Dynamic behavior of an embedded rail track coupled with a tram vehicle

Abstract

This paper presents an investigation into the dynamic behavior of an embedded rail track coupled with a tram vehicle in time domain. A new designed embedded rail track structure firstly introduced into the Chinese tramways is described and the results of vibration tests of the embedded rail track (ERT) and another fastened slab track (FST) are discussed. A three-dimensional (3D) dynamic model of a tram vehicle coupled with an embedded rail track was developed on the basis of the multi-body dynamics approach and the finite element method. In the model, the tram vehicle was modeled as a multi-body system. The embedded rail track was modeled as a two layer system consisting of two rails, filling material, slabs, and adjustment layer beneath slabs. The rails were treated as Timoshenko beams with continuous elastic supports, in which the modal superposition method was used to reduce the order of the partial differential equations of beams. Continuous viscoelastic elements were used to represent the filling material and rail pad that connecting the rails and the slabs. The concrete slabs were modelled using the 3D finite element method, while the modal superposition method was adopted to improve the computational efficiency. Uniformly viscoelastic elements were introduced to model the elastic layer beneath the concrete slabs. The proposed model was then applied to compare the dynamic response of the innovative embedded rail track with respect to a conventional fastened slab track. The numerical results indicate that the innovative embedded rail track has advantages over the fastened slab track for its potentialities to reduce the dynamic wheel/rail force, the vibration level and deformation of the track parts, and the track defects and damages.

Keywords

Embedded rail track tramway dynamic behavior fastened slab track vehicle-track interaction

1. Introduction

Tramways are one of the most environmentally friendly modes for transporting large numbers of people. There are approximately 400 systems in operation worldwide today, among which Europe has the densest level of light rail systems with approximately 170 systems in operation but North America and Canada have also been active in the last decade in opening new systems (Department for Transport, 2011). New schemes are also being built in the Middle East and Asia. With the rise in population and prosperity in China, tramways are being built in many of China’s major cities. Approximate 40 cities plan to build more than 2,000 km of tram lines by 2020, while Shanghai plans to construct more than 800 km of modern tramways. To support the rapid construction of modern tramways in China, the authors have conducted extensive experimental and numerical analyses of the dynamic behaviour and vibration noise of tramway vehicle–track coupling systems. This article presents an investigation of the dynamic response of an innovative embedded rail track, with the main focus on modelling a tram vehicle–embedded rail track system and comparing the difference in the dynamic behaviour of the embedded rail track (ERT) compared to the conventional fastened slab track (FST).

Railway system dynamics researches have been performed for almost a century, producing a great deal published papers and theoretical models (Knothe and Grassie, 1993; Popp et al., 1999). In particular, detailed studies of train-track interactions has been the focus of many researchers and engineers. Numerical methods for simulating train-track interaction dynamics can be found in references (Knothe and Grassie, 1993; Popp et al., 1999; Zhai et al., 2009; Sun and Dhanasekar, 2002; Baeza and Ouyang, 2011; Ju and Li, 2011; Lei and Mao, 2004; Ling et al., 2014b). The coupled vehicle-track models can characterize many basic phenomena of a vehicle coupled with a track, and they are widely used to understand the behaviour of the railway system dynamics. Among the existing dynamic models, the vehicle system is usually modeled by using the multi-body dynamics approach. Several modern multi-body vehicle dynamics simulation models such as NUCARS®, SIMPACK®, VAMPIRE® and GENSYS™ were developed and applied to railway dynamics analysis and designed well matched trains and tracks. These simulations models have in turn, facilitated many modern developments in high-speed passenger, heavy haul freight, and urban rail transit trains (Zeng, et al., 2015; Sun et al., 2003; Lei and Wang, 2014; Liu et al., 2015).

Traditional railway track models can be classified into continuous and discrete models. In the continuous model, the rails are usually represented by the infinite Euler or Timoshenko beams resting on Winkler elastic foundation. The discrete model take into account the discrete rail support by the rail pads, which can be divided into models of infinite and finite length. The infinite ones contain the rails represented by the Euler or Timoshenko beams resting on discrete sleepers or rail pads. Discrete models of finite length are usually based on the finite element method. Embedded rail track is one of the most distinguishing characteristic of a tramway system, which has the advantage of sharing traffic lane with road vehicles in a street. Compared to traditional railway tracks, the structure and mechanics mechanism of the embedded rail track is different, and the research on the dynamic behaviours of this special track seem to be limited.

Man (2002) studied several typical dynamic behavior of the embedded rail structures in terms of track receptance by using a two dimensional and two layer continuously supported slab track based on the finite element method (FEM). In the model, the rail and the slab were continuously supported by means of an elastic compound and a subgrade, only the vertical behavior of the embedded rail track was considered. Shamalta and Metrikine (2003) investigated the steady-state dynamic response of an embedded railway track subjected to moving train loads by using theoretical analysis. In their study, the track model consists of two Euler-Bernoulli beams for the rails, a Kirchhoff’s plate performing vertical vibrations for the track slab, and continuous viscoelastic elements representing the filling material and the elastic foundation that supports the track slabs, the lateral motions of the rails and slabs were neglected. Lakusic and Lazarevic (2007) developed a numerical model of a tram moving on the track, in which the rail was modelled as elastically supported continuous Euler-Bernoulli beam with lumped masses, the vehicle was approximated as a discrete system consisting of three rigid masses connected with linear elastic springs, dissipation mechanisms in the system were modelled with the linear dashpots, but the lateral stiffness of the filling material was not considered. Bezin et al. (2010) investigated the dynamic interaction between a vehicle and two innovative slab tracks and compared the dynamic performance of two innovative slab track designs with respect to conventional ballasted track using a Flexible Track System Model integrated with a multibody dynamics software tool. Real et al. (2011) established an analytical mathematical model for predicting the ground vibrations caused by the passing of trams along a slab track. The model was based on the wave equation and was solved in the frequency domain through the Fourier Transform. The loads caused by the vehicle were calculated using a quarter car model.

