This paper deals with the interaction between a moving wheelset and a slab track due to the short-pitch corrugated rail. The wheelset is modeled using a free-free Timoshenko beam with attached rigid bodies as the axle boxes, wheels and brake discs. The slab track model consists of elastically supported double Euler-Bernoulli beams. In fact, both wheelset and slab track are symmetric structures and the issue of the wheelset/slab track interaction is reduced to the wheel/rail interaction. The nonlinear equations of motion describing the wheelset/slab track interaction due to the short-pitch corrugated rail are solved using the time-domain Green’s functions method and the convolution theorem. The wheelset/slab track interaction due to the short-pitch corrugated rail exhibits a critical velocity when the vibration reaches the maximum leve

Traian MAZILU

Directory of Open Access Journals

  Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     62 A STUDY ON THE WHEELSET/SLAB TRACK  VERTICAL INTERACTION  Associate Professor PhD. eng. Traian MAZILU Department of Railway Vehicles, University Politehnica of Bucharest 313 Splaiul Independentei, sector 6, 77206, Bucharest, Romania  trmazilu@yahoo.com          Abstract This paper deals with the interaction between a moving wheelset and a slab track due to the short-pitch corrugated rail. The wheelset is modeled using a free-free Timoshenko beam with attached rigid bodies as the axle boxes, wheels and brake discs. The slab track model consists of elastically supported double Euler-Bernoulli  beams.  In  fact,  both  wheelset  and  slab  track  are  symmetric  structures  and  the  issue  of  the wheelset/slab  track  interaction  is  reduced  to  the  wheel/rail  interaction.  The  nonlinear  equations  of  motion describing the wheelset/slab track interaction due to the short-pitch corrugated rail are solved using the time-domain Green’s functions method and the convolution theorem. The wheelset/slab track interaction due to the short-pitch corrugated rail exhibits a critical velocity when the vibration reaches the maximum level.       Keywords: wheelset, slab track, short-pitch corrugated rail, Green’s functions     1. Introduction    This paper showcases a study of the interaction between a wheelset and a slab track when  considering  the  input  due  to  the  short-pitch  corrugated  rail.  These  aspects  of  the wheelset/slab  track  interaction  are  critical  in  predicting  the  rolling  noise  [1],  the  ground vibration [2] or the formation and developing of the short-pitch rail corrugation [3].   The slab track is seldom preferred for the high-speed lines crossing region with soft soil because it is able to isolate the rail vibration from the subgrade influence, mainly when the trains’ velocity and the phase velocity of the waves induced by the trains in the track structure are closed. Similarly, the slab track is extensively applied for urban railway transport [4]. In virtue of the symmetry feature, the slab track model is usually reduced to a rail resting on a slab via rail pad, both taken as Euler-Bernoulli or Timoshenko beams [5].    For the middle and high frequency range, the issue of the vehicle/track interaction is solved  by  reducing  the  vehicle  to  a  wheel.  The  lumped  mass  model  [6]  and  two-mass oscillator [7] are the usual models for the wheel. Likewise, there are models that include the bending modes due to the axle or the radial modes of the wheel itself [7]. These models are either discrete-continuous or FE models like the ones used by Szolc [9] and respectively, Diana et al [10] to study the vibration behavior of the ballasted track.    In this paper, the interaction between a moving wheelset and a slab track is investigated using the time-domain Green’s functions for both wheelset and slab track, which are suitable to account for the nonlinear wheel/rail contact [11, 12]. The wheelset is modeled using a free-free uniform Timoshenko beam as the axle, with attached rigid bodies as the wheels, axle boxes and brake discs. Only the symmetric bending modes are taken into account. The model of the slab track consists of two infinite uniform Euler–Bernoulli beams coupled by a Winkler layer as the rail, slab and the rail pad, respectively. Similarly, the ground is taken as a Winkler foundation. Finally, the results from the numerical simulation of the interaction between the wheelset and the slab track due to the short-pitch corrugated rail are presented.     Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     63   2. The vehicle mechanical model   It considers the case of one wheelset that rolls without slip at the constant velocity  V along a tangent slab track as depicted in fig. 1. Both wheelset and slab track (including the irregularities of the rolling surfaces and the loading of the wheel/rail contacts) are presumed as symmetric structures. Subsequently, only half of the slab track needs to be modeled and the dynamics  of  one  wheel  considering  the  symmetric  modes  of  the  wheel  set  have  to  be accounted for.       Fig. 1. The wheelset/slab track mechanical model: (a) wheelset/slab track system; (b) wheelset.   Sticking with the general domain, we refer to one wheelset of a passenger coach with two wheels, two axle boxes and two brake discs, as well. The model of the wheelset consists of a uniform Timoshenko beam as the axle, with attached rigid bodies as the wheels, the axle boxes and the brake discs (fig. 1 (b)). The displacements of the axle are described by the column  vector  qw(y,  t)  =  [v(y,  t)  ( y,  t)]T,  where  v(y,  t)  and  ( y,  t)  are  the  vertical displacement  and  the  rotation  of  the  cross-section;  t  and  y  stand  for  the  time  and  the coordinate along the axle, respectively. The parameters of the axle are: the Young’s modulus E, the shear modulus  the density , t he length l, the mass per length unit ma, the cross-section area Sa, the area moment of inertia Ia and the shear coefficient  a. The rigid body i (i = 1÷6) is attached to the axle at the distance ei from the left end of the axle and has the Mi mass and the Ji mass-moment inertia.    The  wheelset  is  subjected  to  static  loads  F0  applied  on  the  two  axle  boxes  by  the suspended mass of the vehicle, the wheelset weight Gw and two equal contact forces Q(t) including the static component Q0 (Q0 = F0 + Gw) and the dynamic one. It is quite obvious that these two forces act on the wheels and are located at the distances e2 and e5, respectively. For that reason, only the symmetric vibration modes are involved.    Assuming  the  hypothesis  of  the  small  motions  about  the  equilibrium  position  and neglecting the gyroscopic effects and the static and dynamic imbalances of the wheelset, the governing equations of motion may be written as follows:  ) , ( ) ( ) , ( ) , (61, , t y e y t e t y wii iwt iw wt y Q q F q L    ,                                   (1)    Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     64 qw(ei, t) is the column vector which contains the axle displacement in the ei section, Qw(y,t) = (Q(t) –Q0)[y  0]T is the column vector of the forces which act on the wheelset with y=(y–e2)+(y–e5)  and  (.)  as  the  Dirac’s  delta  function,  and  Lwy,t  and  Fi,t  stand  for  the  matrix differentials          a a a a a aa a a a awt yStIyEIySyStmyS22222222, L                           (2)   2222,dd00ddtJtMiit i F .                                                   (3)                          Such  model  gives  enough  accuracy  as  long  as  the  wheel  rim  and  the  disc  vibrate together, which means that the frequency is up to 1500 Hz [23].    The boundary conditions (free-free) require that the shear force and the moment are null for both ends of the axle (y = 0 and y = l)   0   yv, 0  y                                                       (4)    The slab track is considered to be an infinite, homogeneous structure and its model (half track) may be reduced to double Euler–Bernoulli beams coupled with a Winkler layer, due to the symmetric structure hypothesis. The loss factor for both beams is neglected. It is assumed that the wheelset velocity is lower than the minimum phase velocity of the waves excited in the ground and, because of that, the ground is looked at as a Winkler foundation.     One the other hand, it has to be mentioned that the Euler-Bernoulli beam model gives satisfactory  results  as  long  as  the  cross-sectional  dimensions  are  small  compared  to  the bending wavelength. The parameters for the slab track model are: the mass per length unit m1,2 (index 1 for upper beam and index 2 for under beam) and the bending stiffness EI1,2. The two  Winkler  foundations  have  the  elastic  constants  k1,2  per  length  unit  and  the  viscous damping factors c1,2 per length unit.    