Dynamic analysis of three-dimensional high-speed train-track model using moving element method

Abstract

In this article, the dynamic analysis of three-dimensional high-speed train-track model is carried out using moving element method. The train comprises a car-body supported by a secondary suspension system to a bogie. The bogie is in turn connected to wheel-sets through a primary suspension system. A total of 16 degrees of freedom is employed to describe the vertical, lateral, rolling, pitching, and yawing displacements of the car-body, the bogie, and the wheel-sets. The Hertzian contact and Kalker’s linear theory are used to account for the vertical and lateral contact forces between the wheels and rails. Two rails are modeled as two Euler–Bernoulli beams resting on a viscoelastic foundation. The moving element method is extended to establish the coupling formulations of the mass, damping, and stiffness matrices in vertical and lateral directions, where the element matrices are formulated based on a convected coordinate system attached to the moving vehicle. The dynamic amplification factor is defined as the ratio of the maximum dynamic contact force to the static load at the contact point between the wheel and the rail. To illustrate the benefits of the proposed three-dimensional train-track model, several numerical examples are performed in this study to present the effects of the train speed, the resonance phenomenon, the track irregularity, and track stiffness variations along the track and across the track width on the dynamic amplification factor of the high-speed train.

Keywords

dynamic amplification factor Euler–Bernoulli beam moving element method three-dimensional high-speed train-track model viscoelastic foundation

Introduction

In recent decades, the development of high-speed rail (HSR) has gained an important role in travel due to their efficiency and environmentally friendly technologies. This has led to the increasing of train speed, higher axle load, and more frequent train usage. However, these improved train technologies have brought new challenges in solving the dynamic interaction problems of the train-track. A number of studies have been conducted to study the problems of train-track dynamic interaction. The HSR system has been investigated as a rail beam resting on a viscoelastic foundation subjected to moving loads. By means of Fourier transform method (FTM) and a moving coordinate system, Mathews (1958, 1959) solved the dynamic problem of a moving load along an infinitely long beam resting on an elastic foundation. Jezequel (1981) studied the dynamic problem of an Euler–Bernoulli infinite beam resting on elastic supports subjected to a concentrated force moving at constant speed. Similar works using the FTM were carried out by Ono and Yamada (1989), Sun (2002), and Chen and Wang (2006). The FTM may give accurate solutions but it is cumbersome when dealing with complicated coupled systems, such as multi-degrees of freedom (DOFs) system with multiple contact points.

In practice, the finite element method (FEM) has also been widely employed in the moving load problems. Filho (1978), Hino et al. (1984, 1985), and Olsson (1985) are the pioneers in the development of finite element formulations for analyzing structures subjected to moving loads. They presented a review on the use of the FEM for solving the problems of a uniform beam subjected to moving loads. Frýba et al. (1993) presented a stochastic finite element for a beam resting on a random foundation with uncertain damping under moving force. Thambiratnam and Zhuge (1993, 1996) proposed a finite element model for simply-supported beams resting on an elastic foundation subjected to moving loads and applied to the analysis of track structures subjected to point or uniformly distributed loads. Nielsen and Igeland (1995) solved the problem of a beam resting on sleepers and treated a bogie as a moving load. A complex modal superposition technique was employed to study the effects of corrugated rail, wheel-flat, and unsupported sleepers.

In dealing with moving load problems, the FEM encounters difficulties when the moving load approaches the boundary of the finite domain and travels beyond the boundary. These difficulties can be overcome by employing a large enough domain size but the computational cost will increase significantly. To deal with the complication encountered by the FEM, Krenk et al. (1999) proposed the use of convected coordinate in the FEM to analyze the response of an elastic half-space subjected to a moving load. The benefit of this approach is its ability to overcome the problem due to the moving load traveling over a finite domain. On the same thread, Andersen et al. (2001) presented a FEM study of a beam on a Kelvin foundation subjected to a harmonic moving load. Koh et al. (2003) adopted the idea of convected coordinate for solving train-track problems and named the numerical algorithm as moving element method (MEM). The method was subsequently applied in the analysis of in-plane dynamic responses of annular disk (Koh et al., 2006) and moving load on continuum (Koh et al., 2007). Xu et al. (2009) extended the one-dimensional MEM proposed by Koh et al. (2003) to two-dimensional (2D) problems for investigating the dynamic responses of an infinite Kirchhoff plate resting on a Kelvin foundation subjected to a moving vehicle. Recently, Ang and Dai (2013) employed the MEM to investigate the dynamic response of high-speed train-track system for the situation where there is an abrupt change of foundation stiffness. The influence of different factors, including the degree of change of foundation stiffness, traveling velocity of train, and the severity of track irregularity, on the response of the train as well as the track is examined and discussed. Ang et al. (2014) applied the MEM to investigate the “jumping wheel” phenomenon in high-speed train motion at constant velocity over a transition region where there is a sudden change of the foundation stiffness. Tran et al. (2014) analyzed the dynamic response of high-speed train traveling at non-uniform speed using the MEM. In this study, both linear and nonlinear wheel-rail contact models were examined. Dai and Ang (2015) presented an analytical solution for the steady-state response of a curved beam resting on a viscously damped foundation and subjected to a single or sequence of moving loads using MEM. On the same thread, Tran et al. (2016, 2017a, 2017b) employed the MEM to investigate the dynamic response of high-speed train-track system experiencing heavy braking and abrupt braking. Most recently, Tran et al. (2017c) examined the vertical dynamic response of HSRs during sudden deceleration using the MEM. The effects of the wheel sliding, initial train deceleration, initial train speed, and the severity of the railhead roughness on the dynamic response of the HSR were investigated. Dai et al. (2017) proposed a computational scheme in conjunction with the MEM to investigate the dynamic response of a high-speed train-track system in which the discrete sleepers on the sub-grade support the railway track.

Many previous studies have proposed the use of simple 2D train-track models, in which the train has been modeled by 3-DOF or 10-DOF system and only one side of the track is used for calculation. The disadvantages of the 2D train-track models are that these models can be employed to examine the effects of track irregularity and track stiffness variations along the track on the dynamic response of high-speed train only. But the effects of track irregularity and track stiffness variations across the track width (one rail related to the other rail) on the dynamic response of high- speed train cannot be captured using them. This article is hence proposed to fill in the above-mentioned research gap by extending the application of MEM to investigate the full dynamic interaction of the high-speed train-track system with the complex three-dimensional (3D) train-track model. In the proposed model, the train is modeled by 16 DOFs involving vertical, lateral, rolling, pitching, and yawing displacements of the car-body, the bogie, and the wheel-sets. Two rails are modeled as two Euler–Bernoulli beams resting on a viscoelastic foundation. The benefits of the proposed model compared to previous 2D train-track models are that all of motions of the train components and the effects of track irregularity and track stiffness variations across the track width can be captured for calculating the dynamic response of the high-speed train. To achieve the above-mentioned purpose, the Hertzian contact and Kalker’s linear theory are employed to account for the vertical and the lateral contact forces between the wheels and the rails. The coupling formulations for calculating the mass, damping, and stiffness matrices in both vertical and lateral directions of the moving elements are presented. The dynamic amplification factor (DAF) is defined as the ratio of the maximum dynamic contact force to the static load at the contact point between the wheel and the rail. To illustrate the advantages of the proposed 3D train-track model, several numerical examples are performed in this study to present the effects of train speed, resonance phenomenon, track irregularity, and track stiffness variations along the track and across the track width on the DAF of the high-speed train.

Formulation

3D train model

Figure 1 shows the 3D train model including three different views (longitudinal, transverse, and plane). The train model comprises a car-body supported through a secondary suspension system to a bogie. The bogie is in turn connected to two wheel-sets through a primary suspension system. Both suspension systems are modeled with an appropriate number of vertical and horizontal spring-damping units. The pitching and yawing motions of the car-body are neglected due to the fact that the pairs of front and rear wheel-sets supporting the car-body are spaced far apart in a typical high-speed train coach.

Figure 1.

3D train model: (a) longitudinal view, (b) transverse view, and (c) plane view.

A total of 16-DOF system is employed to describe the motions of various components of the train. The displacement vector for the 3D train model can be written as

z_{t} = {y_{c} z_{c} R_{c}^{x} y_{b} z_{b} R_{b}^{x} R_{b}^{z} R_{b}^{y} y_{w 1} z_{w 1} R_{w 1}^{x} R_{w 1}^{y} y_{w 2} z_{w 2} R_{w 2}^{x} R_{w 2}^{y}}^{T}

(1)

where $y_{c}, z_{c}, and R_{c}^{x}$ are the vertical, lateral, and rolling motions of car-body; $y_{b}, z_{b}, R_{b}^{x}, R_{b}^{z}, and R_{b}^{y}$ are the vertical, lateral, rolling, pitching, and yawing motions of bogie; and $y_{wi}, z_{wi}, R_{wi}^{x}, and R_{wi}^{y} (i = 1 \div 2)$ are the vertical, lateral, rolling, and yawing motions of the ith wheel-set.

The equation of motion of the train can be written in the general form as

M_{t} {\overset{\cdot\cdot}{z}}_{t} + C_{t} {\overset{\cdot}{z}}_{t} + K_{t} {\overset{\cdot}{z}}_{t} = P_{t}

(2)

in which $M_{t}$ , $C_{t}$ , $K_{t}$ , and $P_{t}$ are the mass, damping, stiffness matrices, and the force vector of the train, respectively.

The mass matrix of the train is expressed as

M_{t} = diag [\begin{matrix} M_{c} & M_{b} & M_{w 1} & M_{w 2} \end{matrix}]

(3)

where $M_{c}$ , $M_{b}$ , and $M_{wi} (i = 1 \div 2)$ are the sub-mass matrices of the car-body, the bogie, and the ith wheel-set

M_{c} = diag [\begin{matrix} m_{c} & m_{c} & I_{c}^{x} \end{matrix}]

(4)

M_{b} = diag [\begin{matrix} m_{b} & m_{b} & I_{b}^{x} & I_{b}^{z} & I_{b}^{y} \end{matrix}]

(5)

M_{wi} = diag [\begin{matrix} m_{wi} & m_{wi} & I_{wi}^{x} & I_{wi}^{y} \end{matrix}] (i = 1 \div 2)

(6)

in which $m_{c}, m_{b}, and m_{wi}$ are the mass of car-body, the mass of bogie, and the mass of $i th (i = 1 \div 2)$ wheel-set, respectively; $I_{c}^{x}$ is the moment of inertia of car-body in roll; $I_{b}^{x}, I_{b}^{z}, and I_{b}^{y}$ are the moment of inertia of bogie in roll, pitch, and yaw; and $I_{wi}^{x} and I_{wi}^{y} (i = 1 \div 2)$ are the moment of inertia of the ith wheel-set in roll and pitch.

The stiffness matrix of the train is symmetric and can be established as

K_{t} = [\begin{matrix} K_{c} & K_{cb} & 0 & 0 \\ K_{b} & K_{bw 1} & K_{bw 2} \\ K_{w 1} & 0 \\ sym . & K_{w 2} \end{matrix}]

(7)

where $K_{c}$ , $K_{b}$ , and $K_{wi} (i = 1 \div 2)$ are the sub-stiffness matrices of the car-body, the bogie, and the ith wheel-set, respectively; $K_{cb}$ , $K_{bw 1}$ , and $K_{bw 2}$ are the sub-stiffness matrices of the coupling of car-body and bogie, the coupling of bogie and first wheel-set, and the coupling of bogie and second wheel-set, respectively

K_{c} = [\begin{matrix} 2 k_{2 V} & 0 & 0 \\ 2 k_{2 H} & - 2 h_{1} k_{2 H} \\ sym . & 2 h_{1}^{2} k_{2 H} + 2 b_{2}^{2} k_{2 V} \end{matrix}]

(8)

K_{b} = [\begin{matrix} 2 k_{2 V} + 4 k_{1 V} & 0 & 0 & 0 & 0 \\ 2 k_{2 H} + 4 k_{1 H} & 2 k_{2 H} h_{2} - 4 k_{1 H} h_{3} & 0 & 0 \\ 2 k_{2 H} h_{2}^{2} + 4 k_{2 V} b_{2}^{2} + 4 k_{1 H} h_{3}^{2} + 4 k_{1 V} b_{1}^{2} & 0 & 0 \\ 4 k_{1 V} l_{w}^{2} & 0 \\ sym . & 4 k_{1 H} l_{w}^{2} + 4 k_{1 x} b_{1}^{2} \end{matrix}]

(9)

K_{wi} = diag [\begin{matrix} 2 k_{1 V} & 2 k_{1 H} & 2 k_{1 V} b_{1}^{2} & 2 k_{1 x} b_{1}^{2} \end{matrix}] (i = 1 \div 2)

(10)

K_{cb} = [\begin{matrix} - 2 k_{2 V} & 0 & 0 & 0 & 0 \\ 0 & - 2 k_{2 H} & - 2 k_{2 H} h_{2} & 0 & 0 \\ 0 & 2 k_{2 H} h_{1} & 2 k_{2 H} h_{1} h_{2} - 2 k_{2 V} b_{2}^{2} & 0 & 0 \end{matrix}]

(11)

K_{bw 1} = [\begin{matrix} - 2 k_{1 V} & 0 & 0 & 0 \\ 0 & - 2 k_{1 H} \\ 0 & 2 k_{1 H} h_{3} & - 2 k_{1 V} b_{1}^{2} \\ - 2 k_{1 V} l_{w} & 0 & 0 & 0 \\ 2 k_{1 H} l_{w} & 0 & - 2 k_{1 x} b_{1}^{2} \end{matrix}]

(12)

K_{bw 2} = [\begin{matrix} - 2 k_{1 V} & 0 & 0 & 0 \\ 0 & - 2 k_{1 H} \\ 0 & 2 k_{1 H} h_{3} & - 2 k_{1 V} b_{1}^{2} \\ 2 k_{1 V} l_{w} & 0 & 0 & 0 \\ - 2 k_{1 H} l_{w} & 0 & - 2 k_{1 x} b_{1}^{2} \end{matrix}]

(13)

in which $k_{2 V} and k_{2 H}$ are the vertical and horizontal stiffness properties of the secondary suspension, respectively; $k_{1 V}, k_{1 H}, and k_{1 x}$ are the vertical, horizontal, and x-direction stiffness properties of the primary suspension, respectively; and $h_{1}, h_{2}, h_{3}, b_{1}, b_{2}, and l_{w}$ are the distances of train parts as shown in Figure 1.

The damping matrix of the train is symmetric and can be written as

C_{t} = [\begin{matrix} C_{c} & C_{cb} & 0 & 0 \\ C_{b} & C_{bw 1} & C_{bw 2} \\ C_{w 1} & 0 \\ sym . & C_{w 2} \end{matrix}]

(14)

where the sub-damping matrices have the same forms as the sub-stiffness matrices and can be obtained by replacing the stiffness coefficient “k” everywhere by the damping coefficient “c”.

The force vector $P_{t}$ of the train can be expressed as

P_{t} = {\begin{matrix} P_{c} & P_{b} & P_{w 1} & P_{w 2} \end{matrix}}^{T}

(15)

where $P_{c}$ , $P_{b}$ , and $P_{wi} (i = 1 \div 2)$ are the sub-force vectors of the car-body, the bogie, and the ith wheel-set, respectively

P_{c} = {\begin{matrix} - m_{c} g & 0 & 0 \end{matrix}}^{T}

(16)

P_{b} = {\begin{matrix} - m_{b} g & 0 & 0 & 0 & 0 \end{matrix}}^{T}

(17)

P_{w i} = {\begin{matrix} - m_{w i} + F_{w i}^{r 1} + F_{w i}^{r 2} & - (F_{w z i}^{r 1} + F_{w z i}^{r 2}) & (F_{w z i}^{r 1} + F_{w z i}^{r 2}) r_{w} - (F_{w i}^{r 1} - F_{w i}^{r 2}) b_{0} & 0 \end{matrix}}^{T} (i = 1 \div 2)

(18)

in which $r_{w}$ is the wheel radius; $b_{0}$ is the half of track gauge as shown in Figure 1; and $F_{wi}^{rj} and F_{wzi}^{rj}$ are the vertical and horizontal contact forces between $i th (i = 1 \div 2)$ wheel-set and $j th (j = 1 \div 2)$ rail, respectively. The single dot over a symbol denotes the velocity, and the double dot denotes the acceleration.

Wheel-rail contact model

The wheel-rail contact model is shown in Figure 2(a). The Hertzian contact is employed to account for the vertical contact force $F_{wi}^{rj}$ between the $i th (i = 1 \div 2)$ wheel-set and the $j th (j = 1 \div 2)$ rail by

F_{wi}^{rj} = {\begin{matrix} K_{H} Δ y_{rj}^{3 / 2} & for Δ y_{rj} ⩾ 0 \\ 0 & for Δ y_{rj} < 0 \end{matrix}

(19)

where

K_{H} = \frac{2}{3} \sqrt{\frac{E^{2} \sqrt{R_{wheel} R_{railprof}}}{{(1 - ν^{2})}^{2}}}

(20)

in which $K_{H}$ denotes the Hertzian spring constant; $R_{wheel}$ and $R_{railprof}$ are the radii of the wheel and the railhead, respectively; E is the elastic modulus; and $ν$ is the Poisson ratio of the material.

Figure 2.

(a) Wheel-rail contact model; (b) 3D track model; (c) discretization of one typical rail beam into moving elements; and (d) typical frame element with 8 DOFs.

The indentation at the contact force $Δ y_{rj}$ is defined as

Δ y_{rj} = y_{rj} + y_{rj}^{t} - y_{wi} \pm b_{0} R_{wi}^{x}

(21)

in which $y_{rj}, y_{wi}, and y_{rj}^{t}$ denote the vertical displacement of the jth rail, the vertical displacement of the ith wheel-set, and the track irregularity of the jth rail at the contact points, respectively; and $R_{wi}^{x}$ denotes the rotation displacement about x-axis of ith wheel-set.

The track irregularity is a main factor of the dynamic excitation force acting on the train-track system, and it describes the vertical unevenness in the railhead surface which arises due to various factors such as wear, tear, and plastic deformation. The track irregularity profile is widely assumed to take the sinusoidal form as proposed by Nielsen and Abrahamsson (1992) and it is written as follows

y_{rj}^{t} = - a_{rj}^{t} [1 - \exp {(- \frac{x}{x_{c}})}^{3}] \sin \frac{2 π x}{λ_{rj}^{t}}

(22)

where $a_{rj}^{t}$ and $λ_{rj}^{t}$ , respectively, denote the amplitude and the wavelength of the track irregularity of the jth rail; and $x_{c}$ is a constant associated with the condition of the railhead.

When the train travels far away from its initial position, the exponential term in equation (22) will soon become negligible (Ang et al., 2014); thus, for simplicity, the expression for the vertical track irregularity profile can be written in terms of a sinusoidal function as

y_{rj}^{t} = - a_{rj}^{t} \sin \frac{2 π x}{λ_{rj}^{t}}

(23)

The Kalker’s linear theory is employed to account for horizontal contact force $F_{wzi}^{rj}$ between the ith wheel-set and jth rail (as proposed by Iwnicki, 2006) and can be expressed by

F_{wzi}^{rj} = f_{11} ε_{z}^{rj} + f_{12} ε_{12}^{rj}

(24)

where $f_{11} and f_{12}$ are the Kalker’s coefficient; and $ε_{z}^{rj} and ε_{12}^{rj}$ are the creepages in the lateral and rotational directions at the contact point of jth rail and can be expressed by

ε_{z}^{rj} = - \frac{{\overset{\cdot}{z}}_{wi}}{V} + R_{wi}^{y}

(25)

ε_{12}^{rj} = - \frac{\sin γ}{r_{w}} - \frac{{\overset{\cdot}{R}}_{wi}^{y}}{V}

(26)

in which ${\overset{\cdot}{z}}_{wi} and {\overset{\cdot}{R}}_{wi}^{y}$ denote the velocity of lateral displacement and velocity of rotation displacement about y-axis of the ith wheel-set; and $γ$ is the conicity of the wheel profile.

3D track model

The 3D track model is shown in Figure 2(b), in which two rails are assumed as two infinite Euler–Bernoulli beams resting on a viscoelastic foundation subjected to moving vertical and lateral contact forces.

The governing equations of vertical and lateral motions of the $j th (j = 1 \div 2)$ rail beam are written as

\begin{array}{l} E I_{v j} \frac{\partial^{4} y_{r j}}{\partial x^{4}} + m_{r j} \frac{\partial^{2} y_{r j}}{\partial t^{2}} + c_{r j}^{v} \frac{\partial y_{r j}}{\partial t} + k_{r j}^{v} y_{r i} \\ = - \sum_{i = 1}^{2} F_{w i}^{r j} δ (x - S) \end{array}

(27)

\begin{array}{l} E I_{h j} \frac{\partial^{4} z_{r j}}{\partial x^{4}} + m_{r j} \frac{\partial^{2} z_{r j}}{\partial t^{2}} + c_{r j}^{h} \frac{\partial z_{r j}}{\partial t} + k_{r j}^{h} z_{r j} \\ = - \sum_{i = 1}^{2} F_{w z i}^{r j} δ (x - S) \end{array}

(28)

where t denotes the time; S is the distance traveled by the train at any instant time t; $δ$ is the Dirac-delta function; $I_{vj}, I_{hj}, and m_{rj}$ are the vertical and horizontal moments of inertial and mass per unit length of the jth rail beam, respectively; $k_{rj}^{v} and k_{rj}^{h}$ are the vertical and horizontal coefficients of the foundation stiffness per unit length of the jth rail; $c_{rj}^{v} and c_{rj}^{h}$ are the vertical and horizontal coefficients of the foundation viscous damping per unit length of the jth rail; and $y_{rj} and z_{rj}$ are the vertical and horizontal displacements of the jth rail beam.

Figure 2(c) shows the discretization of one typical rail beam into a finite number of moving elements (as proposed by Koh et al., 2003; the length of the rail beam elements are not necessarily equal). Both the upstream and downstream ends of the beam are taken to be sufficiently far from the contact forces such that their end forces and moments are zero. The typical frame element with 8 DOFs is shown in Figure 2(d). The nodal displacement vector for the frame element can be written as follows

z_{e} = {\begin{matrix} y_{i} & θ_{i}^{z} & y_{j} & θ_{j}^{z} & z_{i} & θ_{i}^{y} & z_{j} & θ_{j}^{y} \end{matrix}}^{T}

(29)

By using the interpolation functions, the vertical and lateral displacements in the rail element can be expressed as

y = N_{v} z_{e}, z = N_{h} z_{e}

(30)

in which

N_{v} = [\begin{matrix} N_{1} & N_{2} & N_{3} & N_{4} & 0 & 0 & 0 & 0 \end{matrix}]

(31)

N_{h} = [\begin{matrix} 0 & 0 & 0 & 0 & N_{1} & N_{2} & N_{3} & N_{4} \end{matrix}]

(32)

where $N_{i} (i = 1 \div 4)$ are the shape functions based on Hermitian cubic polynomials.

MEM

The MEM is proposed with the idea of attaching the origin of the spatial coordinate system to the applied point of the moving load (Koh et al., 2003). The relationship between the moving coordinate r and the fixed coordinate x is given by

r = x - S

(33)

In view of equation (33), the governing equations (27) and (28) may be written as

\begin{array}{l} E I_{v j} \frac{\partial^{4} y_{r j}}{\partial r^{4}} + m_{r j} (V^{2} \frac{\partial^{2} y_{r j}}{\partial r^{2}} - 2 V \frac{\partial^{2} y_{r j}}{\partial r \partial t} + \frac{\partial^{2} y_{r j}}{\partial t^{2}}) \\ + c_{r j}^{v} (\frac{\partial y_{r j}}{\partial t} - V \frac{\partial y_{r j}}{\partial r}) + k_{r j}^{v} y_{r j} = - \sum_{i = 1}^{2} F_{w i}^{r j} δ (r) \end{array}

(34)

\begin{array}{l} E I_{h j} \frac{\partial^{4} z_{r j}}{\partial r^{4}} + m_{r j} (V^{2} \frac{\partial^{2} z_{r j}}{\partial r^{2}} - 2 V \frac{\partial^{2} z_{r j}}{\partial r \partial t} + \frac{\partial^{2} z_{r j}}{\partial t^{2}}) \\ + c_{r j}^{h} (\frac{\partial z_{r j}}{\partial t} - V \frac{\partial z_{r j}}{\partial r}) + k_{r j}^{h} z_{r j} = - \sum_{i = 1}^{2} F_{w z i}^{r j} δ (r) \end{array}

(35)

By adopting Galerkin’s approach and procedure of establishing the weak form in terms of the displacement field, the formulations for the mass, damping, and stiffness matrices of the moving element of the $j th (j = 1 \div 2)$ rail beam can be determined as follows

M_{ej} = m_{rj} \int_{0}^{L} {N_{v}}^{T} N_{v} d r + m_{rj} \int_{0}^{L} {N_{h}}^{T} N_{h} d r

(36)

\begin{array}{l} C_{e j} = - 2 m_{r j} V \int_{0}^{L} {N_{v}}^{T} {N_{v}}_{, r} d r + c_{r j}^{v} \int_{0}^{L} {N_{v}}^{T} N_{v} d r \\ - 2 m_{r j} V \int_{0}^{L} {N_{h}}^{T} {N_{h}}_{, r} d r + c_{r j}^{h} \int_{0}^{L} {N_{h}}^{T} N_{h} d r \end{array}

(37)

\begin{matrix} K_{ej} & = E I_{vj} \int_{0}^{L} N_{v, rr}^{T} N_{v, rr} d r + m_{rj} V^{2} \int_{0}^{L} {N_{v}}^{T} N_{v, rr} d r \\ - c_{rj}^{v} V \int_{0}^{L} {N_{v}}^{T} N_{v, r} d r + k_{rj}^{v} \int_{0}^{L} {N_{v}}^{T} N_{v} d r \\ + E I_{hj} \int_{0}^{L} N_{h, rr}^{T} N_{h, rr} d r + m_{rj} V^{2} \int_{0}^{L} {N_{h}}^{T} N_{h, rr} d r \\ - c_{rj}^{h} V \int_{0}^{L} {N_{h}}^{T} N_{h, r} d r + k_{rj}^{h} \int_{0}^{L} {N_{h}}^{T} N_{h} d r \end{matrix}

(38)

where $()_{, r}$ denotes the partial derivative with respect to r.

After assemblage, the equation of motion for the 3D train-track model can be written as

M \overset{\cdot\cdot}{z} + C \overset{\cdot}{z} + Kz = P

(39)

where $z = {\begin{matrix} y_{1} θ_{1}^{z} y_{2} θ_{2}^{z} z_{1} θ_{1}^{y} z_{2} θ_{2}^{y} . . . y_{n} θ_{n}^{z} y_{n + 1} θ_{n + 1}^{z} z_{n} θ_{n}^{y} z_{n + 1} θ_{n + 1}^{y} \\ y_{c} z_{c} R_{c}^{x} y_{b} z_{b} R_{b}^{x} R_{b}^{z} R_{b}^{y} y_{w 1} z_{w 1} R_{w 1}^{x} R_{w 1}^{y} y_{w 2} z_{w 2} R_{w 2}^{x} R_{w 2}^{y} \end{matrix}}^{T}$ is the global displacement vector of the 3D train-track system; M, C, K, and P are the global mass, damping, stiffness matrices, and the global load vector, respectively. The above dynamic equation can be solved by Newmark-β method.

Numerical results

To verify the accuracy of the proposed 3D train-track model and the MEM approach, the results of this study are compared with the results obtained by Koh et al. (2003). The parameters related to the properties of the high-speed train-track system are summarized in Tables 1 and 2.

Table 1.

Parameters of the train model and the fixed rail UIC 60 (Jin et al., 2006).

Description	Notation	Value	Unit
Parameters of the train model
Mass of car-body	$m_{c}$	14,000	kg
Car-body inertia moment	$I_{c}^{x}$	38,000	kg·m²
Mass of bogie	$m_{b}$	1000	kg
Bogie inertia moment	$I_{b}^{x}; I_{b}^{y}; I_{b}^{z}$	2260; 3160; 2710	kg·m²
Secondary suspension stiffness	$k_{2 H}; k_{2 V}$	150,000; 282,000	N/m
Secondary suspension damping	$c_{2 H}; c_{2 V}$	60,000; 17,740	N·s/m
Primary suspension stiffness	$k_{1 H}; k_{1 V}; k_{1 x}$	580,000; 1,260,000; 600,000	N/m
Primary suspension damping	$c_{1 H}; c_{1 V}; c_{1 x}$	10,000; 14,200; 40,000	N·s/m
Mass of wheel-set	$m_{w 1}; m_{w 2}$	700	kg
Conicity of the wheel	$γ$	0.125	rad
Wheel-set inertia moment	$I_{w 1}^{x}; I_{w 2}^{x}; I_{w 1}^{y}; I_{w 2}^{y}$	915; 915; 915; 915	kg·m²
Dimensions	$h_{1}; h_{2}; h_{3}; b_{0}; b_{1}; b_{2}; l_{w}$	1.145; 0.081; 0.14; 0.75; 1.0; 1.21; 1.2	m
Parameters of the fixed rails UIC 60
Mass per unit length	$m_{r 1}; m_{r 2}$	60	kg/m
Moment of inertial of rail	$I_{v 1}, I_{v 2}; I_{h 1}, I_{h 2}$	3.217 × 10⁻⁵; 5.24 × 10⁻⁶	m⁴
Track sub-structure stiffness	$k_{r 1}^{v}, k_{r 2}^{v}; k_{r 1}^{h}, k_{r 2}^{h}$	1 × 10⁷; 5.5 × 10⁶	N/m²
Track sub-structure damping	$c_{r 1}^{v}, c_{r 2}^{v}; c_{r 1}^{h}, c_{r 2}^{h}$	4900; 3600	N·s/m

Table 2.

Parameters of track irregularity (Ang et al., 2014).

Severity	Amplitude	Unit
Near smooth	0.01	mm
Moderate	1÷2	mm
Severe	3÷4	mm

For the purpose of comparison, the train speed $V = 20 m / s$ is employed. The radius of the wheel $R_{wheel}$ , the railhead $R_{railprof}$ , and the Poisson ratio $ν$ of the wheel and the rail material are given by $460 mm$ , $300 mm$ , and 0.3, respectively. The Kalker’s coefficients are $f_{11} = 9.43 \times 10^{6} N$ and $f_{11} = 1200 Nm$ .

One advantage of the MEM is that it allows the use of variable element lengths for better computational efficiency. Figure 3 shows the non-uniform discretization of the rail beams with a total of $60 \times 0.4 m$ elements, $40 \times 0.5 m$ elements, $20 \times 1 m$ elements, and $12 \times 1.5 m$ elements. The equations of motion are solved using the Newmark-β method employing a time step of $Δ t = 0.0005 s$ .

Figure 3.

Discretization of two rail beams into moving elements used in this model.

Figure 4 shows the time history displacements of the rails at four contact points for steady-state condition. The rails are assumed to be smooth in this case ( $a_{r 1}^{t} = a_{r 2}^{t} = 0$ ) and the total calculated time is $t_{total} = 0.5 s$ . The results show that the rail displacements are good convergence, except in the initial time interval of 0–0.2 s, due to the initial conditions.

Figure 4.

Time history displacements of the rails at four contact points for steady-state condition (track irregularity amplitude $a_{r 1}^{t} = a_{r 2}^{t} = 0$ ): (a) Vertical displacement at contact point 1; (b) Vertical displacement at contact point 2; (c) Vertical displacement at contact point 3; (d) Vertical displacement at contact point 4.

Table 3 shows the displacements of the wheel-sets and the displacements of the rail at four contact points when the amplitudes and wavelengths of track irregularity are $a_{r 1}^{t} = a_{r 2}^{t} = 0.05 mm$ and $λ_{r 1}^{t} = λ_{r 2}^{t} = 0.5 m$ , respectively. The results obtained in this study are compared with those obtained by Koh et al. (2003). The displacements obtained from the present 3D train-track model are larger than the displacements obtained from the 2D train-track model by Koh et al. (2003) because most of motions of the train are considered in the 3D train-track model.

Table 3.

Displacements of the wheel-sets and the rail at four contact points.

Displacements (mm)		Present study (3D model)	Koh et al. (2003) (2D model)	Percent difference between 3D model and 2D model (%)
Wheel-sets	Wheel-set 1	1.802	1.798	0.22
	Wheel-set 2	1.799		0.05
Contact points on the rail beams	Contact point 1	1.770	1.768	0.11
	Contact point 2	1.770		0.11
	Contact point 3	1.762		0.34
	Contact point 4	1.762		0.34

The DAF is defined as the ratio of the maximum dynamic contact force to the static wheel load, which is the sum of the self-weight of car-body, bogie, and wheel-sets. As expected, for the case of perfectly smooth track and no dynamic load, the DAF is found to be 1. When the DAF is greater than or equal to 2, there is the momentary loss of contact between the wheel and rail, which is known as the jumping wheel phenomenon (Ang et al., 2014). First, the effects of the track irregularity amplitude on DAF1, DAF2, DAF3, and DAF4 at four contact points are shown in Figure 5. In this example, the track irregularity amplitudes of two rails are the same $a_{r 1}^{t} = a_{r 2}^{t}$ and vary from 0.01mm (smooth track condition) to 4mm (severe track condition). The track irregularity wavelengths of two rails are $λ_{r 1}^{t} = λ_{r 2}^{t} = 0.5 m$ . Figure 5(a) shows that when the train speed is 50 m/s and the track irregularity amplitudes are $a_{r 1}^{t} = a_{r 2}^{t} \leq 1$ (the track irregularity is smooth), the DAF is smaller than 2. Figure 5(b) and (c) shows that when the train speed is higher than 70 m/s, the DAF is much larger than 2 although the tracks are in smooth condition.

Figure 5.

Effect of track irregularity amplitude on the DAF. The train speed of (a) $V = 50 m / s$ , (b) $V = 70 m / s$ , and (c) $V = 90 m / s$ .

Second, the effects of the track irregularity amplitude ratio $a_{r 1}^{t} / a_{r 2}^{t}$ on the DAF at four contact points are shown in Figure 6. All the track irregularity wavelengths of two rails are considered $λ_{r 1}^{t} = λ_{r 2}^{t} = 1 m$ ; the track irregularity amplitude of the second rail is kept at $a_{r 2}^{t} = 2 mm$ ; and the track irregularity amplitude ratio $a_{r 1}^{t} / a_{r 2}^{t}$ is varied from $0.005$ (near smooth track condition) to 2 (severe track condition). As expected, the results show that the DAF is proportional with the track irregularity amplitude ratio $a_{r 1}^{t} / a_{r 2}^{t}$ and the train speed V. Figure 6(a) and (b) shows that when the train speed is lower than 70 m/s and the track irregularity amplitude ratio is $a_{r 1}^{t} / a_{r 2}^{t} \leq 1$ (the track irregularity is smooth), the DAF is smaller than 2. However, Figure 6(c) shows that when the train speed is higher than 90 m/s, the DAF is larger than 2 although the tracks are in smooth conditions $(a_{r 1}^{t} / a_{r 2}^{t} \leq 1)$ .

Figure 6.

Effect of the track irregularity amplitude ratio $a_{r 1}^{t} / a_{r 2}^{t}$ on the DAF. The train speed of (a) $V = 50 m / s$ , (b) $V = 70 m / s$ , and (c) $V = 90 m / s$ .

Third, the effects of the track irregularity wavelength on the DAF are investigated. The track irregularity amplitudes of two rails are $a_{r 1}^{t} = a_{r 2}^{t} = 1 mm$ . The track irregularity wavelengths of two rails are the same $(λ_{r 1}^{t} = λ_{r 2}^{t})$ and vary from $0.5 m$ to $4 m$ . Figure 7(a) shows that when the train speed is lower than 50 m/s, the DAF is smaller than 2 and the jumping wheel does not occur. However, Figure 7(b) and (c) shows that when the train speed is higher than 70 m/s and the track irregularity wavelengths are $λ_{r 1}^{t} = λ_{r 2}^{t} \leq 1 m$ , the DAF is significantly larger than 2. It can be inferred that the higher speed amplifies the strong dynamic response of high-speed train-track system, especially for case of severe track condition.

Figure 7.

Effect of track irregularity wavelength on the DAF. The train speed of (a) $V = 50 m / s$ , (b) $V = 70 m / s$ , and (c) $V = 90 m / s$ .

The resonance phenomenon occurs when the frequency of the dynamic excitation force approaches the natural frequencies of the components of the train. The frequency $f_{e}$ of the dynamic excitation force depends on the train velocity and the track irregularity wavelength and is defined as

f_{e} = \frac{V}{λ_{rj}^{t}}

(40)

The natural frequency $f_{n}$ of the components of the train is defined as

f_{n} = \frac{ω}{2 π}

(41)

in which $ω$ can be obtained from the equation

det (K - ω^{2} M) = 0

(42)

Table 4 shows the numerical results of frequencies of the dynamic excitation forces for the irregularity wavelengths ranging from 0.5 to 4 m and the train speeds of $50, 70$ , and $90 m / s$ . The natural frequencies of the car-body, the bogie, and the wheel-sets are presented in Table 5. It can be seen that the frequencies of the dynamic excitation forces when the track irregularity wavelengths $λ_{r 1}^{t} = λ_{r 2}^{t} = 1.5, 2, 2, 5 m$ to the train speed $V = 50, 70, 90 m / s$ are 33.33, 35, and 36 Hz, respectively. The natural frequencies of the wheel-sets are $38.45 Hz$ . The frequencies of the dynamic excitation forces approach the natural frequencies of the wheel-sets and the resonance phenomenon occurs. This explains that the DAF is larger than 2 at the track irregularity wavelengths $λ_{r 1}^{t} = λ_{r 2}^{t} = 1.5, 2.0, 2.5 m$ corresponding to train speed $V = 50, 70, 90 m / s$ as shown in Figure 7.

Table 4.

Frequency $f_{e}$ of the dynamic excitation force (Hz).

Train speed (m/s)	Track irregularity wavelength (m)
Train speed (m/s)	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0
50	100	50	33.33	25	20	16.67	14.29	12.50
70	140	70	46.67	35	28	23.33	20	17.50
90	180	90	60	45	36	30	25.71	22.50

Table 5.

Natural frequencies $f_{n}$ of the train components (Hz).

Train component	Car-body	Bogie	Wheel-set 1	Wheel-set 2
$f_{n}$	1.09	10	38.45	38.45

Figure 8 shows the effects of the track irregularity wavelength ratio $λ_{r 1}^{t} / λ_{r 2}^{t}$ on the DAF. The track irregularity amplitudes are $a_{r 1}^{t} = a_{r 2}^{t} = 1 mm$ . The track irregularity wavelength of the second rail is kept at $λ_{r 2}^{t} = 1 m$ and the track irregularity wavelength ratio $λ_{r 1}^{t} / λ_{r 2}^{t}$ is varied from 0.5 to 4. The results show that the DAF is significantly larger than 2 when the track irregularity wavelength ratio $λ_{r 1}^{t} / λ_{r 2}^{t}$ is smaller than 1 (severe track condition) and the train speed is higher than 70 m/s. However, it decreases strongly for all cases of the train speeds when the track irregularity wavelength ratio $λ_{r 1}^{t} / λ_{r 2}^{t} = 1$ (the irregularity amplitudes and wavelengths of two rails are the same condition).

Figure 8.

Effect of the track irregularity wavelength ratio $λ_{r 1}^{t} / λ_{r 2}^{t}$ on the DAF. The train speed of (a) $V = 50 m / s$ , (b) $V = 70 m / s$ , and (c) $V = 90 m / s$ .

The effects of the track sub-structure stiffness on the DAF are next investigated. The track irregularity amplitudes and wavelengths of two rails are $a_{r 1}^{t} = a_{r 2}^{t} = 2 mm$ and $λ_{r 1}^{t} = λ_{r 2}^{t} = 1 m$ , respectively. The track sub-structure stiffness of the second rail is kept at $k_{r 2}^{v} = 1 \times 10^{7} N / m^{2}$ and the track sub-structure stiffness ratio $k_{r 1}^{v} / k_{r 2}^{v}$ is varied from 0.5 to 5. Figure 9 shows that the DAF is smaller than 2 for all cases of the train speeds when the track sub-structure stiffness ratio is $k_{r 1}^{v} / k_{r 2}^{v} < 1$ . However, as track sub-structure stiffness ratio is $k_{r 1}^{v} / k_{r 2}^{v} > 2$ , the DAF is extremely larger than 2. It is interesting to note that DAF1 and DAF3 are larger than DAF2 and DAF4 when the track sub-structure stiffness ratio is $k_{r 1}^{v} / k_{r 2}^{v} > 1$ . It can be inferred that the DAF at the contact point on the rail resting on a hard foundation is larger than that on the rail resting on a soft foundation.

Figure 9.

Effect of the track sub-structure stiffness ratio $k_{r 1}^{v} / k_{r 2}^{v}$ on the DAF. The train speed of (a) $V = 50 m / s$ , (b) $V = 70 m / s$ , and (c) $V = 90 m / s$ .

Conclusion

In this article, a numerical study on the dynamic response of the 3D train-track model using MEM was carried out. The train was modeled as a complex 3D train model with 16 DOFs, and the two rails were modeled as two infinite Euler–Bernoulli beams resting on a viscoelastic foundation. The MEM method is extended to establish the coupling formulations of the mass, damping, and stiffness matrices in vertical and lateral directions. The benefits of the proposed model are that all of motions of the train and the effects of the track irregularity and track stiffness variations across the track width can be captured for calculating the dynamic response of the train. The effects of various factors such as the train speed, the resonance phenomenon, the track irregularity, and track stiffness variations along the track and across the track width on the DAF were investigated and some useful results can be drawn as follows:

The DAF is proportional to the ratio of the track irregularity amplitude $a_{r 1}^{t} / a_{r 2}^{t}$ and the train speed V. The DAF is smaller than 2 when the train speed is lower than 70 m/s and the ratio $a_{r 1}^{t} / a_{r 2}^{t} \leq 1$ (the smooth track condition). However, the DAF is larger than 2 when the train speed is higher than 90 m/s although the tracks are in smooth condition $(a_{r 1}^{t} / a_{r 2}^{t} \leq 1)$ .

The track irregularity wavelength also has strong effects on the DAF. When the train speed is lower than 50 m/s, the DAF is smaller than 2 for all cases of track irregularity wavelengths. On the contrary, when the train speed is higher than 70 m/s and the track irregularity wavelength ratio is $λ_{r 1}^{t} / λ_{r 2}^{t} \leq 1$ , the DAF is significantly larger than 2.

The frequency of the dynamic excitation force depends on the train velocity and the track irregularity wavelength. The resonance phenomenon occurs when the frequencies of the dynamic excitation force approach the natural frequencies of the wheel-sets.

Finally, the large change in track sub-structure stiffness tends to induce strong effects on the DAF. As the track sub-structure stiffness ratio is $k_{r 1}^{v} / k_{r 2}^{v} \geq 2$ , the DAF is extremely larger than 2. In contrast, when the track sub-structure stiffness ratio is $k_{r 1}^{v} / k_{r 2}^{v} \leq 1$ , the DAF is smaller than 2 for all cases of the train speeds. It is noted that the DAF at the contact point on the rail resting on a hard foundation is larger than that on the rail resting on a soft foundation.

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: This research was funded by the Vietnam National University–Ho Chi Minh City (VNU-HCM) under grant number B2017-20-01: “Development and application of Moving Element Method for dynamic problems in civil engineering structures.”

References

Andersen

Nielsen

SRK

Kirkegaard

(2001) Finite element modelling of infinite Euler beams on Kelvin foundations exposed to moving loads in convected co-ordinates. Journal of Sound and Vibration 241(4): 587–604.

Ang

Dai

(2013) Response analysis of high-speed rail system accounting for abrupt change of foundation stiffness. Journal of Sound and Vibration 332: 2954–2970.

Ang

Dai

Tran

et al . (2014) Analysis of high-speed rail accounting for jumping wheel phenomenon. International Journal of Computational Methods 11: 1343007.

Chen

Wang

(2006) Steady-state response of a Timoshenko beam on an elastic half-space under a moving load. Acta Mechanica Solida Sinica 19(1): 26–39.

Dai

Ang

(2015) Steady-state response of a curved beam on a viscously damped foundation subjected to a sequence of moving loads. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 229: 375–394.

Dai

Ang

Tran

et al . (2017) Moving element analysis of discretely supported high-speed rail systems. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 232: 783–797.

Filho

(1978) Finite element analysis of structures under moving loads. Shock and Vibration Digest 10(8): 27–35.

Frýba

Nakagiri

Yoshikawa

(1993) Stochastic finite element for a beam on a random foundation with uncertain damping under a moving force. Journal of Sound and Vibration 163(1): 31–45.

Hino

Yoshimura

Ananthanarayana

(1985) Vibration analysis of non-linear beams subjected to moving load using the finite element method. Journal of Sound and Vibration 100(4): 477–491.

10.

Hino

Yoshimura

Konishi

et al . (1984) A finite element method prediction of the vibration of a bridge subjected to a moving vehicle load. Journal of Sound and Vibration 96(1): 45–53.

11.

Iwnicki

(2006) Handbook of Railway Vehicle Dynamics. Boca Raton, FL: CRC Press.

12.

Jezequel

(1981) Response of periodic systems to a moving load. Journal of Applied Mechanics 48(3): 613–618.

13.

Jin

Wen

Wang

et al . (2006) Three-dimensional train-track model for study of rail corrugation. Journal of Sound and Vibration 293: 830–855.

14.

Koh

Chiew

Lim

(2007) A numerical method for moving load on continuum. Journal of Sound and Vibration 300: 126–138.

15.

Koh

Ong

JSY

Chua

DKH

et al . (2003) Moving element method for train-track dynamics. International Journal for Numerical Methods in Engineering 56: 1549–1567.

16.

Koh

Sze

Deng

(2006) Numerical and analytical methods for in-plane dynamic response of annular disk. International Journal of Solids and Structures 43: 112–131.

17.

Krenk

Kellezi

Nielsen

SRK

et al . (1999) Finite elements and transmitting boundary conditions for moving loads. In: Proceedings of the 4th European Conference on Structural Dynamics, (Eurodyn’99), Praha, 1–7 June 1999, pp. 447–452. The Netherlands: A.A./Taylor & Francis.

18.

Mathews

(1958) Vibrations of a beam on elastic foundation. Journal of Applied Mathematics and Mechanics 38(3–4): 105–115.

19.

Mathews

(1959) Vibrations of a beam on elastic foundation II. Journal of Applied Mathematics and Mechanics 39(1–2): 13–19.

20.

Nielsen

JCO

Abrahamsson

TJS

(1992) Coupling of physical and modal components for analysis of moving non-linear dynamic systems on general beam structures. International Journal for Numerical Methods in Engineering 33(9): 1843–1859.

21.

Nielsen

JCO

Igeland

(1995) Vertical dynamic interaction between train and track-influence of wheel and track imperfections. Journal of Sound and Vibration 187(5): 825–839.

22.

Olsson

(1985) Finite element, modal co-ordinate analysis of structures subjected to moving loads. Journal of Sound and Vibration 99(1): 1–12.

23.

Ono

Yamada

(1989) Analysis of railway track vibration. Journal of Sound and Vibration 130(2): 269–297.

24.

Sun

(2002) A closed-form solution of beam on viscoelastic subgrade subjected to moving loads. Computers & Structures 80(1): 1–8.

25.

Thambiratnam

Zhuge

(1993) Finite element analysis of track structures. Journal of Microcomputers in Civil Engineering 8(6): 467–476.

26.

Thambiratnam

Zhuge

(1996) Dynamic analysis of beams on an elastic foundation subject to moving loads. Journal of Sound and Vibration 198(2): 149–169.

27.

Tran

Ang

Luong

(2014) Vertical dynamic response of non-uniform motion of high-speed rails. Journal of Sound and Vibration 333: 5427–5442.

28.

Tran

Ang

Luong

et al . (2016) High-speed trains subject to abrupt braking. Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility. DOI: 10.1080/00423114.2016.1232837.

29.

Tran

Ang

Luong

(2017a) Dynamic response of high-speed rails due to heavy braking. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit231: 701–716.

30.

Tran

Ang

Luong

(2017b) Multiple-railcar high speed train subject to braking. International Journal of Structural Stability and Dynamics 17(7): 1750071 (31pp.).

31.

Tran

Ang

Luong

(2017c) Vertical dynamic response of high-speed rails during sudden deceleration. International Journal of Computational Methods 14(1): 1750014 (24pp.).

32.

Lin

Zhang

et al . (2009) 2D moving element method for random vibration analysis of vehicles on Kirchhoff plate with Kelvin foundation. Latin American Journal of Solids and Structures 6: 169–183.