 m at a speed of 30 km/h. As shown in Figure 4(a) , the lateral force on the flexible track (blue line) is a little lower than that on the rigid track (red line). Further, a periodic fluctuation of the lateral force occurs on the rigid track due to the roll motion of the flexible wheelset while its influence is suppressed on the flexible track be- cause of the track flexibility. Besides, the influence of the fastener spacing on the lateral force can be presented on the flexible track. Figure 4(b) shows that the frequency band of the lateral force on the rigid track, up to 2500 Hz, is much wider than that on the flexible track.

Figure 4: The lateral wheel/rail contact force of the inner wheel on the rigid and flexible tracks: (a) time domain; (b) frequency domain.

Figure 5 compares the mode acceleration amplitudes of the wheelset in the frequency domain when the vehicle operates on the rigid and flexible track respectively. The results show that consid- ering the track flexibility could suppress the wheelset vibration in the low frequency range below 1300 Hz, while it has less influence in the high frequency band. The shapes of dominant modes are also illustrated in this Figure. To describe the wheelset modes (𝑛𝑛, 𝑚𝑚, 𝑥𝑥) , the number of nodal diame- ters 𝑛𝑛 and the number of nodal circles 𝑚𝑚 as well as the direction 𝑥𝑥 are used. Here only axial (a) wheel- set modes are identified in the Figure, hence 𝑥𝑥= 𝑎𝑎 . The most sensitive frequencies are around 2668 Hz, 3645 Hz, 4617 Hz and 4878 Hz, they correspond the (0, 5, 𝑎𝑎) , (1, 3, 𝑎𝑎) , (0, 6, 𝑎𝑎) , (2, 2, 𝑎𝑎) , (0, 7, 𝑎𝑎) , (2, 4, 𝑎𝑎) and (2, 6, 𝑎𝑎) mode of the wheelset. Compared to the rigid track, less modes in lower frequency range are excited in flexible track. Although the track flexibility has a relatively small influence on the wheelset vibration in the high frequency range, some researches also show that the curve squeal can occur in the lower frequencies such as around 1 kHz corresponding mode (0, 3, 𝑎𝑎) . To eliminate the impact effect generated by interaction between flexible wheelset and rigid rails, to consider the damping effect of the rails and rail pads as well as to consider the curve squeal more exact in lower frequency range, the flexibility of the rails should not be neglected.

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Frequency in Hz —Rigid track —Flexible track a - 4 A ‘“ . & @ @ © | © &/@ | 52 Ww ‘| 8 e | \| tai i, | {hy | 2 | lb abl ofl A LL Lu L Lull HW Vy Wer ey j 1 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Figure 5: Mode acceleration amplitude of the wheelset in the frequency domain

3.2. Influence of the friction falling Figure 6 shows the results of the effective friction coefficient and creepage considering the friction falling based on Polach method [11] . The related Polach parameters are 0.38 and 0.70. Figure 6(a, b) describes the development of the creepage and friction coefficient respectively, when the vehicle operates from the tangent track into the curve. Figure 6(c) shows the relationship between friction coefficient and creepage. The friction coefficient and the corresponding creepage increase quickly as the wheelset entries into the curve. Since the creepage on the curve is larger than 0.005, the friction is always in the negative slope region. The impact effect marked in the black circle in Figure 6(a) indicates that the rear bogie entries into the curve at that moment.

a 2 g 2 yo00 uoHDud 0.05, Ss sz ¢ 2 a $3 3 5 qu9!o4y909 UOHOL 2 25 15 Time in s —Entrance from tagent to curve —In the

a oo ‘Slip and adhesion region 005 curve 0.01 Creepage 0.015 0.02 0.025

Figure 6: (a) creepage; (b) friction coefficient; (c) friction coefficient versus the creepage

@ 0.005

Figure 7(a) compares the lateral forces of the inner wheel with and without friction falling in the time domain, while the corresponding responses in the frequency domain are presented in Figure 7(b, c) . At first, the lateral force becomes a little smaller when considering the friction falling, and the

© ~ Friction faling

static force difference is around 600 N. Secondly, the forces become a little higher in the frequency range marked in black circles in Figure 7(c) when considering the friction falling.