The track’s differential equations of motion can be written in matrix form as  ) , ( ) , ( 0 0 , 0 t x t x t x Q w L  ,                                                    (5)  where  t x , 0 L  stands for matrix differential operator   Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     65    ) ( ) ( 2 1 2 1 222 4042 1 11 1 1 1 221 4041, 0k ktc ctmxEI ktcktc ktctmxEIt x L ,    (6)  w(x0,t)=[w1(x0,t)    w2(x0,t)]T  is  the  column  vector  of  the  beams  displacements  and Q(x0,t)=[Q(t)(x0–Vt)  0]T is the column vector of forces on the track.    The boundary conditions are   T0 0 0 ) , ( lim0  t xVt xw                                                     (7)  and all initial conditions are zero, including the Q(0) normal contact force.   The wheel and the rail are solid elastic bodies and the Hertz’s theory of elastic contact may be applied for the wheel/rail deformation at contact point. By neglecting the wheel radius influence on the contact position, the wheel/rail normal force will be expressed as      ) ( ) ( / ) ( 3 / 2 t z t z C t Q H     ,                                    (8)  where CH represents the Hertzian constant, [.] is the Heaviside function and z is wheel/rail deflection, including the irregularity of the rolling surface.    It has to be underlined that the two subsystems – the wheelset and the slab track - are represented  by  the  linear  models  and  this  fact  allows  to  highlight  their  response,  using adequate forms of the Green’s functions.      3. The estimation of the dynamic forces at a railway vehicle    In this section, both the random behavior and the time-domain numerical analysis of a particular  wheelset  that  uniformly  moves  along  a  slab  track  are  presented.  The  model parameters for the wheelset and the slab track are presented in ref. [11] and [12].   Fig. 2. The wheel/rail interaction due to the corrugated rail (amplitude of 30 µm and wave length of 80 mm) at 84 m/s: (a) , rail displacement at the moving contact point, , wheel displacement; (b) , contact force, , static load.   Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     66   The Hertzian constant CH = 9.42∙1010 N/m3/2 is determined by taking the curve radiuses of the UIC 60 rail-type and the radius of 0.46 m for the wheel (conic profile). The stiffness of the wheel/rail contact has the value of 1.338 GN/m.   Fig. 3. Contact force versus wheelset velocity: (a) •, maximum contact force; *, minimum contact force; – – –, static load; (b) o, asymmetry ratio.      As it is known, the rail corrugation is a specific defect of the rail rolling surface with the wavelength  between 3-4 cm  and 2-3  m.  Critical  for the vibration  level  is the short pitch corrugation  with  the  wavelength  of  30-100  mm.  To  illustrate  the  wheelset/slab  track interaction due to the short pitch corrugation, we look at the case of the wheelset rolling on the corrugated rails with the wavelength of 80 mm and the amplitude of 30 m. Figure 2 shows the  numerical results of the  steady-state interaction derived  from the above  model when the wheelset velocity is 84 m/s (302.4 km/h). For this velocity, the vibration frequency due to the short pitch corrugation of the 80 mm wavelength is 1050 Hz. Upon comparing the average value of the wheel displacement and rail’s at the moving contact point, it may be observed that the wheel displacements are higher due to the contact elasticity. On the other hand,  the  rail  amplitude  is  significantly  higher  than  the  wheel’s  one  and  this  aspect  is consistent  with  the  results  obtained  via  the  frequency-domain  analysis.  It  is  worthwhile mentioning  that  the  wheel  and  rail  at  the  moving  contact  point  are  in  antiphase,  which explains why the contact force reaches values so high (see Fig. 2 (b)). The peak of the contact force is 148.4 kN and the minimum contact force value is 7.8 kN. By comparing these values with the static load of 70 kN, the nonlinear character of the wheel/rail vibration is obvious. The asymmetry ratio may be introduced as the ratio between the high and low amplitudes of the contact force during a cycle. The asymmetric ratio reflects the nonlinear character of the vibration.  