In the present paper, a new designed embedded rail structure introduced into the Chinese tramways for the first time is described and the vibration tests of the embedded rail track and a conventional fastened slab track are discussed. A three-dimensional (3D) dynamic model of a tram vehicle coupled with an embedded rail track was developed on the basis of the multi-body dynamics approach and the FEM. The tramcar is composed of three vehicles, which was modeled as a 96 degrees of freedom (DOF) multi-body system with nonlinear suspensions. The embedded rail track was modeled as a 2-layer model, the grooved rails were treated as Timoshenko beams, in which the modal superposition method was introduced to reduce the order of their partial differential equations, continuous viscoelastic elements were used to represent the filling material, the concrete slabs were modelled using FEM and the modal superposition method, and uniformly viscoelastic elements were introduced to model the elastic layer beneath the concrete slabs. The proposed model was then applied to compare the dynamic response of an innovative embedded rail track with respect to a conventional fastened slab track. The results would help to clear the difference in the dynamic behaviour of the ERT with respect to the FST.

2. The embedded rail track system and its vibration test

Embedded rail track is an innovative track system that being much different from the traditional ballast and slab tracks, as the continuous supported rails in embedded rail structures are placed in paved slabs or bridge decks. Although the embedded rail tracks are often classified in the group of slab track structures (Man, 2002), the mechanics mechanism and dynamic behaviour of the embedded rail structures are significantly different from the other slab track structures. As the rail is supported continuously by an elastic compound rather than fastened at regular intervals, which make the rail support stiffness is independent in the lateral and vertical directions, and thus the rail rotation can also be reduced (Iwnicki, 2009). Owing to its advantages over other types of slab tracks, the embedded rail tracks are usually considered as optimum rail structures to reduce the track damage and vibration noise.

A large amount of different embedded rail tracks have been applied in the Netherlands, UK, and Spain (Man, 2002; Esveld, 2003; Iwnicki, 2009). A new designed embedded rail structure consisting of grooved rails, filling material, PVC pipes, rail pads, precast slabs, Self-compacting concrete adjustment layer (SCA layer), and concrete base was firstly introduced into the tramways in china, as shown in Figure 1. For this embedded rail structure, a concrete slab with formed troughs are applied for each rail. The rails are suspended in the trough and the spaces beside and beneath them are filled with polymer compound that provides both electrical isolation and acoustic attenuation. The slab holds the track to gauge, and there are no rail fastenings. To evaluate the dynamic behaviour and vibration characteristics of the innovative track system, a large amount of experimental and numerical analyses were carried out.

On-site vibration tests of the tram vehicle–track system were carried out on a tangent track that including two types of track strctures: embedded rail track and fastened slab track. The vertical and lateral connection stiffness of the ERT was approximately 110 and 52 MN/m per unit length of the track. And the rail pad stiffness of the FST was approximately 25 MN/m (42 MN/m per unit length of the track). The vibration accelerations of the rails, filling material and slabs of the two types of track systems were measured simultaneously. The acceleration sensor locations on the tested track sections were arranged as shown in Figure 2. The Ei (i = 1,3) terms denote acceleration sensors on the ERT, and the Fi (i = 1,3) terms denote sensors on the FST. There were no serious track defects around the test locations. The tram vehicle tested was a vehicle consisting of two power vehicles and one trailing vehicle. The length of the tramcar is 29.5 m. The static axle loads for the power vehicles and the trailing vehicle are approximately 115 and 98 kN, respectively. The vehicle speeds used in the tests were approximately 20 ∼ 70 km/h.

Figures 3 and 4 illustrate the measured RMS values of the accelerations of the railheads (Points E1 and F1 in Figure 2) and the track slabs (Points E3 and F3 in Figure 2) of the two types of track strctures, respectively. The test results show that the vertical and lateral accelerations of the rails in the ERT are lower than that in the FST, while the vibration levels of the slabs in the ERT are larger compared to that tested in the FST for the analyzed speed range. It is noted that the differences in the tested accelerations of the rails and slabs of the types of track system increase with increasing operating speed.

The reason for these differences should be attribute to the using of different rail support stiffness and the different structure mechanism. It is easy to understand that a higher connection stiffness between the rails and slabs will lead to more vibration transmission from rails to slabs. And the vibrational energy of the rails and the vibration transmissibility would increase with an increase in vehicle speed. These two issues mainly results in the differences in the accelerations of the ERT with high connection stiffness and the FST with soft rail pads. In addition, the rail is supported continuously rather than at regular intervals in the embedded rail structure, which will be helpful to reduce the dynamic loads and the vibrational energy of rails (Man, 2002; Esveld, 2003; Iwnicki, 2009). The detailed difference between the ERT and FST with the same parametres will be discussed in Section 4 using the vehicle-track coupling model illustrated in section 3.

3. Tram vehicle-embedded rail track coupling model

A 3D dynamic model of a tramway vehicle coupled with an embedded rail track was developed and is illustrated in Figure 5. The coupled vehicle–track dynamic model consists of three subsystems: the vehicle subsystem, the track subsystem, and the wheel–rail contact subsystem. The interaction of the vehicles and the track is characterized through the wheels–rails in rolling contact. They are, respectively, described in subsections 2.1, 2.2 and 2.3 in detail.

3.1. Modelling of the tram vehicle

A generic tram vehicle used in China’s tramways was selected to be modelled in the present paper. The modelled vehicle is a multicar tram, which is composed of two power vehicles and one trailing vehicle, as shown in Figure 6. Each vehicle consists of two car bodies connected by an intermediate articulated section and three bogies. The two end body sections are coupled to the centre section by spherical joints and therefore the centre bogie supports the end mass of these two sections, together with the centre section. The two powered bogies are normal bogies with solid conventional axles, and the middle bogie is the trailer bogie equipped with independently rotating wheels. Both the motor bogies and trailer bogie are equipped with double suspension systems. The primary suspensions use the rubber springs arrangement acting on the axle-boxes, which connect the bogie frames to the axle-boxes. The secondary suspensions of the motor bogies, including coil springs and lateral and vertical hydraulic dampers, connect the car bodies to the bogie frame via bolsters, whilst the trailer bogie is not equipped with the bolster. In Figure 6, notations C and K with subscripts stand for the coefficients of the equivalent dampers and the stiffness coefficients of the equivalent springs, respectively. Notations M and I with subscripts are the masses and moments of inertia of the vehicle and track components, respectively.

The tram vehicle model is set up by using the multi-body dynamics approach. The tram composing of three vehicles is modelled as a 96 DOF nonlinear multi–body system with nonlinear suspensions. The degrees of freedom of the vehicle model are listed in Table 1. The subscripts ci (i = 1,3) and bi (i = 1,3) indicate the three car bodies and bogie frames, subscripts bsi (i = 1,2) and twi (i = 1,4) denote the bolsters and wheelsets of the two motor bogies, and subscripts iwi (i = 1,2) and wi (i = 1,4) represent the axles and wheels of the trailer bogie. The motion of the vehicle components is defined in the trajectory coordinate system. That is to say, the coordinate system of the vehicle system moves in the track direction at the vehicle speed (Ling et al., 2014a). The bodies are assumed to undergo small displacements relative to the moving reference, thus the kinematic relationships can be linearized. 3D spring-damper elements are used to represent the primary and the secondary suspensions, and the nonlinear dynamic characteristics of the suspension systems are considered, including the nonlinear lateral and vertical dampers, and the nonlinear bump-stops installed on the secondary suspension.

Table 1.

Degrees of freedom of the tram vehicle model.

Vehicle component	Type of Motion
Vehicle component	Longitudinal	Lateral	Vertical	roll	pitch	yaw
Carbody (i = 1,3)	$X_{c i}$	$Y_{c i}$	$Z_{c i}$	$φ_{c i}$	$β_{c i}$	$ψ_{c i}$
Bogie frame (i = 1,3)	$X_{b i}$	$Y_{b i}$	$Z_{b i}$	$φ_{b i}$	$β_{b i}$	$ψ_{b i}$
Bolster (i = 1,2)	—	—	—	—	—	$ψ_{bs i}$
Wheelset (i = 1,4)	$X_{tw i}$	$Y_{tw i}$	$Z_{tw i}$	$φ_{tw i}$	$β_{tw i}$	$ψ_{tw i}$
Axle (i = 1,2)	$X_{iw i}$	$Y_{iw i}$	$Z_{iw i}$	$φ_{iw i}$	—	$ψ_{iw i}$
Independently rotating wheels (i = 1,4)	$X_{w i}$	$Y_{w i}$	$Z_{w i}$	$φ_{w i}$	$β_{w i}$	$ψ_{w i}$

The motion equations of the vehicle model can be written as:

M_{v} {\overset{··}{U}}_{v} (t) + C_{v} ({\overset{\cdot}{U}}_{v}) {\overset{\cdot}{U}}_{v} (t) + K_{v} (U_{v}) U_{v} (t) = F_{wr} ({\overset{\cdot}{U}}_{v}, U_{v}, {\overset{\cdot}{U}}_{T}, U_{T}, v, t) + F_{ext} (v, t)

(1)

where M_v is the mass matrix of the vehicle, C_v and K_v are the damping and the stiffness matrices depending on the current state of the vehicle subsystem to describe nonlinearities within the suspension. U_v(t),

{\overset{\cdot}{U}}_{v} (t)

and

{\overset{··}{U}}_{v} (t)

are the displacement, velocity and acceleration vectors.

F_{wr} ({\overset{\cdot}{U}}_{v}, U_{v}, {\overset{\cdot}{U}}_{R}, U_{R}, v, t)

is the vector of the nonlinear wheel–rail contact forces, which is determined by the motions U_v(t) and

{\overset{\cdot}{U}}_{v} (t)

of the vehicle and U_T(t) and

{\overset{\cdot}{U}}_{T} (t)

of the track. F_ext(v, t) is the vector of external forces including gravitational forces and forces resulting from the centripetal acceleration when the vehicle negotiates the curved track.

3.2. Modelling of the embedded rail track

Figure 7 illustrates the calculation model of the embedded rail track shown in Figure 1. In the model, the grooved rails were treated as Timoshenko beams with continuous elastic supports, the modal superposition method was used to convert the fourth–order partial differential equations of beams into second-order ordinary equations (Timoshenko et al., 1974). Continuous viscoelastic elements were used to represent the filling material and rail pad that connect the rails and the slabs. The lateral and vertical connection stiffness and damping were obtained by laboratory tests. The concrete slabs were modelled using the 3D FEM, while the modal superposition method (Wang, 2003) was adopted to improve the computational efficiency. The dynamic model didn’t consider the joints between slabs as the tested embedded rail track also didn’t install the connectors between the neighboring slabs. Uniformly viscoelastic elements were introduced to model the elastic SCA Layer beneath the concrete slabs. The effects of the motion of the concrete base and subgrade were ignored. If the dynamic behaviour of the substructure is also concerned, the subgrade can also be modeled with FEM, with the modeling and solution similar to that of the slab model.

The deformation of the grooved rails is described by the Timoshenko beam theory, including the bending and torsion. Governing equations for the forced vibrations of the Timoshenko beams with continuous elastic support in different directions can be written as follows:

For the lateral bending motion:

{ρ A \frac{\partial 2 y (x, t)}{\partial t 2} + κ_{y} GA [\frac{\partial ψ_{y} (x, t)}{\partial x} - \frac{\partial 2 y (x, t)}{\partial x 2}] + K_{y r} (y (x, t) - y_{sr} (x, t)) + C_{y r} (\frac{\partial y (x, t)}{\partial t} - \frac{\partial y_{sr} (x, t)}{\partial t}) - \sum_{j = 1}^{N_{w}} F_{wr yj} (t) δ (x - x_{w_{j}}) = 0 ρ I_{z} \frac{\partial 2 ψ_{y} (x, t)}{\partial t 2} + κ_{y} GA [ψ_{y} (x, t) - \frac{\partial y (x, t)}{\partial x}] - {EI}_{z} \frac{\partial 2 ψ_{y} (x, t)}{\partial x 2} = 0

(2)

For the vertical bending motion:

{ρ A \frac{\partial 2 z (x, t)}{\partial t 2} + κ_{z} GA [\frac{\partial ψ_{z} (x, t)}{\partial x} - \frac{\partial 2 z (x, t)}{\partial x 2}] + K_{z r} (z (x, t) - z_{sr} (x, t)) + C_{z r} (\frac{\partial z (x, t)}{\partial t} - \frac{\partial z_{sr} (x, t)}{\partial t}) - \sum_{j = 1}^{N_{w}} F_{wr zj} (t) δ (x - x_{w_{j}}) = 0 ρ I_{y} \frac{\partial 2 ψ_{z} (x, t)}{\partial t 2} + κ_{z} GA [ψ_{z} (x, t) - \frac{\partial z (x, t)}{\partial x}] - {EI}_{y} \frac{\partial 2 ψ_{z} (x, t)}{\partial x 2} = 0

(3)

For the torsional motion:

ρ I_{0} \frac{\partial 2 φ (x, t)}{\partial t 2} - GK \frac{\partial 2 φ (x, t)}{\partial x 2} + K_{φ r} (φ (x, t) - φ_{sr} (x, t)) + C_{φ r} (\frac{\partial φ (x, t)}{\partial t} - \frac{\partial φ_{sr} (x, t)}{\partial t}) - \sum_{j = 1}^{N_{w}} M_{G j} (t) δ (x - x_{w_{j}}) = 0

(4)

In equations (2)-(4), y, z, and φ are, respectively, the lateral, vertical, and torsion deflection of the rail, ψ_y and ψ_z are the slope of the deflection curve of rail with respect to z and y axis. The material properties of the rail are indicated by the density ρ, the shear modulus G and Young's modulus E. The geometry of the cross section of the rail is represented by the area A, the second moments of area I_y and I_z around the y-axis and the z-axis, respectively, and the polar moment of inertia I₀. The shear coefficients κ_y = 0.5257 and κ_z = 0.2892 for the lateral and the vertical bending and the shear coefficient K_s = 2.473346 × 10^-6 are obtained through a finite element analysis of the rail profile of 59R2. K_y_r, K_z_r, K_φ_r and C_y_r, C_z_r, C_φ_r are the stiffness and the damping constant per unit length of the filling material between rails and slabs, respectively. The subscript j indicates wheel j. δ(x) is the Dirac delta function, x_w _j denote the longitudinal positions of the wheel j, and N_W is the total numbers of the wheels within the analyzed rail. M_G _j (t) is the equivalent moments acting on the rail. The wheel–rail forces at the wheel j in the lateral and vertical directions are represented by F_wr _yj (t) and F_wr _zj (t).

Using the modal superposition method (Timoshenko et al., 1974), the fourth–order partial differential equations of rails are converted into second-order ordinary equations as follows:

For the lateral bending motion:

{{\overset{··}{q}}_{yk} (t) + \frac{κ_{y} GA}{m} (\frac{i π}{l_{tim}}) 2 q_{yk} (t) + κ_{y} GA \frac{i π}{l_{tim}} \sqrt{\frac{1}{m ρ I_{z}}} w_{yk} (t) + \frac{1}{m} (K_{y r} q_{yk} (t) + C_{y r} {\overset{\cdot}{q}}_{yk} (t)) - \frac{1}{{mY}_{k}} \int_{0}^{l_{tim}} (K_{y r} y_{sr} (t) + C_{y r} {\overset{\cdot}{y}}_{sr} (t)) d x - \sum_{j = 1}^{N_{w}} F_{wr yj} (t) Y_{k} (x_{w_{j}}) = 0 {\overset{··}{w}}_{yk} (t) + [\frac{κ_{y} GA}{ρ I_{z}} + \frac{{EI}_{z}}{ρ I_{z}} (\frac{i π}{l_{tim}}) 2] w_{yk} (t) - κ_{y} GA \frac{i π}{l_{tim}} \sqrt{\frac{1}{m ρ I_{z}}} q_{yk} (t) = 0 (k = 1 ~ NMY)

(5)

For the vertical bending motion:

{{\overset{··}{q}}_{zk} (t) + \frac{κ_{z} GA}{m} (\frac{i π}{l_{tim}}) 2 q_{zk} (t) + κ_{z} GA \frac{i π}{l_{tim}} \sqrt{\frac{1}{m ρ I_{y}}} w_{zk} (t) + \frac{1}{m} (K_{zr} q_{zk} (t) + C_{z r} {\overset{\cdot}{q}}_{zk} (t)) - \frac{1}{{mZ}_{k}} \int_{0}^{l_{tim}} (K_{z r} z_{sr} (t) + C_{z r} {\overset{\cdot}{z}}_{sr} (t)) d x - \sum_{j = 1}^{N_{w}} F_{wr zj} (t) Z_{k} (x_{w_{j}}) = 0 {\overset{··}{w}}_{zk} (t) + [\frac{κ_{z} GA}{ρ I_{y}} + \frac{{EI}_{y}}{ρ I_{y}} (\frac{i π}{l_{tim}}) 2] w_{zk} (t) - κ_{z} GA \frac{i π}{l_{tim}} \sqrt{\frac{1}{m ρ I_{y}}} q_{zk} (t) = 0 (k = 1 ~ NMZ)

(6)

For the torsional motion:

{\overset{··}{q}}_{Tk} (t) + \frac{GK}{ρ I_{0}} (\frac{i π}{l_{tim}}) 2 q_{Tk} (t) + \frac{1}{ρ I_{0}} (K_{φ r} q_{Tk} (t) + C_{φ r} {\overset{\cdot}{q}}_{Tk} (t)) - \frac{1}{ρ I_{0} Φ_{k}} \int_{0}^{l_{tim}} (K_{φ r} φ_{sr} (t) + C_{φ r} {\overset{\cdot}{φ}}_{sr} (t)) d x - \sum_{j = 1}^{N_{w}} M_{G j} (t) Φ_{k} (x_{w_{j}}) = 0 (k = 1 ~ NMT)

(7)

In equations (5)–(7), q_yk(t), q_zk(t) and q_T _k (t) are the generalized coordinates of the lateral, vertical, and rotational deflection of the rail, respectively, while w_yk(t) and w_zk(t) are the generalized coordinates of the deflection curve of the rail with respect to the z-axis and the y-axis. The calculation length of the rail is denoted by l_tim. m is the mass per unit longitudinal length. The two ends of the rail are hinged, thus the shape functions of the simple supported beam was used. NMY, NMZ, and NMT are the total numbers of the shape functions, and Y_k(x), Z_k(x), and Φ_k(x) are the k-th shape function, which are given by (Timoshenko et al., 1974):

Y_{k} (x) = \sqrt{\frac{2}{ρ Al}} \sin (\frac{k π}{l} x)

(8)

Z_{k} (x) = \sqrt{\frac{2}{ρ Al}} \sin (\frac{k π}{l} x)

(9)

Φ_{k} (x) = \sqrt{\frac{2}{ρ I_{0} l}} \sin (\frac{k π}{l} x)

(10)

The track slab was simulated by a 3D FE model with spring-damper suspension supports, as shown in Figure 7. To reduce computational efforts efficiently, the modal superposition method was adopted to solve the equations of motion of the slabs. Thus, each slab is modeled by its first N-modes and the modal data are normalized to the mass matrix. The equation of the slab i in the global coordinate system can be expressed as (Ling et al., 2014a):

M_{s i} {\overset{··}{U}}_{s i} (t) + C_{s i} {\overset{\cdot}{U}}_{s i} (t) + K_{s i} U_{s i} (t) = F_{rs i} ({\overset{\cdot}{U}}_{s i}, U_{s i}, {\overset{\cdot}{U}}_{r i}, U_{r i}) + F_{gs i} ({\overset{\cdot}{U}}_{s i}, U_{s i}, {\overset{\cdot}{U}}_{g i}, U_{g i}) (i = 1 ~ N_{slab})

(11)

where M_s _i , C_s _i and K_s _i are the global mass, damping and stiffness matrices of the slab i, respectively; U_s _i (t),

{\overset{\cdot}{U}}_{s i} (t)

and

{\overset{··}{U}}_{s i} (t)

are the displacement, velocity and acceleration vectors; F_rs _i and F_gs _i are the vectors of the interaction forces between slab i and the rail, and slab i and the subgrade; N_slab is the total number of the track slabs considered in the track model.

When the modal decomposition method is adopted for the slabs, the displacement vector of slab i can be written as:

U_{s i} (t) = Φ_{s i} X_{s i} (t) = \sum_{k = 1}^{N_{mode}} Φ_{s ik} X_{s ik} (t) (i = 1 ~ N_{slab})

(12)

where, X_s _i (t) is the displacement vector of slab i in modal coordinates; Φ_s _i is the modal matrix containing the first N-modes of slab i; X_s _ik (t) is the k-th generalized coordinate. Φ_s _ik is the k-th mode shape vector, which is obtained through the modal analysis of the track slab using the FEM. N_mode is the total number of mode shape used in the modal superposition calculation. Because the observed structural resonant frequency of the embedded rail structures in site is usually at frequencies below 80 Hz (Man, 2002). Thus this calculation considered 6 rigid-body modes and 4 flexible modes, as shown in Figure 8. The frequency of the highest mode was 109 Hz, which is enough to simulate the dynamic behaviour of the track slabs in the embedded rail structures. Using the normalized shape functions of the slab, the government equation (12) is transformed to a set of uncoupled equations:

{\overset{··}{X}}_{s ik} + 2 ξ_{s ik} ω_{s ik} {\overset{\cdot}{X}}_{s ik} + ω_{s ik}^{2} X_{s ik} = P_{s ik} (i = 1 ~ N_{slab}, k = 1 ~ N_{mode})

(13)

where ξ_s _ik and ω_s _ik are the damping ratio and the circular frequency, and P_s _ik is the generalized node load, corresponding to the k-th mode of the slab i.

3.3. Modelling of wheel-rail contact

The interaction of the vehicle and the track is characterized through the wheel–rail in rolling contact. The wheel–rail contact model includes the geometric relationship and the contact forces between the wheel and the rail. A spatial and on-line wheel–rail contact model (Chen and Zhai, 2004; Xiao et al., 2011) was used in this study to characterize the geometry of the wheel–rail rolling contact. Two point contact behaviour was considered when the flange root of the wheel contact with grooved head of the rail.

The calculation of wheel–rail contact forces includes a normal model and a tangent one. The normal model, which characterizes the relationship law of the normal load and deformation between the wheel and rail, is described by a Hertzian nonlinear contact spring with a unilateral restraint, reads:

F_{n} (t) = {[\frac{1}{G_{hertz}} Z_{wrnc} (t)] 3 / 2, Z_{wrnc} (t) > 0 0, Z_{wrnc} (t) \leq 0

(14)

where G_hertz is the wheel–rail contact constant (m/N^2/3), which can be determined by the following equation:

G_{Hertz} = \frac{K (e)}{π m} [\frac{9}{2} (A + B) G * 2] 1 / 3

(15)

where

K (e)

is the first kind of complete elliptic integral, reads:

K (e) = \int_{0}^{π / 2} (1 - e 2 \sin 2 θ_{1}) - 1 / 2 d θ_{1}

(16)

e is the ellipticity of the wheel/rail contact spot, which can be expressed as follows:

e 2 = 1 - (b / a) 2

(17)

m = [\frac{2 E (e)}{π (1 - e 2)}] 1 / 3

(18)

E (e)

is the second kind of complete elliptic integral, reads:

E (e) = \int_{0}^{π / 2} (1 - e 2 \sin 2 θ_{1}) 1 / 2 d θ_{1}

(19)

G *

is the wheel/rail material coefficient, which can be determined by:

G * = \frac{1 - ν_{w}^{2}}{E_{w}} + \frac{1 - ν_{r}^{2}}{E_{r}}

(20)

A and B are constants of wheel/rail contact, which can be expressed as follows:

A + B = \frac{1}{2} (\frac{1}{R_{w 1}} + \frac{1}{R_{w 2}} + \frac{1}{R_{r 1}} + \frac{1}{R_{r 2}})

(21)

B - A = \frac{1}{2} [(\frac{1}{R_{w 1}} - \frac{1}{R_{w 2}}) 2 + (\frac{1}{R_{r 1}} - \frac{1}{R_{r 2}}) 2 + 2 (\frac{1}{R_{w 1}} - \frac{1}{R_{w 2}}) (\frac{1}{R_{r 1}} - \frac{1}{R_{r 2}}) \cos 2 η] 1 / 2

(22)

where, a and b are the semi-axil lengths of the wheel-rail contact patches, E_w and E_r are the Young’s modulus of wheel/rail material, v_w and v_r are the Poisson ratios, R_w1 and R_w2 are the longitudinal and lateral curvature radii of contact point on the wheel, R_r1 and R_r2 are the longitudinal and lateral curvature radii of contact point on the rail, and η is contact angle at R_w1 and R_r1.

