A hybrid solution for studying vibrations of coupled train–track

Abstract

This article develops a hybrid model to analyse the dynamic interactions between a train, tracks and a bridge. The model couples the train and track subsystems to form an integrated time-dependent subsystem through a vertically interacting wheel–rail model. In turn, this time-dependent subsystem is coupled with the bridge subsystem by enforcing the compatibility of forces at the contact points between the track and the bridge. A new hybrid solution algorithm is proposed which combines the strongly coupled method and the loosely coupled method to numerically solve the equation of motion of the coupled train–track–bridge system in the time domain. The integrated time-dependent equation of motion of the train–track subsystem is solved by applying the strongly coupled method. The equilibrium equations of the train–track subsystem and bridge subsystem are then solved via the loosely coupled method using the Newmark integration scheme. Significantly faster convergence can be achieved by avoiding the iterative equilibrium calculations between the wheel and the rail, and the total computational efficiency increases significantly because of the considerably smaller size of the time-dependent equations of motion and larger integration time step. The accuracy and computational cost of the proposed method are validated and compared to the existing models using a case study on the vibration of a cable-stayed bridge.

Keywords

cable-stayed bridge dynamic analysis hybrid model iteration solution method time-dependent system train–track–bridge interaction

Introduction

Numerous studies have addressed the dynamic response of a railway bridge subjected to a travelling train (Azimi et al., 2013; Lou et al., 2012). Developing efficient and robust algorithms to analyse train–bridge interactions remains a critical task, especially given the current trend of designing longer bridge spans which has been enabled by the development of new construction materials and methods (Salcher and Adam, 2015). For example, main spans of the Tsing Ma suspension bridge in Hong Kong and the Hutong Yangtze River railway cable-stayed bridge in mainland China both exceed 1 km in length (Chen et al., 2011, 2014).

Because of the relative movements between the subsystems and the associated constraint equations which relate the train and bridge displacements, the methods used to analyse the vehicle–bridge interaction problem can generally be subdivided into two categories: the strongly coupled method (SCM) and loosely coupled method (LCM; Liu et al., 2014; Zhai et al., 2013). In the SCM, the vehicle and bridge are treated as a single integrated system, in which the forces acting at the contact interface are internal forces (Zhai et al., 2009). The equation of motions of the coupled train–bridge system (TBS) can be solved using the direct integration method (DIM), and a solution is produced at each time step without any iteration (Liu et al., 2014). However, this procedure may be computationally demanding because the global matrices are time dependent. Accordingly, they must be updated and factorized based on a train’s location on the bridge (Neves et al., 2014; Yang and Hwang, 2016).

To overcome these limitations, most investigators use the LCM approach, which partitions the train–bridge interaction system into a train subsystem and a bridge subsystem. The two subsystems are coupled by enforcing the compatibility of the displacements at the contact points between the train and the bridge (Liu et al., 2014). The contact forces are considered explicitly, but they are not treated as unknowns in the governing equilibrium equations. An iterative procedure is used to ensure coupling between the two subsystems at each time step (Nguyen et al., 2009; Zhang and Xia, 2013). A primary advantage of the LCM is that the dynamic property matrices in the two sets of equation of motions remain constant. However, the maximum time interval of iteration that is required for convergence in the LCM is determined by the highest frequency of the system, which may be rather high for cases in which the bridge has many degrees of freedom (DOFs), thus requiring an iterative procedure to achieve convergence (Zhang et al., 2010).

Two approaches have commonly been used in previous studies to achieve higher computational efficiency for both SCM and LCM. One approach involves simplifying the TBS model. Lou et al. (2012) developed a rail–bridge coupling element of unequal lengths, with the length of a bridge element longer than that of a rail element. Yang and Wu (2001) proposed a new contact element based on a condensation technique that eliminates the DOFs at the contact interface. The model superposition method has also been applied by many researchers to reduce the number of DOFs of the TBS model (Jin et al., 2015; Salcher and Adam, 2015). However, the number of modes must be carefully chosen to adjust the high-frequency content of the acceleration response (Zhang et al., 2010). The other approach involves the use of a new numerical technique to solve the TBS vibration equation. Zhang and Xia (2013) proposed an inter-system method, which avoids iteration within time steps.Neves et al. (2014) used an optimized block factorization algorithm to solve the integrated equation of motion of the train–structure system.