Figure 7: The lateral wheel/rail contact force of the inner wheel with and without friction falling: (a) time domain; (b) frequency domain; (c) zoom plot

Figure 8 shows the results of the mode acceleration states over the frequency range. The blue line represents the situation without friction falling, while the red line represents considering its effect. Overall, the responses in the frequency domain are similar. The modes that dominate the wheelset vibration remain unchanged except that a higher amplitude around 3650 Hz is observed when con- sidering the friction falling effect. This amplitude difference might be caused by the force difference described in Figure 7(c) . It means the modes around this frequency are more sensitive to the change in excitation under considering the friction falling effect. Vibrations of these modes are more easily excited.

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Figure 8: Mode acceleration amplitude of the wheelset in the frequency domain

(05a) (0,6) (0,7,a) (26.] ®& @ Es E (a3 2a) asa e @ r) 82 1 2 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency in Hz th fiction falling —Without friction falling

4. CONCLUSIONS

To explore the generation mechanism of the curve squeal, a vehicle-track coupled model considering the wheelset and track flexibilities has been developed to investigate the influence of track flexibility and friction condition on the wheelset vibration, when the vehicle is operating on the curve with a radius of 100 m. The results show that considering track flexibility can improve the wheel/rail contact and reduce the number of dominant vibration modes in the low frequency range. When the friction falling is considered, the main sensitive modes of wheelset remain unchanged, but the amplitudes of some modes change significantly. 5. REFERENCES

1. Thompson, D.J., Squicciarini, G., Ding, B., Baeza, L. A state-of-the-art review of curve squeal noise: Phenomena, mechanisms, modelling and mitigation. Notes on Numerical Fluid Mechanics and Multidisciplinary Design , 3–41(2018). 2. Rudd, M.J. Wheel/Rail Noise—part II: Wheel squeal. Journal of Sound and Vibration , 46(3) , 381–394 (1976). 3. Othman, Y. B., Kurvenquietschen: Untersuchung des Quietschvorgangs und Wege der Minde- rung. Technische Universität Berlin , 2009. (Doctoral thesis) 4. Hoffmann, N., Gaul, L. Effects of damping on mode-coupling instability in friction induced os- cillations. Journal of applied mathematics and mechanics: Zeitschrift für angewandte Mathe- matik und Mechanik , 83(8) , 524–534 (2003). 5. Glocker, C., Cataldi-Spinola, E., Leine, R.I. Curve squealing of trains: Measurement, modelling and Simulation. Journal of Sound and Vibration , 324(1–2) , 365–386 (2009). 6. Pieringer, A. A numerical investigation of curve squeal in the case of constant wheel/rail friction.

Journal of Sound and Vibration , 333(18) , 4295–4313 (2014). 7. Mustapha, A., Mohamed, R., Ludovic, C., Olivier, C., Guy-Leon, K. Flexible wheelset models in dynamic interaction with track. Proceedings of the 7th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry , pp. 679–682. Stuttgart, Germany, October 11-13, 2017. 8. Iwnicki, S. Manchester benchmarks for rail vehicle simulation. Vehicle System Dynamics , 30(3) , 295–313 (1998). 9. Kalker J.J. Three-Dimensional Elastic Bodies in Rolling Contact, 1st Edition, Springer, 1990. 10. Kalker J.J. A fast algorithm for the simplified theory of rolling contact. Vehicle System Dynamics , 11(1) , 1–13 (1982). 11. Polach, O. Influence of locomotive tractive effort on the forces between wheel and rail. Vehicle System Dynamics , sup(35) , 7–22 (2001).

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