In the present investigation, G_hertz was set as:

G_{hertz} = 3.86 R - 0.115 \times 10 - 8

(23)

Z_wrnc(t) is the normal compressing amount at wheel/rail contact point. Z_wrnc(t) > 0 indicates the wheel/rail in contact, and Z_wrnc(t) ≤ 0 stands for their separation. Z_wrnc(t) is strictly defined as an approach between the two far points, the one belonging to the wheel, the other one belonging to the rail. The wheel and the rail are, respectively, assumed to be an elastic half–space. This approach is confined in the normal direction at the contact point of the wheel and the rail. Z_wrnc(t) is determined by the following equation:

Z_{wrnc} (t) = \frac{Z_{w} (t) - Z_{r} (t) + δ_{0} (t) - Z_{0} (t)}{\cos (δ (t) \pm φ (t))}

(24)

where Z_w(t) is the vertical displacement of the wheel center; Z_r(t) is the vertical displacement of the centroid of the rail cross section; δ₀(t) is the approach between the wheel and rail caused by the static normal load; Z₀(t) is the time function of the track vertical irregularity; δ (t) is the wheel–rail contact angle and φ (t) denotes the relative roll angle between the rail head and the wheelset. These quantities can be determined by solving the system equations.

The model by Shen et al. (1983) is used as the tangent model determining the relationship between the creepages and the total creep forces of the wheel/rail. First, the wheel–rail creep forces are calculated using Kalker’s linear creep theory (Kalker, 1967) for small amounts of creep. For large amounts of creep, saturation occurs, resulting in a nonlinear relation that is described using the Shen model. The description of the creep force calculation model was not shown due to a detailed derivation was given in the published paper (Ling et al., 2014c) by the authors.

3.4. Simulation program

From equations (1)–(24), it is obvious that the tram vehicle–track system equations form a large-scale nonlinear system. These motion equations were solved by using an explicit integral algorithm based on Newmark’s method developed by Zhai (1996) and called, ‘new fast numerical integration for dynamics analysis of large systems’. The scheme of this time-stepping integral method to calculate the n-step dynamic responses of system parts (including displacement U and the velocity $\overset{\cdot}{U}$ ) can be expressed as follows (Zhai, 1996, Zhai et al., 2009):

{U_{n} = U_{n - 1} + {\overset{\cdot}{U}}_{n - 1} Δ t + (0.5 + ψ) {\overset{··}{U}}_{n - 1} Δ t 2 - ψ {\overset{··}{U}}_{n - 2} Δ t 2 {\overset{\cdot}{U}}_{n} = {\overset{\cdot}{U}}_{n - 1} + (1 + ϕ) {\overset{··}{U}}_{n - 1} Δ t - ϕ {\overset{··}{U}}_{n - 2} Δ t

(25)

where Δt is the time step, and Δt = 1.4E-5 s was selected in the present investigation; ψ and ϕ are free parameters that control the stability and numerical dissipation of the algorithm. Zhai’s research (Zhai et al., 2009) shows that a value of 0.5 could be assigned to ψ and ϕ to achieve a good compatibility between numerical stability and accuracy.

Based on the mathematical model described in section 3, a computer simulation program with FORTRAN, named tram-embedded rail system dynamics (TERSD), was developed to analyze the dynamics of the coupled tram vehicle-embedded rail track system. The flow chart of the complete simulation process is illustrated in Figure 9. TERSD can be used to investigate the vertical-lateral coupling dynamic behaviour of embedded rail track structures including the vibration responses, stability, dynamics performance, etc. It also can be used to evaluate the dynamics performances of tram vehicles, including the curving performance, ride comfort, lateral stability, etc.

3.5. Experiment validation

Once the tram vehicle–track model has been built, the reliability and accuracy of the simulation results must be calibrated firstly. In this section, the dynamic results obtained using the numerical model were compared with those obtained from the field experiments. Figure 10 shows the time histories of the vertical accelerations of the rails and slabs when the test vehicle negotiated a tangent track section without welded joints and any other severe track irregularities. The track irregularities were measured from the test line, and the vehicle speed was approximately 67 km/h. The other simulation and test conditions are the same as that described in section 2.

The comparison results in Figure 10 indicate that the calculated accelerations more or less correspond to those obtained by field tests. And a good fit can be easily observed between the peaks from the axles as well as in the attenuation of the wave. It should be also noted the maximum and minimum accelerations measured in site are larger than that obtained using the numerical simulations. Several issues should be in charge of these differences. One possible reason is that the measurement points of the measured and calculated accelerations were different, as a result of the way in which the rails were modelled. In fact, the vertical acceleration of the railhead (Points E1 and F1 in Figure 2) was captured in the field tests, while the continuous Timoshenko beam model used for the rail in this paper cannot differentiate the dynamic behavior of the rail head, web, and bottom as the cross-sectional twist occurring. Another main cause should be attributed to the neglecting the nonlinearity of the filling material (as well as the rail pad) that connecting the rails and slabs. Actually, these high molecular compound (filling material) shows a remarkable dependency on their deformation and the vibration frequency, which was not considered in the present ERT modelling. In addition, many unavoidable deviations occur in the degree of irregularities on the wheel and rail surfaces, the calculation of wheel–rail rolling contact, as well as in the parameters used in the calculation. But, the comparisons shown in Figure 10 indicate that the errors between the measured results and the calculated results are acceptable for the engineering application, and the proposed model can be used to investigate the dynamics problems occurring in the tram vehicle–embedded rail track system.

4. Numerical simulation and results analysis

The experimental results discussed in section 2 indicate the vibration response of the ERT are quite different from that occurring in the FST. However, the differences between the two types of tracks are not only existed in the structural mechanism, but also in the connection material and track slab parameters. To clear the difference in the dynamic behaviour of the ERT with respect to the FST, this section investigates several key dynamic behavior between these two types of track structures with the same calculation conditions. In the simulations, the only difference is that the ERT model using beams with continuous supports while the FST model using beams with supported discretely at regular intervals. A generic tram vehicle used in China is considered in this study, which is composed of two power vehicles and one trailing vehicle. The maximum speed of the tramcar is 70 km/h, and tangent tracks are considered. The main physical parameters of the tram vehicle and embedded rail track used in the simulations are given in Table 2 and 3.

Table 2.

Main parameters of tram vehicle.

Notations	Parameters	Values (end)	Values (centre)
M/I	Body inertia
M _c	Car body mass (kg)	12.50 × 10³	4.25 × 10³
M _b	Bogie mass (kg)	2.87 × 10³	1.20 × 10³
M _w	Wheelset mass (kg)	1.05 × 10³	1.10 × 10³
I _cx	Car body roll moment of inertia (kg·m²)	1.94 × 10⁴	0.85 × 10⁴
I _cy	Car body pitch moment of inertia (kg·m²)	0.23 × 10⁶	0.12 × 10⁵
I _cz	Car body yaw moment of inertia (kg·m²)	0.22 × 10⁶	0.71 × 10⁴
I _bx	Bogie roll moment of inertia (kg·m²)	0.85 × 10³	0.51 × 10³
I _by	Bogie pitch moment of inertia (kg·m²)	1.73 × 10³	0.42 × 10³
I _bz	Bogie yaw moment of inertia (kg·m²)	2.50 × 10³	0.89 × 10³
I _wx	Wheelset roll moment of inertia (kg·m²)	0.40 × 10³	0.35 × 10³
I _wy	Wheelset pitch moment of inertia (kg·m²)	27.57	23.71
I _wz	Wheelset yaw moment of inertia (kg·m²)	0.40 × 10³	0.35 × 10³
K/C	Primary suspension
K _px	Longitudinal stiffness (MN/m)	15.70	6.30
K _py	Lateral stiffness (MN/m)	2.80	6.30
K _pz	Vertical stiffness (MN/m)	0.98	1.80
C _pz	Vertical damping coefficient (kN·s/m)	5.0	10.0
K/C	Secondary suspension
K _sx	Longitudinal stiffness (MN/m)	0.17	0.35
K _sy	Lateral stiffness (MN/m)	0.17	0.35
K _sz	Vertical stiffness (MN/m)	0.62	0.98
C _sy	Lateral damping coefficient (kN·s/m)	50.0	50.0
C _sz	Vertical damping coefficient (kN·s/m)	45.0	45.0
L/D/H	Dimension
R _w	Wheel radius (m)	0.33
L _v	Vehicle length (m)	29.5
L _b	Distance between two axles of a bogie (m)	1.9	1.8
L _c	Distance between bogie centers and joints (m)	8.2	2.2
D _ps	Lateral span of primary suspensions (m)	1.87	1.11
D _ss	Lateral span of secondary suspensions (m)	2.07	2.06
H _bw	Height of bogie centre from wheelset centre (m)	0.011	0.029
H _cb	Height of car body centre from bogie centre (m)	1.20	0.86

Table 3.

Main parameters of embedded rail track.

Notations	Parameters	Values
Rail
D _r	Gauge (m)	1.435
ρ	Density of rail (kg/m³)	7860
G	Young’s modulus (N/m²)	2.059 × 10¹¹
A	Cross-sectional area (m²)	7.753 × 10^-3
γ	Poisson’s ratio	0.3
I ₀	Flexural moment of inertia (m⁴)	3.74 × 10^-5
I _y	Bending moment around lateral axis (m⁴)	3.217 × 10^-5
I _z	Bending moment around vertical axis (m⁴)	5.24 × 10^-6
κ_y	Lateral shear coefficient	0.5257
κ_z	Vertical shear coefficient	0.2892
Filling material
K_z _r	Vertical stiffness (MN/m)	102.7
K_y _r	Lateral stiffness (MN/m)	41.3
C_z _r	Vertical damping coefficient (kN·s/m)	32.0
C_y _r	Lateral damping coefficient (kN·s/m)	20.0
Concrete slab
G _s	Young’s modulus (N/m²)	3.9 × 10¹⁰
ρ _s	Density (kg/m³)	2.5 × 10³
L/D/H	Dimensions (m³)	6.0 × 2.5 × 0.46
K _sv	Vertical supported stiffness (MN/m)	300.0
K _sh	Lateral supported stiffness (MN/m)	150.0
C _sv	Vertical supported damping coefficient (kN·s/m)	30.0
C _sh	Lateral supported damping coefficient (kN·s/m)	15.0

4.1. Steady state response

To make clear the differences in the dynamic performances obtained by ERT and FST, the steady state response of the wheel/rail normal force and the rail deflection are firstly compared. In the calculation, the tram vehicle operating speed was 70 km/h, and no track irregularities were considered. The time histories of the wheel/rail normal force and the rail vertical deflection in the ERT and FST are shown in Figure 11.

Figure 11 shows the oscillating amplitudes of the wheel/rail normal force and rail deflection in the FST are much larger than that in the ERT. This is because the track stiffness at the sleepers is larger with respect to the sleeper bays in the FST, and the dynamic variation in the rail support stiffness causes the larger oscillating amplitude of the normal load and rail deflection. But this behavior is practically impossible in embedded rail due to the continuous support mechanism.

The big peaks shown in Figure 11(b) should be caused by the gaps between neighboring slabs of the fastened slab track and the pitch motion of the slabs. For the actual fastened slab track in high-speed railway lines or metro lines, there are usually installed joints between slabs, which can certainly reduce the rail deflection peaks (shown in Figure 11(b)) at the slab connection area. But in the present investigation, the tested embedded rail track didn’t install connectors between neighboring slabs. To clear the differences in the dynamic behaviour between the embedded rail track and the fastened slab track, all the simulation conditions including the slab connection area were set to the same. So to some extent the result shown in Figure 10(b) overestimates the rail deflection at slab transitional area of actual fastened slab tracks. If the numerical model considered the joints between neighboring slabs, the peaks shown in Figure 11(b) wouldn’t be so obvious.

In addition, the histories of the wheel/rail normal force and the rail vertical deflection in the FST are dominated by two harmonic components with wavelengths of 0.6 m and 6 m, which refer to the sleeper span and the length of track slabs. But the steady state response of the wheel/rail normal force and the rail vertical deflection in the ERT only present the periodic waves related to the length of track slabs. It is noted that these harmonic components in the steady state response of the normal load and the rail deflection may result in serious rail defects and track irregularities. The periodic waves related to the sleeper span could cause rail corrugation (Esveld, 2003; Jin et al., 2008). And the oscillating waves caused by the discrete slabs would lead to the longitudinal and cross level irregularities forming the predominant components with wavelengths around the slab length, which were observed in the slab tracks on Chinese high-speed railway lines (Zhang, 2009). From Figure 11, it can be easily concluded that the track irregularities caused by the physical dimensions of the track components in the ERT will be much slighter than that in FST.