Previous studies have demonstrated that while the track plays an important role in the train–bridge interaction, the dynamic behaviour of the track should also be considered in addition to that of the train and bridge to more accurately assess the safety and passenger comfort of a train on a bridge (Guo et al., 2012; Sadeghi et al., 2016b). When a sophisticated track model is integrated into the model used to analyse train–bridge interactions, the train–track–bridge interaction problem becomes considerably more complex than the typical train–bridge interaction problem (Biondi et al., 2005). There are two primary reasons for the increased complexity. First, the entire system is typically modelled using the direct stiffness method (DSM) because it provides a feasible way to couple the train, track and bridge subsystems (Li et al., 2015; Olmos and Astiz, 2013), and it is difficult to create a track–bridge model that simultaneously models the track and bridge as one system via the model superposition method. In this case, many DOFs are unavoidably generated, rendering the entire system cumbersome. Second, the time step length is typically less than 10⁻⁴ s to ensure the stability of the numerical time-domain response when the Hertzian contact model is used between the wheel and the rail (Chen et al., 2015). However, an excessively short time step size will lead to an excessive number of time steps, thereby further reducing the computational efficiency (Sadeghi et al., 2016a). These issues highlight the need for further study on the subject.

In this study, a hybrid model (HM) with the corresponding solution algorithm is presented to improve the computation efficiency of the dynamic interaction of coupled train–track–bridge systems (CTTBSs). The model divides the CTTBS into a train–track subsystem and a bridge subsystem. In the HM, the SCM is first used to obtain an integrated coupled time-dependent equation of motion for the train–track subsystem. Then, the LCM is used to couple the train–track subsystem and the bridge subsystem. At each time step, only the matrices of the train–track subsystem require updating, whereas the matrices of the bridge subsystem remain unchanged throughout the entire numerical simulation process. This significantly reduces computational costs. The iteration equilibrium procedure is modified from the wheel-and-rail type to the track-and-bridge type. Because the stiffness between the track and the bridge is lower than wheel–rail contact stiffness, maximum time requirements step for stability and equilibrium conditions in the iterative calculation can be more easily satisfied. The HM is validated by comparing the HM results with the SCM and LCM results using a numerical example. The proposed methodology is implemented in the MATLAB environment. The track and bridge are modelled using ANSYS, with their structural matrices extracted via a MATLAB program.

Train, track and bridge models

Train model

In this study, the travelling train is assumed to consist of $N_{v}$ identical vehicles. The car body, bogie frames and wheelsets of each vehicle are assumed to behave as rigid components without elastic deformation during vibration, and the vehicles are assumed to undergo only in-plane motion, as shown in Figure 1(a). The connections between the car body and each bogie and those between each bogie and its wheelsets are represented by linear springs and viscous dashpots, whose stiffness (damping) coefficients are $k_{s} \cdot (c_{s})$ and $k_{p} \cdot (c_{p})$ , respectively. Under this assumption, stiffness, damping and mass matrices with 10 DOFs can be constructed in the same manner as suggested by Biondi et al. (2005). Downward vertical displacements and clockwise rotations of the vehicle are taken to be positive and they are measured with reference to their respective static equilibrium positions before entering the section of track under consideration. All wheelsets of the train are numbered from 1 to $4 \times N_{v}$ according to the travel direction of the train. Thus, for the jth vehicle, the numbers of the four wheelsets from front to back are $(4 j - 3), (4 j - 2), (4 j - 1) and 4 j$ .

Figure 1.

Train–track–bridge interaction system: (a) elevation and (b) wheel–rail contact model.

Track and bridge models

The track and bridge can be accurately modelled using finite elements (FEs) of the beam, shell, link and spring-damper types, with the choice of each element type depending on the particular bridge configuration. Beam elements are frequently used for the main girders and piers, shell elements are often used for the bridge deck and link elements are typically used for the stay cables.

Track behaviour is affected by many factors, including the wheel–rail contact model, the bending stiffness of the rails, the shear deformation of the rails and the mechanical behaviour of the track components, such as fasteners. The majority of the track components can be feasibly modelled using FEs. Thus, the track model is constructed as a three-dimensional (3D) single-layer track which consists of a rail and fasteners, as shown in Figure 1(a). The top rail and the sleepers are modelled as beam elements, and the elasticity and damping properties of the fasteners and ballast are represented by discrete springs and dampers. Each fastener is modelled as four spring–damper elements: a longitudinal spring, a lateral spring, a vertical spring and a torsion spring with stiffnesses of $k_{pd}$ , $k_{pl}$ , $k_{pv}$ and $k_{pt}$ , respectively, and damping coefficients of $c_{pd}$ , $c_{pl}$ , $c_{pv}$ and $c_{pt}$ , respectively. The wheel–rail contact is modelled as Hertzian spring. It should be noted that for simplicity the vehicles are assumed to undergo only in-plane motion while the track and bridge are established as 3D spatial models; therefore, each wheelset is connected with two rails by two identical Hertzian springs of stiffness $k_{h}$ , as shown in Figure 1(b). The Hertzian stiffness $k_{h}$ is linearized from Hertz non-linear contact theory (Code, 2005; Esveld, 2001) and can be written as

k_{h} = \frac{3}{2 G} P_{0}^{1 / 3}

(1)

where G represents the wheel–rail contact constant $(m / N^{2 / 3})$ and $P_{0}$ represents the static wheel axle load.

Using the simplified linear or non-linear wheel–rail contact relationship shown in Zeng et al. (2016) and Salcher and Adam (2015), the method developed in this study may also be used for the 3D train–track–bridge coupled dynamic analysis. Let $r (x)$ denote the initial top surface irregularities of the rail, where these irregularities are measured with reference to a smooth rail profile, that is, $r (x) = 0$ if the top surface of the rail is smooth.

Equation of motion for the CTTBS

The CTTBS model shown in Figure 1 can be analysed using the HM, in which the system is separated into a train–track subsystem and a bridge subsystem. To this end, the train and track are interconnected by Hertzian springs and modelled as a coupled system, whereas the train–track subsystem and bridge subsystem are connected by the springs and dampers used to model the fasteners of the track.

The equations of motion for the CTTBS can now be represented as follows

[\begin{matrix} M_{v v} & 0 \\ 0 & M_{r r} \end{matrix}] {\begin{matrix} {\overset{\cdot\cdot}{X}}_{v} \\ {\overset{\cdot\cdot}{X}}_{r} \end{matrix}} + [\begin{matrix} C_{v v} & 0 \\ 0 & C_{r r} \end{matrix}] {\begin{matrix} {\dot{X}}_{v} \\ {\dot{X}}_{r} \end{matrix}} + [\begin{matrix} K_{v v} & K_{v r} \\ K_{r v} & K_{r r} \end{matrix}] {\begin{matrix} X_{v} \\ X_{r} \end{matrix}} = {\begin{matrix} F_{v} \\ F_{r} \end{matrix}}

(2a)

M_{b b} {\overset{\cdot\cdot}{X}}_{b} + C_{b b} {\dot{X}}_{b} + K_{b b} X_{b} = F_{b}

(2b)

where equations (2a) and (2b) represent the dynamic equations of the train–track subsystem and the bridge subsystem, respectively. The two subsystems are coupled through the force vectors $F_{r}$ and $F_{b}$ which are explained in detail in section ‘Load vectors of the vehicles, rail and bridge’. M , C and K represent the mass, damping and stiffness matrix, respectively. X represents the displacement vectors, and F represents the load vectors. The subscripts v, r and b denote the vehicles, rail and bridge, respectively. Because the wheel–rail contact is modelled as a Hertzian spring, the train and track are coupled only through the stiffness matrices. The displacement vectors, the mass, stiffness and damping matrices and the load vectors of the vehicles, rail and bridge are explained in the following.

Displacement vectors

The vehicle displacement vector $X_{v}$ , which is of order $(10 \times N_{v}) \times 1$ , can be written as

X_{v} = {\begin{matrix} X_{v 1} & X_{v 2} & \begin{matrix} \dots & X_{v N_{v}} \end{matrix} \end{matrix}}^{T}

(3)

where the superscript T denotes the transpose of the matrix and $X_{vj} (j = 1, 2, \dots, N_{v})$ denotes the displacement vector of the $j th$ vehicle. $X_{vj}$ , which is of order $1 \times 10$ , can be written as follows

X_{v j} = {\begin{matrix} z_{c j} & θ_{c j} & z_{b 1 j} & θ_{b 1 j} & z_{b 2 j} & θ_{b 2 j} & z_{w (4 j - 3)} & z_{w (4 j - 2)} & z_{w (4 j - 1)} & z_{w 4 j} \end{matrix}}

(4)

The rail displacement vector $X_{r}$ , which is of order $N_{r} \times 1$ , is obtained by collecting all the rail node displacements ${\begin{matrix} x & y & \begin{matrix} z & rot x & \begin{matrix} rot y & rot z \end{matrix} \end{matrix} \end{matrix}}$ , of order $6 \times 1$ , and the bridge displacement vector $X_{b}$ , which is of order $N_{b} \times 1$ , can be derived from the bridge FE model. Here, $N_{r}$ and $N_{b}$ denote the total number of DOFs of the rail and bridge, respectively.

Matrices for the vehicles

The matrices for the vehicles are denoted by the subscript vv. The mass matrix $M_{vv}$ of the vehicles, which is of order $(10 \times N_{v}) \times (10 \times N_{v})$ , can be written as

M_{vv} = diag [\begin{matrix} M_{v 1} & M_{v 2} & \begin{matrix} \dots & M_{v N_{v}} \end{matrix} \end{matrix}]

(5)

where $M_{vj}$ , of order $10 \times 10$ , denotes the mass matrix of the $j th$ vehicle and can be expressed as

M_{vj} = diag [\begin{matrix} m_{c} & J_{c} & m_{t} & J_{t} & m_{t} & J_{t} & \begin{matrix} m_{w} & m_{w} & \begin{matrix} m_{w} & m_{w} \end{matrix} \end{matrix} \end{matrix}]

(6)

The vehicle stiffness matrix $K_{vv}$ of the vehicles, which is of order $(10 \times N_{v}) \times (10 \times N_{v})$ , can be written as

K_{vv} = diag [\begin{matrix} K_{v 1} & K_{v 2} & \begin{matrix} \dots & K_{v N_{v}} \end{matrix} \end{matrix}]

(7)

where $K_{vj}$ , of order $10 \times 10$ , denotes the stiffness matrix of the $j th$ vehicle and can be expressed as

K_{vj} = K_{vj 1} + K_{vj 2}

(8)

where $K_{vj 1}$ , which is of order $10 \times 10$ , represents the stiffness matrix of the $j th$ vehicle itself and can be written as

K_{vj 1} = [\begin{matrix} 2 k_{s} \\ 0 \\ \begin{matrix} - k_{s} \\ 0 \\ \begin{matrix} - k_{s} \\ 0 \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \begin{matrix} 0 \\ 2 k_{s} L_{c}^{2} \\ \begin{matrix} - k_{s} L_{c} \\ 0 \\ \begin{matrix} k_{s} L_{c} \\ 0 \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \begin{matrix} - k_{s} \\ - k_{s} L_{c} \\ \begin{matrix} k_{s} + 2 k_{p} \\ 0 \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} - k_{p} \\ - k_{p} \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 2 k_{p} L_{b}^{2} \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} - k_{p} L_{b} \\ k_{p} L_{b} \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \begin{matrix} - k_{s} \\ k_{s} L_{c} \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} k_{s} + 2 k_{p} \\ 0 \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} - k_{p} \\ - k_{p} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 2 k_{p} L_{b}^{2} \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} - k_{p} L_{b} \\ k_{p} L_{b} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \begin{matrix} 0 \\ 0 \\ \begin{matrix} - k_{p} \\ - k_{p} L_{b} \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} k_{p} \\ 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \begin{matrix} 0 \\ 0 \\ \begin{matrix} - k_{p} \\ k_{p} L_{b} \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ k_{p} \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \begin{matrix} 0 \\ 0 \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} - k_{p} \\ - k_{p} L_{b} \\ \begin{matrix} 0 \\ 0 \\ \begin{matrix} k_{p} \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}]

(9)

$K_{vj 2}$ , also of order $10 \times 10$ , represents the stiffness matrix of the jth vehicle induced by the wheel–rail contact and can be written as

K_{vj 2} = diag [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 2 k_{h} & 2 k_{h} & 2 k_{h} & 2 k_{h} \end{matrix}]

(10)

In the above expressions, $k_{h}$ denotes the Hertzian spring coefficient $L_{c}$ and $L_{b}$ denote the half the distance between the front bogie’s centre of gravity and the rear bogie’s centre of gravity and half the length of the bogie axle base, respectively.

The damping matrix of the vehicles, $C_{vv}$ , which is of order $(10 \times N_{v}) \times (10 \times N_{v})$ , can be written as

C_{vv} = diag [\begin{matrix} C_{v 1} & C_{v 2} & \begin{matrix} \dots & C_{v N_{v}} \end{matrix} \end{matrix}]

(11)

where $C_{vj}$ , of order $10 \times 10$ , represents the damping matrix of the $j th$ vehicle and can be obtained by replacing k in the corresponding stiffness matrix $K_{vj 1}$ with c.

Matrices for the rail

Matrices for the rail are denoted by the subscript rr. The consistent mass matrix $M_{rr}$ of the rail, which is of order $N_{r} \times N_{r}$ , and the stiffness matrix $K_{rr 1}$ of the rail itself, also of order $N_{r} \times N_{r}$ , can be obtained using the DSM.

The total stiffness matrix for the rail $K_{rr}$ can be written as

K_{rr} = K_{rr 1} + K_{rr 2} + K_{rr 3}

(12)

with

K_{rr 2} = \sum_{i = 1}^{4 \times N_{v}} \sum_{j = 1}^{2} k_{h} \cdot N_{ij}^{T} \cdot N_{ij}

(13)

where $K_{rr 2}$ , of order $N_{r} \times N_{r}$ , represents the overall stiffness matrix induced by all vehicles. Only the vertical displacements (z) and the rotations around the y-axis ( $rot y$ ) of the nodes of the rail element upon which a wheelset is acting are coupled with the displacement of the wheelset because there is only vertical contact between the wheelset and the rail. $N_{ij}$ , which is of order $1 \times N_{r}$ , represents the shape function vector of a rail element in contact with a wheel

N_{ij} = {[\begin{matrix} 0 & 0 & \begin{matrix} 0 & N_{1} & \begin{matrix} 0 & N_{2} & \begin{matrix} 0 & 0 & \begin{matrix} 0 & N_{3} & \begin{matrix} 0 & N_{4} & \begin{matrix} 0 & 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] |}_{x = x_{i}}

(14)

N_{1} (x) = 1 - 3 {(\frac{x}{l})}^{2} + 2 {(\frac{x}{l})}^{3}

(14a)

N_{2} (x) = x {(1 - \frac{x}{l})}^{2}

(14b)

N_{3} (x) = 3 {(\frac{x}{l})}^{2} - 2 {(\frac{x}{l})}^{3}

(14c)

N_{4} (x) = \frac{x^{2}}{l} (\frac{x}{l} - 1)

(14d)

where the 0 represents the DOFs of all parts of the rail except for the DOFs of the $j th$ rail element upon which the $i th$ wheelset is acting. $x_{i}$ represents the local coordinate of the $i th$ wheelset on the rail element, and l is the length of the rail element.

$K_{rr 3}$ , which is of order $N_{r} \times N_{r}$ , represents the overall stiffness matrix induced by the stiffness of all fasteners and is obtained by combining all the node stiffness matrices to obtain $diag [\begin{matrix} k_{pd} & k_{pl} & \begin{matrix} k_{pv} & k_{pt} & \begin{matrix} 0 & 0 \end{matrix} \end{matrix} \end{matrix}]$ , of order $6 \times 1$ .

With the omission of the rail’s damping, the damping matrix of the rail, $C_{rr}$ , which is of $N_{r} \times N_{r}$ , represents the overall damping matrix induced by the damping of all fasteners, which can be obtained by replacing k in the corresponding stiffness matrix $K_{rr 3}$ with c.

Matrices for the bridge

Matrices for the bridge are denoted by the subscript bb. The consistent mass matrix of the bridge, $M_{bb}$ , which is of order $N_{b} \times N_{b}$ , and the stiffness matrix of the bridge, $K_{bb},$ , which is of order $N_{b} \times N_{b}$ , can be extracted from the FE model using the commercial FE software ANSYS (Zhihui et al., 2014).

When the bridge is directly modelled using FEs, the damping matrix of the bridge, $C_{bb}$ , is constructed based on Rayleigh damping and is expressed as a linear combination of the mass matrix and stiffness matrix as follows

C_{bb} = α \cdot M_{bb} + β \cdot K_{bb}

(15)

Given a damping ratio of $ξ_{r}$ , the two coefficients $α$ and $β$ can be calculated as $α = 2 ξ_{r} ω_{1} ω_{2} / (ω_{1} + ω_{2})$ and $β = 2 ξ_{r} / (ω_{1} + ω_{2})$ , where $ω_{1}$ and $ω_{2}$ are the first two natural circular frequencies of vibration of the bridge.

Matrices of the vehicle–rail interaction

The matrices induced by the vehicle–rail interaction are denoted by the subscripts vr or rv; the stiffness matrices $K_{vr}$ , of order $(10 \times N_{v}) \times N_{r}$ , and $K_{rv}$ , of order $N_{r} \times (10 \times N_{v})$ , that are induced by the interaction between the vehicles and the rail can be written as

K_{vr} = \sum_{i = 1}^{4 \times N_{v}} \sum_{j = 1}^{2} K_{vr}^{ij}

(16)

K_{rv} = \sum_{i = 1}^{4 \times N_{v}} \sum_{j = 1}^{2} K_{rv}^{ij}

(17)

with

K_{rvT}^{ij} = K_{vr}^{ij} = [\begin{matrix} 0 & 0 & \dots & - k_{h} N_{ij} & \dots & 0 & 0 \end{matrix}]_{(10 \times N_{v}) \times N_{r}}

(18)

where $K_{rv}^{ij}$ and $K_{vr}^{ij}$ represent the stiffness matrices induced by the interaction between the ith wheelset and the jth rail element upon which the wheelset is acting. Each row in $K_{vr}^{ij}$ is a zero vector except for the row corresponding to the ith wheelset, and each column in $K_{rv}^{ij}$ is a zero vector except for the column corresponding to the ith wheelset.

Load vectors of the vehicles, rail and bridge

The load vector of the vehicles, $F_{v}$ , which is of order $(10 \times N_{v}) \times 1$ , can be written as

F_{v} = {\begin{matrix} F_{v 1} & F_{v 2} & \begin{matrix} \dots & F_{v N_{v}} \end{matrix} \end{matrix}}^{T}

(19)

where $F_{vj}$ , of order $1 \times 10$ , is the load vector of the $j th$ vehicle

\begin{matrix} F_{v j} = {\begin{matrix} m_{c} g & 0 & m_{b} g & 0 & m_{b} g & 0 \end{matrix} \\ \begin{matrix} m_{w} g + F_{(4 j - 3)}^{i r r} & m_{w} g + F_{(4 j - 2)}^{i r r} & m_{w} g + F_{(4 j - 2)}^{i r r} & m_{w} g + F_{4 j}^{i r r}} \end{matrix} \end{matrix}

(20)

Here, $F_{(4 j - 3)}^{irr}$ , $F_{(4 j - 2)}^{irr}$ , $F_{(4 j - 1)}^{irr}$ and $F_{4 j}^{i rr}$ represent the loads on the wheelsets of the jth vehicle that are induced by the irregularity of the rail; for the ith wheelset, $F_{irr}^{i}$ can be written as

F_{i}^{irr} = 2 k_{h} r (x_{i})

(21)

The load vector of the rail, $F_{r}$ , which is of order $N_{r} \times 1$ , can be written as

F_{r} = F_{rv} + F_{rb}

(22)

where $F_{rv}$ , of order $N_{r} \times 1$ , represents the load induced by all wheelsets and can be expressed as