It  has  to  be  observed  that  the  nonlinear  Hertzian  contact  has  a  hardening nonlinearity and, because of that, when the wheel is moving downwards, the contact force increases higher than it decreases during the upwards motion of the wheel.       Finally, the influence of the wheelset velocity on the maximum/minimum contact force value when the wheelset rolls over the corrugated rail is displayed in Figure 3 for velocities between 40 and 100 m/s. In this figure, the asymmetry ratio may be also found for the same velocity interval. The level of the vibration behavior increases along the velocity and reaches the highest level at 84 m/s. Then, at higher velocities, the vibration level decreases. This fact corresponds  to  the  results  derived  from  the  frequency-analysis,  where  the  contact  force exhibits a relative flattened peak around 1050 Hz. The nonlinear character of the vibration becomes more evident at the velocity of 84 m/s.      Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     67   6. Conclusions    Here, the time-domain analysis of the wheelset/slab track interaction behavior has been performed by taking into account the nonlinearity of the wheel/rail Hertzian contact.    The wheelset is modeled as a free-free Timoshenko beam as the axle, attached rigid bodies as the axle boxes, wheels and brake discs. The mechanical structure of the wheelset is symmetric  and  consequently,  the  symmetric  bending  modes  of  the  wheelset  only  are considered. Also, the slab track is supposed to be a symmetric infinite structure and its model consists of elastically supported double Euler-Bernoulli beams. The upper beam models the rail, the  lower  is  for the  slab, while the two elastic  layers represent the rail pad and the subgrade.     The  wheelset/slab  track  interaction  might  be  strongly  influenced  by  the  contact nonlinearities, such as  the non-linear contact stiffness according to the Hertz theory. When the  wheelset  rolls  over  a  short-pitch  corrugated  rail,  the  level  of  the  wheelset/slab  track vibration increases along the wheelset velocity and reaches a maximum level at a particular velocity.  The  vibration  nonlinear  character  is  visible  thanks  to  the  asymmetry  ratio  that exhibits the maximum value when the vibration level is the highest.    References  [1] Remington, J.P., Wheel/rail noise—part I: theoretical analysis, J. Acoust. Soc. Am. 81, (1987), 1805–1823. [2] Metrikine, A.V., Popp, K., Steady-state vibrations of an elastic beam on a visco-elastic layer undermoving load, Arch. Appl. Mech. 70, (2000), 399–408.  [3]  Nielsen,  J.C.O.,  Lundén,  R.  Johansson,  A.  Vernersson,  T.,  Train–track  interaction  and mechanisms of irregular wear on wheel and rail surfaces, Veh. Syst. Dyn. 40, (2003), 3–54. [4] Esveld, C., Track Structures in an Urban Environment, TU Delft, Symposium K. U. Leuven (1997). [5] Hussein, M.F.M., Hunt, H.E.M., Modelling of floating-slab tracks with continuous slabs under oscillating moving loads, J. Sound Vib. 297, (2006), 37–54.  [6] Grassie, S.L., Gregory, R.W., Harrison, D., Johnson, K.L., The dynamic response of railway track to high frequency vertical excitation, J. Mech. Eng. Sci. 24, (1982), 77–90. [7]  Nielsen,  J.C.O.,  High-frequency  vertical  wheel–rail  contact  forces-validation  of  a  prediction model by field testing, Wear 265, (2008), 1465–1471. [8]  Thompson,  D.J.,  Wheel–rail  noise  generation,  part  II:  wheel  vibration,  J.  Sound  Vib.,  161, (1993), 401–419. [9] Szolc, T., Simulation of bending-torsional-lateral vibration of the railway wheelset-track system in the medium frequency range, Veh. Syst. Dyn., 30, (1998), 473–508. [10] G. Diana, F. Cheli, S. Bruni, A. Collina, Exprimental and numerical investigation on subway short pitch corrugation, Veh. Syst. Dyn. Suppl., 28, (1998), 234–245. [11] Mazilu, T., Dumitriu, M., Tudorache, C., Sebeşan, M., Using the Green’s functions method to study wheelset/ballasted track vertical interaction, Math. Comp. Mod. 54, (2011), 261–279. [12]  Mazilu,  T., Interaction  between  a  moving  two  mass  oscillator  and  an infinite  homogeneous structure: Green’s functions method, Arch. Appl. Mech., 80, (2010), 909-927. 

A STUDY ON THE WHEELSET/SLAB TRACK VERTICAL INTERACTION

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals