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Simulation of vertical dynamic vehicle–track interaction in a railway crossing using Green's functions
Journal of Sound and Vibration 410 (2017) 318–329
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Simulation of vertical dynamic vehicle–track interaction in a railway crossing using Green’s functions X. Li * , P.T. Torstensson, J.C.O. Nielsen Department of Applied Mechanics/CHARMEC, Chalmers University of Technology, SE-412 96, Gothenburg, Sweden
article info
abstract
Article history: Received 29 November 2016 Revised 11 August 2017 Accepted 16 August 2017 Available online XXX
Vertical dynamic vehicle–track interaction in the through route of a railway crossing is simulated in the time domain based on a Green’s function approach for the track in combination with an implementation of Kalker’s variational method to solve the non-Hertzian, and potentially multiple, wheel–rail contact. The track is described by a linear, three-dimensional and non-periodic finite element model of a railway turnout accounting for the variations in rail cross-sections and sleeper lengths, and including baseplates and resilient elements. To reduce calculation time due to the complexity of the track model, involving a large number of elements and degrees-of-freedom, a complex-valued modal superposition with a truncated mode set is applied before the impulse response functions are calculated at various positions along the crossing panel. The variation in three-dimensional contact geometry of the crossing and wheel is described by linear surface elements. In each time step of the contact detection algorithm, the lateral position of the wheelset centre is prescribed but the contact positions on wheel and rail are not, allowing for an accurate prediction of the wheel transition between wing rail and crossing rail. The method is demonstrated by calculating the wheel–rail impact load and contact stress distribution for a nominal S1002 wheel profile passing over a nominal crossing geometry. A parameter study is performed to determine the influence of vehicle speed, rail pad stiffness, lateral wheelset position and wheel profile on the impact load generated at the crossing. It is shown that the magnitude of the impact load is more influenced the wheel–rail contact geometry than by the selection of rail pad stiffness. © 2017 Published by Elsevier Ltd.
Keywords: Railway crossing Vehicle–track interaction Impact load Green’s functions
1. Introduction Turnouts (switches and crossings, S&C) are critical components of a railway track requiring regular maintenance and generating high life cycle costs. Main drivers for the high maintenance costs are needs to repair and replace switch rails and crossing rails due to plastic deformation, wear and rolling contact fatigue. A wheel passing over the crossing in the facing move (from the switch panel towards the crossing panel) will first make contact with the outwards deviating wing rail, see Fig. 1(a). The wheel–rail contact then moves towards the field side of the wheel profile. For a typical conical wheel profile, the rolling radius decreases and the wheel moves downwards unless the wing rail is elevated. When the wheel reaches and makes contact with the crossing nose, the contact load is transferred from the wing rail to the crossing nose. The rolling radius will then increase as the new contact is closer to the wheel flange. The dynamic vehicle–track interaction typically results in an impact load on the crossing nose as the downward motion of the vertical wheel
* Corresponding author. E-mail address: [email protected] (X. Li).
http://dx.doi.org/10.1016/j.jsv.2017.08.037 0022-460X/© 2017 Published by Elsevier Ltd.
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Fig. 1. (a) Rail configuration at crossing illustrating parts of two wing rails and one crossing nose. The crossing transition takes place at around 47.2 m (the theoretical crossing point, TCP) from front of turnout. From Ref. [2]. (b) Measured vertical wheel–rail contact force during a crossing transition. The total vertical force on the wheel was measured by strain gauges on the wheel disc. Traffic in facing move of the through route at 70 km/h. From Ref. [6].
trajectory is reversed and the wheel is accelerated upwards by the crossing nose, see Fig. 1(b), where the theoretical crossing point (TCP) is at 47.2 m from the front of the turnout [1]. Impact loads with high magnitudes, as caused by non-optimal wheel– rail contact geometry in the transition zone, may lead to severe plastic deformation of the rails and to subsurface-initiated rolling contact fatigue resulting in breakouts of material due to the merging of cracks, see Fig. 2. Particularly severe impact loads are generated if wheel and rail profiles are not correctly maintained as will be demonstrated by the model presented in this paper. The influence of the rail profile and track stiffness on the impact load, wear and rolling contact fatigue in a crossing panel has been studied previously, see Refs. [2–4]. In Refs. [2] and [3], a so-called moving track model was employed to simulate
Fig. 2. Examples of damage in a railway crossing: (a) plastic deformation of the wing rail and (b) rolling contact fatigue of the crossing nose (spalling). Photos published with permission from (a) voestalpine VAE GmbH and (b) Jay Jaiswal, private communication.
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the dynamic behaviour of the track. Such a track model, which includes a simple mass-spring-damper model of the track based on a few degrees-of-freedom (DOFs), does not account for the variation in properties related to bending and torsion of the rails and is typically only valid up to about 20 Hz. This is a severe restriction as field measurements presented in Refs. [5,6] have shown that the impact load at the crossing contains significant contributions at high frequencies (≫20 Hz), see Fig. 1(b). For applications at higher frequencies than can be represented by a moving track model, continuous beam models of the track with lumped parameter supports are typically applied for the simulation of dynamic vehicle–track interaction, see e.g. Refs. [4,7,8]. However, for this type of track model, to account for the continuous variation in dynamic track properties in the crossing panel requires the prescription of a large number of space-variant input data. This problem can be overcome by describing the turnout with a finite element (FE) model. Such FE models of different levels of detail, and using non-Hertzian springs for the wheel–rail contact, have been applied to predict the high-frequency impact load at the crossing [9,10]. Computationally efficient time-domain models can be achieved by using Green’s functions describing the dynamic behaviour of linear time-invariant structures. This approach has previously been utilised for the simulation of vehicle–track dynamics in different frequency ranges [11], for studies of vehicle–track dynamics under the influence of rail irregularities [12,13], for investigation of unstable vibrations generated by vehicle–track interaction [14], and for the prediction of squeal noise [15,16] and rolling contact fatigue damage [17] on standard track. Green’s functions, describing the dynamic behaviour of the track in a moving contact point, are generated by transforming receptance functions (frequency domain) evaluated at several positions along the track to the time domain. This approach is versatile because it can account for parametric excitation due to a vehicle moving on a discretely supported track as well as the excitation by periodic and discrete wheel/rail irregularities. Wheel/track models represented by Green’s functions can be used without changing the mathematical formulation of the interaction model, which makes it possible to account for non-linear contact mechanics in the crossing panel. The magnitude of the wheel–rail impact load in Fig. 1(b) is very sensitive to the wheel transition point, at which the wheel moves between wing rail and crossing rail. The multiple point contact condition shown in Fig. 1(a) (here two-point contact) is an indication of when the wheel transition occurs. For an accurate prediction of the magnitude and position of the timevariant contact force acting in the direction normal to the wheel–rail contact during the transition of the wheel in the crossing panel, a model that accounts for the case of multiple-point and elastic non-Hertzian contact is required. Various categories of approximate solutions for solving multiple-point contact problems exist, see Refs. [18–20]. In Ref. [18], multiple and nonHertzian contact patches are replaced by a set of Hertzian ellipses and then further approximated by a single equivalent ellipse. This method has been shown to be adequate for static stress analysis but less suitable for dynamic simulations [19]. Another type of method presented in Refs. [19] and [20] estimates the contact area from the interpenetration area that is obtained by virtually penetrating the two undeformed surfaces. However, the application of these methods requires that the contact conditions do not deviate much from Hertzian contact conditions [21]. Kalker’s variational method [22] is a preferred alternative for the calculation of non-Hertzian and multiple-point wheel–crossing contact. It deals with the normal and tangential rolling contact problem for arbitrary geometries as long as the half-space assumption is valid, and it provides both the steady-state and transient solutions. On the other hand, the calculation cost is relatively high [23]. In the present paper, a set of Green’s functions are pre-calculated to represent the dynamics of the crossing panel and to account for the motion of the wheel along the crossing. The Green’s functions of the crossing panel are based on an assembly of time-domain impulse response functions, transformed from vertical point and cross receptances calculated in a stationary reference frame using a finite element model of the turnout. For vertical train–track dynamics, a simplified vehicle model including the primary suspension, and a rigid model of one wheel and half of the axle is used. The Green’s function of the vehicle model is equivalent with the impulse response function due to radial excitation at the nominal contact point on the wheel. In the interaction model, the vertical displacement of the rail is calculated by convoluting the vertical contact force with the Green’s function of the rail. In each time step of the contact detection algorithm, the lateral position of the wheelset centre is prescribed but the contact positions on wheel and rail are not, allowing for an accurate prediction of the wheel transition between wing rail and crossing nose. Further, Kalker’s variational method is used for the calculation of non-Hertzian wheel–rail contact. In this paper, the influence of the wheel–rail contact load from the second wheel in the wheelset, making contact with one of the outer stock rails, on the response of the crossing is neglected. 2. Harmonic response analysis of track model based on modal parameters A model of a complete 60E1–760–1:15 turnout (with curve radius 760 m and turnout angle 1:15) has been modelled with finite elements (FE) using the commercial software ABAQUS. The model is linear, three-dimensional and non-periodic, and it was validated in Ref. [24] by comparing calculated responses with field measurements of sleeper strains in a turnout at Eslöv, Sweden. The full description of the model can be found in Ref. [1] but will be shortly summarised below. The turnout model includes the switch panel, closure panel and crossing panel of the turnout, see Fig. 3. The total length of the model is 140 sleeper bays and the nodes at the rail ends are locked in all degrees of freedom (DOFs). The rails, sleepers and baseplates are modelled by Euler-Bernoulli beam elements with DOFs in vertical translation (z) and rotation with respect to y (rails) or x (sleepers and baseplates). The beam cross-sectional properties (equivalent moment of inertia and cross-sectional area) are varied from one element to the next to account for the continuous variation in combined cross-sections of stock rail and switch rail, and of crossing rail and wing rails. There are 40 rail elements in each sleeper bay (discretisation length 0.015 m at sleeper spacing LS = 0.6 m). The variation in sleeper length and sleeper spacing is accounted for. The concrete sleepers have density 2.4 × 103 kg / m3 , cross sectional area 5.0 × 104 mm2 and moment of inertia around the bending axis 2.61 × 108 mm4 .
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Fig. 3. Section of FE model illustrating parts of crossing panel including crossing rail with wing rails, two stock rails and two check rails. BEAM and JOINT stand for different element types in the ABAQUS input.
The length of the 69th to 89th sleeper from the front of the turnout varies from 3731 mm to 4445 mm. The rail pads, baseplate pads and track bed are modelled by discrete, linear spring and viscous damper elements. The rail pads above the 71st to 87th sleepers, where the crossing transition zone lies, have vertical stiffness kp = 120 MN/m and viscous damping cp = 18 kNs/m. The rail pads above the rest of the sleepers in the crossing panel have vertical stiffness kp = 100 MN/m and viscous damping cp = 15 kNs/m. In the crossing panel, the baseplate pad is considered as rigid. The track bed modulus (stiffness per area unit) is taken as 150 MN/m3 . As the FE model is linear, track receptance (displacement over force) is calculated using the procedure of modal decomposition described in Ref. [25]. The equations of motion for the track model with a non-proportional distribution of viscous damping are written as
𝐌𝐳̈ (t) + 𝐂𝐳( ̇ t) + 𝐊𝐳(t) = 𝐐(t)
(1)
where M, C and K are the symmetric mass, viscous damping and stiffness matrices of the track, Q is the vertical wheel load vector and z is the track displacement vector. All matrices/vectors are of size/length N FEM , where N FEM = 40578 is the number of DOFs in the model. The real-valued eigenvectors V of the track model are obtained by solving the associated undamped eigenvalue problem
(𝐊 − 𝜔2 𝐌)𝐕 = 0
(2)
Based on the calculated eigenvectors, Eq. (1) is reduced to a system of N mod coupled second-order differential equations (N mod ≪ N FEM ) written as
𝐕T 𝐌𝐕𝐮̈ + 𝐕T 𝐂𝐕𝐮̇ + 𝐕T 𝐊𝐕𝐮 = 𝐕T 𝐐 = V−1 z.
where u is expressed as
By introducing the state-space vector 𝐝 = {𝐮
⎡𝐅 ⎤ 𝐀𝐝̇ + 𝐁𝐝 = ⎢ ⎥ , ⎢0 ⎥ ⎣ ⎦ ⎡𝐂 𝐀=⎢ ⎢𝐌 ⎣
𝐌 ⎤ ⎥ ⎥ 0 ⎦
(3)
𝐮} ̇ and F = V Q, the system of first-order equations of motion T
T
(4a)
⎡𝐊 𝐁=⎢ ⎢0 ⎣
⎤ ⎥ −𝐌 ⎥⎦
0
(4b,c)
For real and symmetric matrices M, C and K, the solutions of the two adjoint eigenvalue problems coincide and yield N mod pairs of complex-conjugated eigenvectors (with angular eigenfrequency 𝜔(n) and eigenvector 𝝀(n) for the nth eigenvector, complex-valued quantities are indicated by an underbar). Through a new modal transformation, the coupled equations of
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motion become fully decoupled as
⎡𝐅 ⎤ 𝚲T 𝐀𝚲 𝐪̇ + 𝚲T 𝐁𝚲 𝐪 = 𝚲T ⎢ ⎥ , ⎢0 ⎥ ⎣ ⎦
(5a)
⎡ 𝝀(1) 𝚲=⎢ ⎢ i𝜔(1) 𝝀(1) ⎣
(5b)
… …
⎤ ⎥ 2N ⎥ i𝜔(2Nmod ) 𝝀( mod ) ⎦
𝝀(2Nmod )
where 𝚲 is a matrix with 2N mod eigenvectors stored as columns and the modal displacement vector 𝐪 = 𝚲−1 𝐝. The resulting diagonal matrices 𝐚 = 𝚲T 𝐀𝚲 and 𝐛 = 𝚲T 𝐁𝚲 contain the modal normalisation constants, with each diagonal element related to the nth eigenfrequency as −i𝜔(n) a(n) = b(n) . Note that the eigenvalue problem associated with Eq. (1) has here been solved in two steps to speed up the calculation time. 𝐅(𝜔)ei𝜔t , the amplitude ̂ 𝐝 of the physical displacement 𝐝 = ̂ 𝐝(𝜔)ei𝜔t can be calcuAssuming a stationary harmonic load 𝐅 = ̂ lated as
̂ 𝐝=𝚲
[
]
⎡𝐅 ̂ ⎤ 𝚲T ⎢ ⎥ ⎢0 ⎥ i𝐚(𝜔 − 𝝎) ⎣ ⎦ 1
(6)
The complex-valued receptance function ejk , which represents the response in DOF j due to a harmonic force of unit amplitude in DOF k is then obtained as
ejk (𝜔) =
∑
2Nmod
n=1
𝜆(j n) 𝜆(kn) ia(n) (𝜔 − 𝜔(n) )
(7)
In the present case of vertical track dynamics, indices j and k correspond to vertical DOFs at excitation point x and response point x + 𝜒 , where x is the track following coordinate. A convergence study was performed to determine the number of eigenvectors required for an accurate calculation of the receptance. For the present track model, it was concluded that to consider frequencies up to 500 Hz, N mod = 1200 eigenvectors need to be included. Further, for the turnout structure studied here, it was found that 32 sleeper bays (16 sleeper bays, or 10 m, on either side of the crossing) are sufficient to reach a condition at mid-region of the track model where the effects of the boundary conditions at the rail ends are negligible. 3. Time-domain vehicle–track interaction model The description of the applied time-domain model for simulation of vehicle–track interaction in a railway crossing was also presented in Ref. [1], but is repeated here in order to present all features and components of the complete model. 3.1. Vehicle and track models described by Green’s functions A previously validated computational code RAVEN [17] based on the Green’s function methodology in Refs. [11,21] is applied. A brief summary of the procedure is given below. The vertical rail displacement 𝜉 R (t) is calculated by a convolution integral as
𝜉R (t) =
t
∫0
Q (𝜏 )GR,𝑣 (𝑣𝜏, t − 𝜏 )d𝜏
(8)
where Q(t) is the vertical wheel–rail contact force and GR,𝜐 (𝜐𝜏 , t − 𝜏 ) is the Green’s function. For vertical excitation of the rail (index R) at position x = 𝜐𝜏 at time 𝜏 , GR,𝜐 is describing the vertical displacement response of the rail at a point moving at vehicle speed 𝜐 away from the excitation. The Green’s functions are derived by a systematic sampling from a set of inverse Fourier transforms, which are based on point and cross receptances calculated for the track in a stationary reference frame. This assembly procedure is described in Ref. [26]. As the current track model is not periodic, one separate Green’s function needs to be calculated for each excitation position (longitudinal position of the wheel given by the selected time step) along the track. In the current study, the Green’s functions are calculated for a length corresponding to the distance the wheel (train) is travelling at 100 km/h during 0.72 s, starting from 10 m in front of the theoretical crossing point (TCP). Using Matlab, the computation time for generating the Green’s functions for 1000 excitation positions is around 8.5 h on a computer with 32 GB of RAM and Intel Core i7-2700K CPU. Examples of Green’s functions for the rail at three positions in the turnout, at −10 m in front of the crossing where the track structure is close to nominal track, at the crossing where the sleepers are longer and the rail mass is higher, and at 3 m after the crossing where the rail mass is reduced compared to the crossing and the sleepers are longer to accommodate for the two tracks, are illustrated in Fig. 4(a). It is observed that the Green’s function for the crossing (x = 47.2 m)
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Fig. 4. (a) Examples of Green’s functions for the track of a point moving at vehicle speed 100 km/h. Excitation on the rail at 37.2 m, 47.2 m (crossing) and 50.2 m from front of turnout. (b) Green’s function for the wheel.
has a longer wavelength and a lower magnitude compared to the function for the nominal track (x = 37.2 m), indicating a larger mass and higher dynamic stiffness at the crossing due to the rail arrangement and the longer sleepers. The dynamic properties of the track vary throughout the crossing panel due to the variations in rail and sleeper properties. The vehicle is here modelled as a single-DOF system with a rigid mass (MW = 1400 / 2 kg), representing half of the wheelset mass, and a spring-damper system representing the primary suspension (kW = 10.5 kN/mm, cW = 50 kNs/m). To account for the part of the vehicle above the primary suspension, a prescribed displacement corresponding to an axle load of 20 tonnes is included. This simplification is justified for high-frequency vertical wheel–rail interaction because the bogie and car body are isolated from the unsprung mass by the primary suspension [27]. Since the wheel rotation is not accounted for, the Green’s function of the vehicle model is equivalent with the impulse response function of the radial displacement due to radial excitation at the nominal contact point on the wheel, see Fig. 4(b). The vertical wheel displacement 𝜉 W (t) is calculated based on the Green’s function GW (t) as
𝜉W (t) =
t
∫0
Q (𝜏 )GW (t − 𝜏 )d𝜏
(9)
3.2. Contact model To account for the multiple-point and non-Hertzian contacts occurring during the wheel transition from wing rail to crossing rail in the crossing panel, Kalker’s variational method [22] with an active-set algorithm is used. This method was implemented by Pieringer [28] and is based on a discretisation of the Boussinesq-Cerruti integral equations using a set of rectangular elements, in which surface traction in each element is taken as constant. It can be used for arbitrary non-Hertzian contact geometries as long as the half-space assumption is valid. The method formulates the contact problem as a minimisation of an energy potential over a potential wheel–rail contact area P moving in the longitudinal direction x at vehicle speed. The potential contact area, which is larger than the actual contact area, is divided into N P rectangular elements with side lengths Δx and Δy in the longitudinal (x) and lateral directions (y), respectively. For element I in the potential contact area, the contact condition is formulated as
dIz pIz = 0 subjected to
dIz ≥ 0;
pIz ≥ 0
(10)
where dIz is the distance between the two deformed bodies (wheel and rail) and pIz is the constant contact pressure in the Ith element. The elements in the potential contact area are divided into two sets: the in-contact set C (dIz = 0, pIz ≥ 0) and the active set O (dIz ≥ 0, pIz = 0), where C ⊂ P , O ⊂ P and C ∪ O = P. Negative values of pIz or dIz are not allowed. To determine the wheel–rail contact area (the elements that are in contact) and the contact pressure distribution, an iterative procedure is applied at each time step using the two inequality conditions. The kinematic constraint of the deformed wheel and rail at time step t gives
dIz = 𝜉R − 𝜉W + uIz + zR,I − zW,I
(11)
where zR,I and zW,I are the rail and wheel profiles, 𝜉 R and 𝜉 W are the global vertical displacements of rail and wheel calculated by Eq. (8) and Eq. (9). The local vertical displacement uIz (the displacement difference between rail and wheel) is obtained by
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summing the contributions due to the contact pressure on each loaded element J as
uIz =
NP ∑
AIzJz pJz ,
I ∈ P,
J∈P
(12)
J=1
The elements AIzJz of the influence coefficient matrix (piecewise linear) are given in Kalker [22] for the case of elastic half-spaces (the smallest radius of wheel and rail curvature in contact is significantly larger than the largest semi-axis of the contact area) and semi-identical contact bodies (same material). The element AIzJz of the influence matrix gives the vertical displacement at the centre of element I due to a unit normal pressure in element J. Although for contact on the crossing rail, the smallest radius of rail curvature is similar to the largest semi-axis of the contact area, it was found in Refs. [29,30] that the half-space assumption can be used for crossing rail contact as long as no plastification of the material occurs. Summation of contact pressure from all elements in the calculated contact area yields the total vertical contact force Q at time step t as
Q=
NP ∑
pIz ΔxΔy
(13)
I=1
4. Results and discussion The presented method is demonstrated for a 60E1–760–1:15 turnout subjected to freight traffic (axle load 20 tonnes) in the facing move of the through route. The track geometry is assumed to be nominal and no track irregularity except the variation in rail profile is considered in the example. The three-dimensional geometry of the rail surface is linearly interpolated from rail profiles given at a discretisation distance of 50 mm. The vertical wheel trajectory, wheel–rail impact load and contact stress distribution are first presented for a reference case with a nominal S1002 wheel profile passing over the nominal crossing geometry at vehicle speed 100 km/h. For simulating the distance covered by the vehicle model during 0.72 s at 100 km/h, the simulation time for the vehicle–track interaction problem is about 1 h on a computer with 32 GB of RAM and Interl Core i72700K CPU. In the reference case, the average rail pad stiffness kp = 120 kN/mm and the lateral position of the wheelset centre is prescribed to zero, implying that the wheelset centre is aligned with the track centre of the through route. Parameter studies are then performed to determine the influence of vehicle speed, rail pad stiffness, wheel profile and lateral wheelset position on the impact load generated at the crossing. 4.1. Reference case The calculated vertical wheel trajectory and vertical wheel–rail contact force for the reference case are shown in Fig. 5(a) and (b). In front of the crossing (up to 41.5 m from front of turnout), a parametric excitation caused by the variation in track stiffness due to the discrete sleeper supports is observed. From 41.5 m from front of turnout, the wheel will experience an upward displacement due to the increase in stiffness in the crossing panel. In the facing move, the wheel first makes contact with the outwards deviating wing rail and the contact moves towards the field side of the wheel profile. As discussed in Section 1, the rolling radius decreases and the wheel moves downwards, as can be seen at x = 47 m in Fig. 5(a). When the wheel reaches and makes contact with the crossing rail, the contact load is transferred from the wing rail to the crossing nose. The rolling radius
Fig. 5. (a) Vertical wheel trajectory and (b) summed vertical wheel–rail contact force in the crossing panel. Traffic in facing move of through route at vehicle speed 100 km/h. Nominal S1002 wheel profile on nominal wing rail and crossing rail profiles.
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then increases as the new contact point is close to the wheel flange. The dip in wheel trajectory during the crossing transition seen in Fig. 5(a) agrees with findings in literature [3,8,9,31]. The impact load is observed right after the TCP at 47.2 m from front of turnout, shortly after the position for reversal in vertical wheel trajectory when the wheel is accelerated upwards by the crossing rail. The calculated magnitude of the impact load is in acceptable agreement with the measurements reported in Refs. [5,6]. Apart from the impact load, a transient response is observed throughout the crossing transition (cf. the high-frequency contents in the force before and after the impact in Fig. 1(b)). The wheel–rail contact geometry, contact positions and distributions of contact pressure at four different locations in the crossing are illustrated in Fig. 6. At x = 47.05 m, the wheel is in contact with the wing rail. For the given wheel–rail profiles, the conformal contact geometry leads to a relatively low contact pressure distributed over a contact area that is elongated in the lateral direction. At x = 47.17 m, the wheel makes contact with both the wing rail and the crossing rail, thus a two-point
Fig. 6. Illustration of contact geometry and contact pressure between wheel and rail during wheel transition from wing rail to crossing rail. (a) Vertical wheel–rail contact force (circles mark selected points in the time history). Contact geometries shown for contact at (b) x = 47.05 m, (c) x = 47.17 m, (d) x = 47.22 m and (e) x = 47.36 m from front of turnout. Vertical wheel–rail contact pressure shown for contacts at (f) x = 47.05 m and (g) x = 47.36 m.
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contact can be observed. The vertical contact pressure on the crossing rail is much higher than on the wing rail because of the small contact area. This corresponds to the position where the local minimum in vertical wheel–rail contact force is observed. At x = 47.22 m, the wheel has transferred to the crossing rail and the high contact pressure is distributed over a thin section on the crossing rail. This corresponds to the position where the local maximum in impact load is observed. Finally, at x = 47.36 m, the wheel continues to roll on the crossing rail developing a non-Hertzian distribution of contact pressure. It is pointed out in Ref. [30] that plastification of the rail material may occur under such high contact pressure and that a finite element model accounting for non-linear material properties of wheel and rail may need to be included to accurately simulate the contact. 4.2. Parameter study A parameter study is performed to determine the influence of vehicle speed, rail pad stiffness, wheel profile and lateral wheelset position on the impact load generated at the crossing. First, the influence of vehicle speed is investigated. The calculated vertical wheel–rail contact force is shown in Fig. 7. As expected, it is observed that the magnitude of the wheel–rail impact load at the crossing increases with increasing vehicle speed. This is in agreement with the field measurements reported in Ref. [6]. Next, the influence of rail pad stiffness (and associated rail pad damping, here taken as proportional to pad stiffness) is investigated. The four studied values of rail pad stiffness are kp = 60 kN/mm, kp = 120 kN/mm kp = 240 kN/mm and kp = 480 kN/mm. In all four cases, results have been calculated for vehicle speed 100 km/h. The calculated static track stiffness for a stationary load applied on the crossing rail at different distances from front of turnout is increasing with increasing pad stiffness, see Fig. 8(a). Further, the influence of pad stiffness on the Green’s function for the track due to excitation on the crossing rail are shown in Fig. 8(b). It is seen that the difference in Green’s functions evaluated for kp = 240 kN/mm and kp = 480 kN/mm is small. This is because when the rail pad stiffness is much higher than the ballast stiffness, the impulse response of the track is dominated by the ballast properties. Fig. 8(c) shows the calculated vertical wheel–rail contact force. The magnitude of impact load at the crossing is larger for kp = 240 kN/mm than for kp = 120 kN/mm. The impact load magnitudes calculated for kp = 240 kN/mm and kp = 480 kN/mm are similar due to the similarity in dynamic track response. Note that similar impact loads are also evaluated for kp = 60 kN/mm and kp = 120 kN/mm. The influence of wheel–rail contact geometry is studied by varying the prescribed lateral position of the wheelset centre and by changing the nominal S1002 wheel profiles to two different hollow worn wheel profiles. In all of the cases, results have been calculated for vehicle speed 100 km/h and rail pad stiffness 120 kN/mm. In Fig. 9, the results of simulations for three prescribed lateral wheelset positions Δy are shown. A negative value of Δy corresponds to a lateral displacement of the wheel towards the track centre. This means that the wheel flange is moved away from the crossing. As observed in Fig. 9(a), the dip in the vertical wheel trajectory increases with decreasing Δy because the contact between wheel and wing rail is taking place further out on the field side. For an opposite prescribed wheelset position, the wheel makes contact with the crossing at a position closer to the wheel flange leading to an earlier transition to the crossing rail and a smaller dip in vertical wheel trajectory. In the current study, as flange contact does not occur for Δy = 2 mm, the smooth transition leads to a lower wheel–rail impact load on the crossing compared to when the wheel is further away from flange contact, see Fig. 9(b). When Δy > 2 mm (not shown here), flange contact between wheel and crossing will appear ahead of the TCP, causing an earlier transition and a higher impact load. For large positive values of Δy, loss of wheel–rail contact has been observed. Similar findings are found in Refs. [8,31]. In Fig. 10, the results from simulations with wheel profiles with two different magnitudes (2 mm and 4 mm) of hollow wear are compared with the results for S1002. The three wheel profiles are compared in Fig. 10(a). As can be observed in Fig. 10(b),
Fig. 7. Influence of vehicle speed on vertical wheel–rail contact force.
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Fig. 8. Influence of rail pad stiffness on (a) static track stiffness along inner rail of through route, (b) Green’s functions for the track of a point moving at vehicle speed 100 km/h. Excitation on the rail at 47.2 m (crossing) from front of turnout and (c) vertical wheel–rail contact force. – · – kp = 60 kN∕mm, kp = 120 kN/mm, – – – kp = 240 kN/mm, … kp = 480 kN∕mm
Fig. 9. Influence of prescribed lateral wheelset position Δy on vertical (a) wheel trajectory and (b) wheel–rail contact force.
the vertical position of the two hollow worn wheels is lifted upwards during contact with the wing rail as the rolling radius of these wheels is increased when rolling on the ‘false flange’ at the field side of the wheel tread. The wheel lift continues along the contact with the wing rail until the wheel falls down on the crossing. The hollow wear leads to that these wheels make a significantly later contact with the crossing rail than the nominal S1002 wheel profile. The wheel profile with the most hollow wear (hollow worn 4 mm) generated the largest magnitude wheel–rail impact load at the crossing, see Fig. 10(c). These conclusions are in agreement with the study presented in Ref. [2].
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Fig. 10. (a) Three different wheel profiles used in the study. Influence of wheel profile on vertical (b) wheel trajectory and (c) wheel–rail contact force.
5. Conclusions A versatile and computationally efficient method for the simulation of high-frequency dynamic vehicle–track interaction in a railway crossing has been presented. Based on linear and time-invariant finite element models, the dynamic properties of wheelset and crossing panel are represented by Green’s functions (impulse response functions). Since the structural properties of the crossing panel can be described in greater detail, this approach offers a significant benefit compared to so-called moving or lumped parameter track models traditionally being used in vehicle–track interaction software. The three-dimensional surface geometries of crossing and wheel are described by four-noded linear elements. In each time-step of the simulation, vertical non-Hertzian (potentially multiple) wheel–rail contact is solved by an implementation of Kalker’s variational method. The method was demonstrated for a crossing subjected to freight traffic in the facing move of the through route. It was shown that the transition point from wing rail to crossing rail can be predicted with good precision. Parameter studies were performed to determine the influence of vehicle speed, rail pad stiffness, lateral wheelset position and wheel profile on the wheel–rail impact load generated at the crossing. It can be argued that an implementation of resilient (low stiffness) rail pads in the crossing panel offers an attractive approach to reduce impact loads as long as rail bending stresses are below the fatigue limit. However, it can also be concluded that wheel–rail contact geometry, as would be influenced by crossing rail and wing rail design, check rail gap width, and wear and plastic deformation of wheel and rail profiles (not studied here with the exception of wheel profile), seem to influence impact loads more than the selection of rail pad stiffness. In future work, the presented method will be implemented in an iterative calculation scheme to predict long-term differential track settlement in the crossing panel [32]. The influence of modelling the wheelset based on a lumped parameter or a finite element model on the impact load will also be investigated.
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Acknowledgements This work was performed as part of the activities within the Centre of Excellence CHARMEC (Chalmers Railway Mechanics), see www.charmec.chalmers.se. Discussions with Dr Björn Pålsson and his sharing of input data to the track model are gratefully acknowledged. This work was partly funded by Trafikverket (the Swedish National Traffic Administration), the turnout manufacturers VAE GmbH and Vossloh Nordic Switch Systems AB, and the sleeper manufacturer Abetong AB. References [1] X. Li, P. Torstensson, J.C.O. Nielsen, Vertical dynamic vehicle–track interaction in a railway crossing predicted by moving Green’s functions, in: Proceedings of the 24th Symposium of the International Association for Vehicle System Dynamics – the Dynamics of Vehicles on Roads and Tracks, 2016, pp. 1319–1327. [2] B.A. Pålsson, Optimisation of railway crossing geometry considering a representative set of wheel profiles, Veh. Syst. Dyn. 53 (2) (2015) 274–301. [3] C. Wan, V. Markine, I. Shevtsov, Improvement of vehicle–turnout interaction by optimising the shape of crossing nose, Veh. Syst. Dyn. 52 (11) (2014) 1517–1540, http://dx.doi.org/10.1080/00423114.2014.944870. [4] I. Grossoni, Y. Bezin, S. Neves, Optimization of support stiffness at a railway crossing panel, Civil-Comp. Proc. 110. [5] E. Kassa, J.C.O. Nielsen, Dynamic interaction between train and railway turnout: full-scale field test and validation of simulation models, Veh. Syst. Dyn. 46 (suppl 1) (2008) 521–534. [6] B.A. Pålsson, J.C.O. Nielsen, Dynamic vehicle–track interaction in switches and crossings and the influence of rail pad stiffness – field measurements and validation of a simulation model, Veh. Syst. Dyn. 53 (6) (2015) 734–755, http://dx.doi.org/10.1080/00423114.2015.1012213. [7] J.C.O. Nielsen, A. Igeland, Vertical dynamic interaction between train and track – influence of wheel and track imperfections, J. Sound Vib. 187 (5) (1995) 825–839, http://dx.doi.org/10.1006/jsvi.1995.0566. [8] Y. Bezin, I. Grossoni, A. Alonso, The assessment of system maintenance and design conditions on railway crossing performance, Civil-Comp. Proceed. 104. [9] C. Andersson, T. Dahlberg, Wheel/rail impacts at a railway turnout crossing, In: Proceedings of the Institution of Mechanical Engineers, Part F, J. Rail Rapid Transit vol. 212 (2) (1998) 123–134. [10] S. Alfi, S. Bruni, Mathematical modelling of train–turnout interaction, Veh. Syst. Dyn. 47 (5) (2009) 551–574, http://dx.doi.org/10.1080/ 00423110802245015. [11] A. Nordborg, Wheel/rail noise generation due to nonlinear effects and parametric excitation, J. Acoust. Soc. Am. 111 (4) (2002) 1772–1781, http://dx.doi. org/10.1121/1.1459463. [12] T. Mazilu, Green’s functions for analysis of dynamic response of wheel/rail to vertical excitation, J. Sound Vib. 306 (1–2) (2007) 31–58, http://dx.doi.org/ 10.1016/j.jsv.2007.05.037. [13] T. Mazilu, Interaction between moving tandem wheels and an infinite rail with periodic supports - green’s matrices of the track method in stationary reference frame, J. Sound Vib. 401 (2017) 233–254. [14] T. Mazilu, Instability of a train of oscillators moving along a beam on a viscoelastic foundation, J. Sound Vib. 332 (19) (2013) 4597–4619, http://dx.doi.org/ 10.1016/j.jsv.2013.03.022 cited By 5. [15] A. Pieringer, A numerical investigation of curve squeal in the case of constant wheel/rail friction, J. Sound Vib. 333 (18) (2014) 4295–4313. [16] I. Zenzerovic, W. Kropp, A. Pieringer, An engineering time-domain model for curve squeal: tangential point-contact model and Green’s functions approach, J. Sound Vib. 376 (2016) 149–165. [17] R. Andersson, P.T. Torstensson, E. Kabo, F. Larsson, An efficient approach to the analysis of rail surface irregularities accounting for dynamic train–track interaction and inelastic deformations, Veh. Syst. Dyn. 53 (11) (2015) 1667–1685, http://dx.doi.org/10.1080/00423114.2015.1081701. [18] J.P. Pascal, G. Sauvage, Available methods to calculate the wheel/rail forces in non-hertzian contact patches and rail damaging, Veh. Syst. Dyn. 22 (3–4) (1993) 263–275. [19] J. Piotrowski, W. Kik, A simplified model of wheel/rail contact mechanics for non-Hertzian problems and its application in rail vehicle dynamics, Veh. Syst. Dyn. 46 (1–2) (2008) 27–48. [20] J.B. Ayasse, H. Chollet, Determination of the wheel rail contact patch in semi-Hertzian conditions, Veh. Syst. Dyn. 43 (3) (2005) 161–172. [21] A. Pieringer, Time-domain Modelling of High-frequency Wheel/rail Interaction, Ph.D. thesis, Department of Civil and Environmental Engineering, Division of Applied Acoustics, Vibroacoustic Group, Gothenburg, Sweden, 2011. [22] J.J. Kalker, Contact mechanical algorithms, Commun. Appl. Numer. Methods 4 (1) (1988) 25–32. [23] K. Knothe, R. Wille, W. Zastrau, Advanced contact mechanics – road and rail, Veh. Syst. Dyn. 35 (4–5) (2001) 361–407. [24] R. Bolmsvik, J.C.O. Nielsen, P. Kron, B.A. Pålsson, Switch Sleeper Specification, Tech. Rep. 2010-03, Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden, 2010. [25] T.J.S. Abrahamsson, Modal Analysis and Synthesis in Transient Vibration and Structural Optimization Problems, Ph.D. thesis, Department of Solid Mechanics, Chalmers University of Technology, Gothenburg, Sweden, 1990. [26] P. Andersson, Modelling Interfacial Details in Tyre/road Contact : Adhesion Forces and Non-linear Contact Stiffness, Ph.D. thesis, Department of Civil and Environmental Engineering, Division of Applied Acoustics, Vibroacoustic Group, Gothenburg, Sweden, 2005. [27] K. Knothe, S.L. Grassie, Modelling of railway track and vehicle/track interaction at high frequencies, Veh. Syst. Dyn. 22 (3–4) (1993) 209–262. [28] A. Pieringer, W.A. Kropp, D.J. Thompson, Investigation of the dynamic contact filter effect in vertical wheel/rail interaction using a 2D and a 3D non-Hertzian contact model, Wear 271 (1–2) (2011) 328–338. [29] W. Yan, F.D. Fischer, Applicability of the Hertz contact theory to rail-wheel contact problems, Archive Appl. Mech. 70 (4) (2000) 255–268. [30] M. Wiest, E. Kassa, W. Daves, J.C.O. Nielsen, H. Ossberger, Assessment of methods for calculating contact pressure in wheel–rail/switch contact, Wear 265 (9–10) (2008) 1439–1445, http://dx.doi.org/10.1016/j.wear.2008.02.039. [31] B.A. Pålsson, J.C.O. Nielsen, Wheel–rail interaction and damage in switches and crossings, Veh. Syst. Dyn. 50 (1) (2012) 43–58. [32] X. Li, J.C.O. Nielsen, B.A. Pålsson, Simulation of track settlement in railway turnouts, Veh. Syst. Dyn. 52 (supp1) (2014) 421–439, http://dx.doi.org/10.1080/ 00423114.2014.904905.
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Simulation of vertical dynamic vehicle–track interaction in a railway crossing using Green’s functions X. Li * , P.T. Torstensson, J.C.O. Nielsen Department of Applied Mechanics/CHARMEC, Chalmers University of Technology, SE-412 96, Gothenburg, Sweden
article info
abstract
Article history: Received 29 November 2016 Revised 11 August 2017 Accepted 16 August 2017 Available online XXX
Vertical dynamic vehicle–track interaction in the through route of a railway crossing is simulated in the time domain based on a Green’s function approach for the track in combination with an implementation of Kalker’s variational method to solve the non-Hertzian, and potentially multiple, wheel–rail contact. The track is described by a linear, three-dimensional and non-periodic finite element model of a railway turnout accounting for the variations in rail cross-sections and sleeper lengths, and including baseplates and resilient elements. To reduce calculation time due to the complexity of the track model, involving a large number of elements and degrees-of-freedom, a complex-valued modal superposition with a truncated mode set is applied before the impulse response functions are calculated at various positions along the crossing panel. The variation in three-dimensional contact geometry of the crossing and wheel is described by linear surface elements. In each time step of the contact detection algorithm, the lateral position of the wheelset centre is prescribed but the contact positions on wheel and rail are not, allowing for an accurate prediction of the wheel transition between wing rail and crossing rail. The method is demonstrated by calculating the wheel–rail impact load and contact stress distribution for a nominal S1002 wheel profile passing over a nominal crossing geometry. A parameter study is performed to determine the influence of vehicle speed, rail pad stiffness, lateral wheelset position and wheel profile on the impact load generated at the crossing. It is shown that the magnitude of the impact load is more influenced the wheel–rail contact geometry than by the selection of rail pad stiffness. © 2017 Published by Elsevier Ltd.
Keywords: Railway crossing Vehicle–track interaction Impact load Green’s functions
1. Introduction Turnouts (switches and crossings, S&C) are critical components of a railway track requiring regular maintenance and generating high life cycle costs. Main drivers for the high maintenance costs are needs to repair and replace switch rails and crossing rails due to plastic deformation, wear and rolling contact fatigue. A wheel passing over the crossing in the facing move (from the switch panel towards the crossing panel) will first make contact with the outwards deviating wing rail, see Fig. 1(a). The wheel–rail contact then moves towards the field side of the wheel profile. For a typical conical wheel profile, the rolling radius decreases and the wheel moves downwards unless the wing rail is elevated. When the wheel reaches and makes contact with the crossing nose, the contact load is transferred from the wing rail to the crossing nose. The rolling radius will then increase as the new contact is closer to the wheel flange. The dynamic vehicle–track interaction typically results in an impact load on the crossing nose as the downward motion of the vertical wheel
* Corresponding author. E-mail address: [email protected] (X. Li).
http://dx.doi.org/10.1016/j.jsv.2017.08.037 0022-460X/© 2017 Published by Elsevier Ltd.
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Fig. 1. (a) Rail configuration at crossing illustrating parts of two wing rails and one crossing nose. The crossing transition takes place at around 47.2 m (the theoretical crossing point, TCP) from front of turnout. From Ref. [2]. (b) Measured vertical wheel–rail contact force during a crossing transition. The total vertical force on the wheel was measured by strain gauges on the wheel disc. Traffic in facing move of the through route at 70 km/h. From Ref. [6].
trajectory is reversed and the wheel is accelerated upwards by the crossing nose, see Fig. 1(b), where the theoretical crossing point (TCP) is at 47.2 m from the front of the turnout [1]. Impact loads with high magnitudes, as caused by non-optimal wheel– rail contact geometry in the transition zone, may lead to severe plastic deformation of the rails and to subsurface-initiated rolling contact fatigue resulting in breakouts of material due to the merging of cracks, see Fig. 2. Particularly severe impact loads are generated if wheel and rail profiles are not correctly maintained as will be demonstrated by the model presented in this paper. The influence of the rail profile and track stiffness on the impact load, wear and rolling contact fatigue in a crossing panel has been studied previously, see Refs. [2–4]. In Refs. [2] and [3], a so-called moving track model was employed to simulate
Fig. 2. Examples of damage in a railway crossing: (a) plastic deformation of the wing rail and (b) rolling contact fatigue of the crossing nose (spalling). Photos published with permission from (a) voestalpine VAE GmbH and (b) Jay Jaiswal, private communication.
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the dynamic behaviour of the track. Such a track model, which includes a simple mass-spring-damper model of the track based on a few degrees-of-freedom (DOFs), does not account for the variation in properties related to bending and torsion of the rails and is typically only valid up to about 20 Hz. This is a severe restriction as field measurements presented in Refs. [5,6] have shown that the impact load at the crossing contains significant contributions at high frequencies (≫20 Hz), see Fig. 1(b). For applications at higher frequencies than can be represented by a moving track model, continuous beam models of the track with lumped parameter supports are typically applied for the simulation of dynamic vehicle–track interaction, see e.g. Refs. [4,7,8]. However, for this type of track model, to account for the continuous variation in dynamic track properties in the crossing panel requires the prescription of a large number of space-variant input data. This problem can be overcome by describing the turnout with a finite element (FE) model. Such FE models of different levels of detail, and using non-Hertzian springs for the wheel–rail contact, have been applied to predict the high-frequency impact load at the crossing [9,10]. Computationally efficient time-domain models can be achieved by using Green’s functions describing the dynamic behaviour of linear time-invariant structures. This approach has previously been utilised for the simulation of vehicle–track dynamics in different frequency ranges [11], for studies of vehicle–track dynamics under the influence of rail irregularities [12,13], for investigation of unstable vibrations generated by vehicle–track interaction [14], and for the prediction of squeal noise [15,16] and rolling contact fatigue damage [17] on standard track. Green’s functions, describing the dynamic behaviour of the track in a moving contact point, are generated by transforming receptance functions (frequency domain) evaluated at several positions along the track to the time domain. This approach is versatile because it can account for parametric excitation due to a vehicle moving on a discretely supported track as well as the excitation by periodic and discrete wheel/rail irregularities. Wheel/track models represented by Green’s functions can be used without changing the mathematical formulation of the interaction model, which makes it possible to account for non-linear contact mechanics in the crossing panel. The magnitude of the wheel–rail impact load in Fig. 1(b) is very sensitive to the wheel transition point, at which the wheel moves between wing rail and crossing rail. The multiple point contact condition shown in Fig. 1(a) (here two-point contact) is an indication of when the wheel transition occurs. For an accurate prediction of the magnitude and position of the timevariant contact force acting in the direction normal to the wheel–rail contact during the transition of the wheel in the crossing panel, a model that accounts for the case of multiple-point and elastic non-Hertzian contact is required. Various categories of approximate solutions for solving multiple-point contact problems exist, see Refs. [18–20]. In Ref. [18], multiple and nonHertzian contact patches are replaced by a set of Hertzian ellipses and then further approximated by a single equivalent ellipse. This method has been shown to be adequate for static stress analysis but less suitable for dynamic simulations [19]. Another type of method presented in Refs. [19] and [20] estimates the contact area from the interpenetration area that is obtained by virtually penetrating the two undeformed surfaces. However, the application of these methods requires that the contact conditions do not deviate much from Hertzian contact conditions [21]. Kalker’s variational method [22] is a preferred alternative for the calculation of non-Hertzian and multiple-point wheel–crossing contact. It deals with the normal and tangential rolling contact problem for arbitrary geometries as long as the half-space assumption is valid, and it provides both the steady-state and transient solutions. On the other hand, the calculation cost is relatively high [23]. In the present paper, a set of Green’s functions are pre-calculated to represent the dynamics of the crossing panel and to account for the motion of the wheel along the crossing. The Green’s functions of the crossing panel are based on an assembly of time-domain impulse response functions, transformed from vertical point and cross receptances calculated in a stationary reference frame using a finite element model of the turnout. For vertical train–track dynamics, a simplified vehicle model including the primary suspension, and a rigid model of one wheel and half of the axle is used. The Green’s function of the vehicle model is equivalent with the impulse response function due to radial excitation at the nominal contact point on the wheel. In the interaction model, the vertical displacement of the rail is calculated by convoluting the vertical contact force with the Green’s function of the rail. In each time step of the contact detection algorithm, the lateral position of the wheelset centre is prescribed but the contact positions on wheel and rail are not, allowing for an accurate prediction of the wheel transition between wing rail and crossing nose. Further, Kalker’s variational method is used for the calculation of non-Hertzian wheel–rail contact. In this paper, the influence of the wheel–rail contact load from the second wheel in the wheelset, making contact with one of the outer stock rails, on the response of the crossing is neglected. 2. Harmonic response analysis of track model based on modal parameters A model of a complete 60E1–760–1:15 turnout (with curve radius 760 m and turnout angle 1:15) has been modelled with finite elements (FE) using the commercial software ABAQUS. The model is linear, three-dimensional and non-periodic, and it was validated in Ref. [24] by comparing calculated responses with field measurements of sleeper strains in a turnout at Eslöv, Sweden. The full description of the model can be found in Ref. [1] but will be shortly summarised below. The turnout model includes the switch panel, closure panel and crossing panel of the turnout, see Fig. 3. The total length of the model is 140 sleeper bays and the nodes at the rail ends are locked in all degrees of freedom (DOFs). The rails, sleepers and baseplates are modelled by Euler-Bernoulli beam elements with DOFs in vertical translation (z) and rotation with respect to y (rails) or x (sleepers and baseplates). The beam cross-sectional properties (equivalent moment of inertia and cross-sectional area) are varied from one element to the next to account for the continuous variation in combined cross-sections of stock rail and switch rail, and of crossing rail and wing rails. There are 40 rail elements in each sleeper bay (discretisation length 0.015 m at sleeper spacing LS = 0.6 m). The variation in sleeper length and sleeper spacing is accounted for. The concrete sleepers have density 2.4 × 103 kg / m3 , cross sectional area 5.0 × 104 mm2 and moment of inertia around the bending axis 2.61 × 108 mm4 .
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Fig. 3. Section of FE model illustrating parts of crossing panel including crossing rail with wing rails, two stock rails and two check rails. BEAM and JOINT stand for different element types in the ABAQUS input.
The length of the 69th to 89th sleeper from the front of the turnout varies from 3731 mm to 4445 mm. The rail pads, baseplate pads and track bed are modelled by discrete, linear spring and viscous damper elements. The rail pads above the 71st to 87th sleepers, where the crossing transition zone lies, have vertical stiffness kp = 120 MN/m and viscous damping cp = 18 kNs/m. The rail pads above the rest of the sleepers in the crossing panel have vertical stiffness kp = 100 MN/m and viscous damping cp = 15 kNs/m. In the crossing panel, the baseplate pad is considered as rigid. The track bed modulus (stiffness per area unit) is taken as 150 MN/m3 . As the FE model is linear, track receptance (displacement over force) is calculated using the procedure of modal decomposition described in Ref. [25]. The equations of motion for the track model with a non-proportional distribution of viscous damping are written as
𝐌𝐳̈ (t) + 𝐂𝐳( ̇ t) + 𝐊𝐳(t) = 𝐐(t)
(1)
where M, C and K are the symmetric mass, viscous damping and stiffness matrices of the track, Q is the vertical wheel load vector and z is the track displacement vector. All matrices/vectors are of size/length N FEM , where N FEM = 40578 is the number of DOFs in the model. The real-valued eigenvectors V of the track model are obtained by solving the associated undamped eigenvalue problem
(𝐊 − 𝜔2 𝐌)𝐕 = 0
(2)
Based on the calculated eigenvectors, Eq. (1) is reduced to a system of N mod coupled second-order differential equations (N mod ≪ N FEM ) written as
𝐕T 𝐌𝐕𝐮̈ + 𝐕T 𝐂𝐕𝐮̇ + 𝐕T 𝐊𝐕𝐮 = 𝐕T 𝐐 = V−1 z.
where u is expressed as
By introducing the state-space vector 𝐝 = {𝐮
⎡𝐅 ⎤ 𝐀𝐝̇ + 𝐁𝐝 = ⎢ ⎥ , ⎢0 ⎥ ⎣ ⎦ ⎡𝐂 𝐀=⎢ ⎢𝐌 ⎣
𝐌 ⎤ ⎥ ⎥ 0 ⎦
(3)
𝐮} ̇ and F = V Q, the system of first-order equations of motion T
T
(4a)
⎡𝐊 𝐁=⎢ ⎢0 ⎣
⎤ ⎥ −𝐌 ⎥⎦
0
(4b,c)
For real and symmetric matrices M, C and K, the solutions of the two adjoint eigenvalue problems coincide and yield N mod pairs of complex-conjugated eigenvectors (with angular eigenfrequency 𝜔(n) and eigenvector 𝝀(n) for the nth eigenvector, complex-valued quantities are indicated by an underbar). Through a new modal transformation, the coupled equations of
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motion become fully decoupled as
⎡𝐅 ⎤ 𝚲T 𝐀𝚲 𝐪̇ + 𝚲T 𝐁𝚲 𝐪 = 𝚲T ⎢ ⎥ , ⎢0 ⎥ ⎣ ⎦
(5a)
⎡ 𝝀(1) 𝚲=⎢ ⎢ i𝜔(1) 𝝀(1) ⎣
(5b)
… …
⎤ ⎥ 2N ⎥ i𝜔(2Nmod ) 𝝀( mod ) ⎦
𝝀(2Nmod )
where 𝚲 is a matrix with 2N mod eigenvectors stored as columns and the modal displacement vector 𝐪 = 𝚲−1 𝐝. The resulting diagonal matrices 𝐚 = 𝚲T 𝐀𝚲 and 𝐛 = 𝚲T 𝐁𝚲 contain the modal normalisation constants, with each diagonal element related to the nth eigenfrequency as −i𝜔(n) a(n) = b(n) . Note that the eigenvalue problem associated with Eq. (1) has here been solved in two steps to speed up the calculation time. 𝐅(𝜔)ei𝜔t , the amplitude ̂ 𝐝 of the physical displacement 𝐝 = ̂ 𝐝(𝜔)ei𝜔t can be calcuAssuming a stationary harmonic load 𝐅 = ̂ lated as
̂ 𝐝=𝚲
[
]
⎡𝐅 ̂ ⎤ 𝚲T ⎢ ⎥ ⎢0 ⎥ i𝐚(𝜔 − 𝝎) ⎣ ⎦ 1
(6)
The complex-valued receptance function ejk , which represents the response in DOF j due to a harmonic force of unit amplitude in DOF k is then obtained as
ejk (𝜔) =
∑
2Nmod
n=1
𝜆(j n) 𝜆(kn) ia(n) (𝜔 − 𝜔(n) )
(7)
In the present case of vertical track dynamics, indices j and k correspond to vertical DOFs at excitation point x and response point x + 𝜒 , where x is the track following coordinate. A convergence study was performed to determine the number of eigenvectors required for an accurate calculation of the receptance. For the present track model, it was concluded that to consider frequencies up to 500 Hz, N mod = 1200 eigenvectors need to be included. Further, for the turnout structure studied here, it was found that 32 sleeper bays (16 sleeper bays, or 10 m, on either side of the crossing) are sufficient to reach a condition at mid-region of the track model where the effects of the boundary conditions at the rail ends are negligible. 3. Time-domain vehicle–track interaction model The description of the applied time-domain model for simulation of vehicle–track interaction in a railway crossing was also presented in Ref. [1], but is repeated here in order to present all features and components of the complete model. 3.1. Vehicle and track models described by Green’s functions A previously validated computational code RAVEN [17] based on the Green’s function methodology in Refs. [11,21] is applied. A brief summary of the procedure is given below. The vertical rail displacement 𝜉 R (t) is calculated by a convolution integral as
𝜉R (t) =
t
∫0
Q (𝜏 )GR,𝑣 (𝑣𝜏, t − 𝜏 )d𝜏
(8)
where Q(t) is the vertical wheel–rail contact force and GR,𝜐 (𝜐𝜏 , t − 𝜏 ) is the Green’s function. For vertical excitation of the rail (index R) at position x = 𝜐𝜏 at time 𝜏 , GR,𝜐 is describing the vertical displacement response of the rail at a point moving at vehicle speed 𝜐 away from the excitation. The Green’s functions are derived by a systematic sampling from a set of inverse Fourier transforms, which are based on point and cross receptances calculated for the track in a stationary reference frame. This assembly procedure is described in Ref. [26]. As the current track model is not periodic, one separate Green’s function needs to be calculated for each excitation position (longitudinal position of the wheel given by the selected time step) along the track. In the current study, the Green’s functions are calculated for a length corresponding to the distance the wheel (train) is travelling at 100 km/h during 0.72 s, starting from 10 m in front of the theoretical crossing point (TCP). Using Matlab, the computation time for generating the Green’s functions for 1000 excitation positions is around 8.5 h on a computer with 32 GB of RAM and Intel Core i7-2700K CPU. Examples of Green’s functions for the rail at three positions in the turnout, at −10 m in front of the crossing where the track structure is close to nominal track, at the crossing where the sleepers are longer and the rail mass is higher, and at 3 m after the crossing where the rail mass is reduced compared to the crossing and the sleepers are longer to accommodate for the two tracks, are illustrated in Fig. 4(a). It is observed that the Green’s function for the crossing (x = 47.2 m)
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Fig. 4. (a) Examples of Green’s functions for the track of a point moving at vehicle speed 100 km/h. Excitation on the rail at 37.2 m, 47.2 m (crossing) and 50.2 m from front of turnout. (b) Green’s function for the wheel.
has a longer wavelength and a lower magnitude compared to the function for the nominal track (x = 37.2 m), indicating a larger mass and higher dynamic stiffness at the crossing due to the rail arrangement and the longer sleepers. The dynamic properties of the track vary throughout the crossing panel due to the variations in rail and sleeper properties. The vehicle is here modelled as a single-DOF system with a rigid mass (MW = 1400 / 2 kg), representing half of the wheelset mass, and a spring-damper system representing the primary suspension (kW = 10.5 kN/mm, cW = 50 kNs/m). To account for the part of the vehicle above the primary suspension, a prescribed displacement corresponding to an axle load of 20 tonnes is included. This simplification is justified for high-frequency vertical wheel–rail interaction because the bogie and car body are isolated from the unsprung mass by the primary suspension [27]. Since the wheel rotation is not accounted for, the Green’s function of the vehicle model is equivalent with the impulse response function of the radial displacement due to radial excitation at the nominal contact point on the wheel, see Fig. 4(b). The vertical wheel displacement 𝜉 W (t) is calculated based on the Green’s function GW (t) as
𝜉W (t) =
t
∫0
Q (𝜏 )GW (t − 𝜏 )d𝜏
(9)
3.2. Contact model To account for the multiple-point and non-Hertzian contacts occurring during the wheel transition from wing rail to crossing rail in the crossing panel, Kalker’s variational method [22] with an active-set algorithm is used. This method was implemented by Pieringer [28] and is based on a discretisation of the Boussinesq-Cerruti integral equations using a set of rectangular elements, in which surface traction in each element is taken as constant. It can be used for arbitrary non-Hertzian contact geometries as long as the half-space assumption is valid. The method formulates the contact problem as a minimisation of an energy potential over a potential wheel–rail contact area P moving in the longitudinal direction x at vehicle speed. The potential contact area, which is larger than the actual contact area, is divided into N P rectangular elements with side lengths Δx and Δy in the longitudinal (x) and lateral directions (y), respectively. For element I in the potential contact area, the contact condition is formulated as
dIz pIz = 0 subjected to
dIz ≥ 0;
pIz ≥ 0
(10)
where dIz is the distance between the two deformed bodies (wheel and rail) and pIz is the constant contact pressure in the Ith element. The elements in the potential contact area are divided into two sets: the in-contact set C (dIz = 0, pIz ≥ 0) and the active set O (dIz ≥ 0, pIz = 0), where C ⊂ P , O ⊂ P and C ∪ O = P. Negative values of pIz or dIz are not allowed. To determine the wheel–rail contact area (the elements that are in contact) and the contact pressure distribution, an iterative procedure is applied at each time step using the two inequality conditions. The kinematic constraint of the deformed wheel and rail at time step t gives
dIz = 𝜉R − 𝜉W + uIz + zR,I − zW,I
(11)
where zR,I and zW,I are the rail and wheel profiles, 𝜉 R and 𝜉 W are the global vertical displacements of rail and wheel calculated by Eq. (8) and Eq. (9). The local vertical displacement uIz (the displacement difference between rail and wheel) is obtained by
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summing the contributions due to the contact pressure on each loaded element J as
uIz =
NP ∑
AIzJz pJz ,
I ∈ P,
J∈P
(12)
J=1
The elements AIzJz of the influence coefficient matrix (piecewise linear) are given in Kalker [22] for the case of elastic half-spaces (the smallest radius of wheel and rail curvature in contact is significantly larger than the largest semi-axis of the contact area) and semi-identical contact bodies (same material). The element AIzJz of the influence matrix gives the vertical displacement at the centre of element I due to a unit normal pressure in element J. Although for contact on the crossing rail, the smallest radius of rail curvature is similar to the largest semi-axis of the contact area, it was found in Refs. [29,30] that the half-space assumption can be used for crossing rail contact as long as no plastification of the material occurs. Summation of contact pressure from all elements in the calculated contact area yields the total vertical contact force Q at time step t as
Q=
NP ∑
pIz ΔxΔy
(13)
I=1
4. Results and discussion The presented method is demonstrated for a 60E1–760–1:15 turnout subjected to freight traffic (axle load 20 tonnes) in the facing move of the through route. The track geometry is assumed to be nominal and no track irregularity except the variation in rail profile is considered in the example. The three-dimensional geometry of the rail surface is linearly interpolated from rail profiles given at a discretisation distance of 50 mm. The vertical wheel trajectory, wheel–rail impact load and contact stress distribution are first presented for a reference case with a nominal S1002 wheel profile passing over the nominal crossing geometry at vehicle speed 100 km/h. For simulating the distance covered by the vehicle model during 0.72 s at 100 km/h, the simulation time for the vehicle–track interaction problem is about 1 h on a computer with 32 GB of RAM and Interl Core i72700K CPU. In the reference case, the average rail pad stiffness kp = 120 kN/mm and the lateral position of the wheelset centre is prescribed to zero, implying that the wheelset centre is aligned with the track centre of the through route. Parameter studies are then performed to determine the influence of vehicle speed, rail pad stiffness, wheel profile and lateral wheelset position on the impact load generated at the crossing. 4.1. Reference case The calculated vertical wheel trajectory and vertical wheel–rail contact force for the reference case are shown in Fig. 5(a) and (b). In front of the crossing (up to 41.5 m from front of turnout), a parametric excitation caused by the variation in track stiffness due to the discrete sleeper supports is observed. From 41.5 m from front of turnout, the wheel will experience an upward displacement due to the increase in stiffness in the crossing panel. In the facing move, the wheel first makes contact with the outwards deviating wing rail and the contact moves towards the field side of the wheel profile. As discussed in Section 1, the rolling radius decreases and the wheel moves downwards, as can be seen at x = 47 m in Fig. 5(a). When the wheel reaches and makes contact with the crossing rail, the contact load is transferred from the wing rail to the crossing nose. The rolling radius
Fig. 5. (a) Vertical wheel trajectory and (b) summed vertical wheel–rail contact force in the crossing panel. Traffic in facing move of through route at vehicle speed 100 km/h. Nominal S1002 wheel profile on nominal wing rail and crossing rail profiles.
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then increases as the new contact point is close to the wheel flange. The dip in wheel trajectory during the crossing transition seen in Fig. 5(a) agrees with findings in literature [3,8,9,31]. The impact load is observed right after the TCP at 47.2 m from front of turnout, shortly after the position for reversal in vertical wheel trajectory when the wheel is accelerated upwards by the crossing rail. The calculated magnitude of the impact load is in acceptable agreement with the measurements reported in Refs. [5,6]. Apart from the impact load, a transient response is observed throughout the crossing transition (cf. the high-frequency contents in the force before and after the impact in Fig. 1(b)). The wheel–rail contact geometry, contact positions and distributions of contact pressure at four different locations in the crossing are illustrated in Fig. 6. At x = 47.05 m, the wheel is in contact with the wing rail. For the given wheel–rail profiles, the conformal contact geometry leads to a relatively low contact pressure distributed over a contact area that is elongated in the lateral direction. At x = 47.17 m, the wheel makes contact with both the wing rail and the crossing rail, thus a two-point
Fig. 6. Illustration of contact geometry and contact pressure between wheel and rail during wheel transition from wing rail to crossing rail. (a) Vertical wheel–rail contact force (circles mark selected points in the time history). Contact geometries shown for contact at (b) x = 47.05 m, (c) x = 47.17 m, (d) x = 47.22 m and (e) x = 47.36 m from front of turnout. Vertical wheel–rail contact pressure shown for contacts at (f) x = 47.05 m and (g) x = 47.36 m.
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contact can be observed. The vertical contact pressure on the crossing rail is much higher than on the wing rail because of the small contact area. This corresponds to the position where the local minimum in vertical wheel–rail contact force is observed. At x = 47.22 m, the wheel has transferred to the crossing rail and the high contact pressure is distributed over a thin section on the crossing rail. This corresponds to the position where the local maximum in impact load is observed. Finally, at x = 47.36 m, the wheel continues to roll on the crossing rail developing a non-Hertzian distribution of contact pressure. It is pointed out in Ref. [30] that plastification of the rail material may occur under such high contact pressure and that a finite element model accounting for non-linear material properties of wheel and rail may need to be included to accurately simulate the contact. 4.2. Parameter study A parameter study is performed to determine the influence of vehicle speed, rail pad stiffness, wheel profile and lateral wheelset position on the impact load generated at the crossing. First, the influence of vehicle speed is investigated. The calculated vertical wheel–rail contact force is shown in Fig. 7. As expected, it is observed that the magnitude of the wheel–rail impact load at the crossing increases with increasing vehicle speed. This is in agreement with the field measurements reported in Ref. [6]. Next, the influence of rail pad stiffness (and associated rail pad damping, here taken as proportional to pad stiffness) is investigated. The four studied values of rail pad stiffness are kp = 60 kN/mm, kp = 120 kN/mm kp = 240 kN/mm and kp = 480 kN/mm. In all four cases, results have been calculated for vehicle speed 100 km/h. The calculated static track stiffness for a stationary load applied on the crossing rail at different distances from front of turnout is increasing with increasing pad stiffness, see Fig. 8(a). Further, the influence of pad stiffness on the Green’s function for the track due to excitation on the crossing rail are shown in Fig. 8(b). It is seen that the difference in Green’s functions evaluated for kp = 240 kN/mm and kp = 480 kN/mm is small. This is because when the rail pad stiffness is much higher than the ballast stiffness, the impulse response of the track is dominated by the ballast properties. Fig. 8(c) shows the calculated vertical wheel–rail contact force. The magnitude of impact load at the crossing is larger for kp = 240 kN/mm than for kp = 120 kN/mm. The impact load magnitudes calculated for kp = 240 kN/mm and kp = 480 kN/mm are similar due to the similarity in dynamic track response. Note that similar impact loads are also evaluated for kp = 60 kN/mm and kp = 120 kN/mm. The influence of wheel–rail contact geometry is studied by varying the prescribed lateral position of the wheelset centre and by changing the nominal S1002 wheel profiles to two different hollow worn wheel profiles. In all of the cases, results have been calculated for vehicle speed 100 km/h and rail pad stiffness 120 kN/mm. In Fig. 9, the results of simulations for three prescribed lateral wheelset positions Δy are shown. A negative value of Δy corresponds to a lateral displacement of the wheel towards the track centre. This means that the wheel flange is moved away from the crossing. As observed in Fig. 9(a), the dip in the vertical wheel trajectory increases with decreasing Δy because the contact between wheel and wing rail is taking place further out on the field side. For an opposite prescribed wheelset position, the wheel makes contact with the crossing at a position closer to the wheel flange leading to an earlier transition to the crossing rail and a smaller dip in vertical wheel trajectory. In the current study, as flange contact does not occur for Δy = 2 mm, the smooth transition leads to a lower wheel–rail impact load on the crossing compared to when the wheel is further away from flange contact, see Fig. 9(b). When Δy > 2 mm (not shown here), flange contact between wheel and crossing will appear ahead of the TCP, causing an earlier transition and a higher impact load. For large positive values of Δy, loss of wheel–rail contact has been observed. Similar findings are found in Refs. [8,31]. In Fig. 10, the results from simulations with wheel profiles with two different magnitudes (2 mm and 4 mm) of hollow wear are compared with the results for S1002. The three wheel profiles are compared in Fig. 10(a). As can be observed in Fig. 10(b),
Fig. 7. Influence of vehicle speed on vertical wheel–rail contact force.
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Fig. 8. Influence of rail pad stiffness on (a) static track stiffness along inner rail of through route, (b) Green’s functions for the track of a point moving at vehicle speed 100 km/h. Excitation on the rail at 47.2 m (crossing) from front of turnout and (c) vertical wheel–rail contact force. – · – kp = 60 kN∕mm, kp = 120 kN/mm, – – – kp = 240 kN/mm, … kp = 480 kN∕mm
Fig. 9. Influence of prescribed lateral wheelset position Δy on vertical (a) wheel trajectory and (b) wheel–rail contact force.
the vertical position of the two hollow worn wheels is lifted upwards during contact with the wing rail as the rolling radius of these wheels is increased when rolling on the ‘false flange’ at the field side of the wheel tread. The wheel lift continues along the contact with the wing rail until the wheel falls down on the crossing. The hollow wear leads to that these wheels make a significantly later contact with the crossing rail than the nominal S1002 wheel profile. The wheel profile with the most hollow wear (hollow worn 4 mm) generated the largest magnitude wheel–rail impact load at the crossing, see Fig. 10(c). These conclusions are in agreement with the study presented in Ref. [2].
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Fig. 10. (a) Three different wheel profiles used in the study. Influence of wheel profile on vertical (b) wheel trajectory and (c) wheel–rail contact force.
5. Conclusions A versatile and computationally efficient method for the simulation of high-frequency dynamic vehicle–track interaction in a railway crossing has been presented. Based on linear and time-invariant finite element models, the dynamic properties of wheelset and crossing panel are represented by Green’s functions (impulse response functions). Since the structural properties of the crossing panel can be described in greater detail, this approach offers a significant benefit compared to so-called moving or lumped parameter track models traditionally being used in vehicle–track interaction software. The three-dimensional surface geometries of crossing and wheel are described by four-noded linear elements. In each time-step of the simulation, vertical non-Hertzian (potentially multiple) wheel–rail contact is solved by an implementation of Kalker’s variational method. The method was demonstrated for a crossing subjected to freight traffic in the facing move of the through route. It was shown that the transition point from wing rail to crossing rail can be predicted with good precision. Parameter studies were performed to determine the influence of vehicle speed, rail pad stiffness, lateral wheelset position and wheel profile on the wheel–rail impact load generated at the crossing. It can be argued that an implementation of resilient (low stiffness) rail pads in the crossing panel offers an attractive approach to reduce impact loads as long as rail bending stresses are below the fatigue limit. However, it can also be concluded that wheel–rail contact geometry, as would be influenced by crossing rail and wing rail design, check rail gap width, and wear and plastic deformation of wheel and rail profiles (not studied here with the exception of wheel profile), seem to influence impact loads more than the selection of rail pad stiffness. In future work, the presented method will be implemented in an iterative calculation scheme to predict long-term differential track settlement in the crossing panel [32]. The influence of modelling the wheelset based on a lumped parameter or a finite element model on the impact load will also be investigated.
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