4.2. Dynamic response to track irregularities

Figure 12 illustrates the short and middle track irregularities measured from the tram test line (with the vertical defects used here as an example), which were introduced into the dynamic response analysis for the ERT and FST. The short track defects filtered with wavelengths of 0.02–1 m were measured using the corrugation analysis trolley, and the middle irregularities with wavelengths from 1 to 50 m were measured using a portable track detecting instrument. Figure 12(b) shows that there are many severe rail welds on the tested tracks.

Figures 13–17 illustrate the maximum wheel/rail normal force, rail acceleration and deflection, and slab acceleration and deformation in the ERT and FST systems in the cases of the tram vehicle-track interaction excited by the measured random track irregularities. The tram vehicle speeds considered in the simulations were 30 ∼ 70 km/h. Two types of filling material (or rail pad) were also investigated, the connection stiffness of which were obtained by laboratory tests. The vertical and lateral static connection stiffness of the soft material are approximately 51.5 and 24.0 MN/m per unit length of the ERT. And the vertical and lateral connection stiffness of the stiff one are approximately 128.4 and 51.6 MN/m per unit length of the track. With regard to the FST, the equivalent vertical and lateral static connection stiffness are approximately 30.9 and 14.4 MN/m for the soft fastenings with a sleeper span of 0.6 m, and the equivalent vertical and lateral stiffness are approximately 77.0 and 31.0 MN/m for stiff one.

Figure 13 shows the maximum values of the dynamic wheel/rail normal forces of all the wheelsets as the tram vehicle running on the ERT are smaller than the cases of the tram car operating on the FST. And the difference of the results obtained with the two models increases with increasing operating speed. When the vehicle speed is greater than 40 km/h, the maximum normal forces calculated using the model including the FST are greater than that obtained using the model including the ERT regardless of the stiffness of filling material (or rail pad). This means that the ERT can reduce the dynamic wheel/rail loads compared to the FST, which will be helpful to eliminate the severe track defects and damages due to the abnormal track loads. It is also interesting to note that a stiffer filling material (or rail pad) tends to increase the amplitude of the wheel/rail contact force, this phenomenon is in accord with that reported in (Fermér and Nielsen, 1995; Recuero and Escalona, 2014). As a result, an increase in the stiffness of filling material (or rail pad) is likely to lead to increased fatigue damage of the vehicle-track system.

Figures 14 and 15 compare the maximum rail vertical acceleration and deflection in the ERT and FST when the tram vehicle passes over a tangent track section with random track irregularities. Overall, the rail vertical acceleration and deflection in the ERT are smaller than that in the FST for all the cases considered in this study. Figure 14 indicates that the rail vibration acceleration in the FST are greater than that in the ERT regardless of the stiffness of filling material. This may be the main reason for that the embedded rail are usually considered as an optimum rail structure to reduce the wheel/rail vibration noise in practical engineering application. Furthermore, a stiffer filling material (or rail pad) results in a lower rail acceleration, this is in accord with experimental results reported in section 2.

Figure 15 shows the rail deflection is mainly affected by the rail support stiffness, the vertical deflection of rails with soft supported material is much larger than that in the rails with stiff supported material. It is also interesting to see that the difference in the rail deflection of the two types of track structures increases with increasing operating speed. From the above analysis, it can be predicated the rail irregularities in a FST would be more serious than that in an ERT at high speed.

Figure 16 shows the maximum vertical accelerations of the slabs in the ERT are lower than that in the FST, which are different from the results shown in Figure 4(a). This is because the same equivalent stiffness was assumed for the material that connecting the rails and slabs in the current cases and the accelerations were measured from the locations under the rail pad. In this situation, the local connection stiffness between the rails and slabs in where the rail pad installed in the FST is larger than that in the ERT, and a higher connection stiffness will lead to a higher vibration energy from rails to slabs. Figure 16 also indicates a softer filling material (or rail pad) results in a lower slab acceleration, this phenomenon agrees well with the test results illustrated in Figure 4.

Figures 17 illustrates the maximum slab deformation under rail pad in the ERT and FST. It shows that the slab local deformation under rail pad in the FST are greater than that in the ERT regardless of the stiffness of rail pad. This is also caused by the difference in the local connection stiffness between the rails and slabs as discussed above. And as expected, a stiffer filling material will lead to a larger slab deformation. Based on the above analysis, it can be concluded that the track slab damages in a FST would be more serious than that in an ERT. In fact, the track slab damage under rail pad are severe on the slab tracks of the Chinese high-speed railway lines (Zhai et al., 2014). This implies that the embedded rail track can be considered as an optimum track structure to reduce the track slab damage on high-speed railway lines if the above results and the present method are correct and reliable. Further study on this issue is under way.

5. Conclusions

A new designed embedded rail track structure firstly introduced into the Chinese tramways was described, and the results of vibration tests of the embedded rail track and a conventional fastened slab track were discussed. A 3D dynamic model for a tram vehicle coupled with an embedded rail track based on the multi-body dynamics approach and FEM was developed to investigate the dynamic behavior of an embedded rail track under the moving tram vehicles. The proposed model was then applied to compare the dynamic response of the innovative embedded rail track with respect to a conventional fastened slab track. Through extensive vibration tests conducted on site and numerical simulations, the dynamic response, including rail deflection and acceleration, track slab deformation and acceleration, as well as wheel/rail normal force, in the two types of track structures were compared and discussed in detail. The results show that the innovative embedded rail track structure with continuous supported rails has the advantages over the fastened slab track for its potentialities to reduce the dynamic wheel/rail force, the vibration level and deformation of the rails and slabs, and the developing of track defects and damages.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The National Natural Science Foundation of China (U1361117, 51305360), the Chengdu Xinzhu Road & Bridge Machinery Co., Ltd. Project titled “Study on the dynamic performance and vibration & noise characteristics of a modern tram vehicle and an embedded rail track”, and the 2013 Cultivation Program for the Excellent Doctoral Dissertation of Southwest Jiaotong University.

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