F_{rv} = \sum_{i = 1}^{4 \times N_{v}} \sum_{j = 1}^{2} - k_{h} r (x_{i}) \cdot N_{ij}

(23)

and $F_{rb}$ , also of order $N_{r} \times 1$ , represents the load induced by the bridge response and can be expressed as

F_{rb} = K_{rr 3} X_{rb}^{r} + C_{rr} {\overset{\cdot}{X}}_{rb}^{r}

(24)

Here, $X_{r b}^{r}$ and ${\overset{\cdot}{X}}_{rb}^{r}$ , which are both of order $N_{r} \times 1$ , represent the displacements and velocities, respectively, of the bridge nodes that support the rail nodes. Each rail node corresponds to a supporting node, and the positions of the DOFs of a supporting node in $X_{r b}^{r}$ or ${\overset{\cdot}{X}}_{rb}^{r}$ are the same as the positions of the DOFs of the corresponding rail node in $X_{r}$ .

The load vector $F_{b}$ of the bridge, which is of order $N_{b} \times 1$ , can be written as

F_{b} = F_{rb} = K_{br} (X_{br}^{b} - X_{rb}^{b}) + C_{br} ({\overset{\cdot}{X}}_{br}^{b} - {\overset{\cdot}{X}}_{rb}^{b})

(25)

where $F_{rb}$ represents the load induced by track response; $X_{br}^{b}$ and ${\overset{\cdot}{X}}_{br}^{b}$ , both of order $N_{b} \times 1$ , represent the displacements and velocities, respectively, of the rail nodes; the positions of the DOFs of a rail node in $X_{br}^{b}$ or ${\overset{\cdot}{X}}_{br}^{b}$ are the same as the positions of the DOFs of the corresponding supporting node in $X_{b}$ . $X_{rb}^{b}$ and ${\overset{\cdot}{X}}_{rb}^{b}$ represent the displacements and velocities, respectively, of the supporting nodes. $K_{br}$ , which is of order $N_{b} \times N_{b}$ , represents the stiffness matrix induced by the stiffness of all fasteners and can be derived by combining the node stiffness matrices to obtain $diag [\begin{matrix} k_{pd} & k_{pl} & \begin{matrix} k_{pv} & k_{pt} & \begin{matrix} 0 & 0 \end{matrix} \end{matrix} \end{matrix}]$ , of order $6 \times 1$ . $C_{br}$ , which is also of order $N_{b} \times N_{b}$ , represents the damping matrix induced by the damping of all fasteners and can be obtained by simply replacing k in the corresponding stiffness matrix $K_{br}$ with c.

Hybrid solution algorithm

A solution method named the hybrid solution algorithm (HSA) is developed here for the HM. First, the Newmark-β method is used to solve the equations of motion of the train–track subsystem and the bridge subsystem separately. The shape function $N_{ij}$ and the irregularities of the rail $r (x)$ with respect to the local coordinate x in the train–track subsystem are time dependent; therefore, the matrices and load vectors that contain $N_{ij}$ and $r (x)$ , that is, the matrices $K_{rr}$ , $K_{rv}$ and $K_{vr}$ and the load vectors $F_{v}$ and $F_{rv}$ , are also time dependent. These time-dependent quantities must be updated at each time step. Second, the iterative equilibrium procedure proposed in Nguyen et al. (2009) is adopted to determine the rail–bridge interaction forces between the train–track subsystem and the bridge subsystem, and a specified tolerance ε (with a value that is assumed to be between 1.0E−5 and 1.0E−8) is used to check the convergence of the iterative solution. The force vector $F_{rb}$ in equation (24) is a function of $X_{rb}^{r}$ and ${\overset{\cdot}{X}}_{rb}^{r}$ , which represent the displacements and velocities, respectively, of the supporting nodes of the rail; the force vector $F_{b}$ in equation (25) is a function of $X_{br}^{b}$ and ${\overset{\cdot}{X}}_{br}^{b}$ , which represent the displacements and velocities, respectively, of the rail nodes. Finally, if the convergence criterion is satisfied, the calculation proceeds to the next time step.

The detailed operations of the HSA consist of the following procedures:

Step 1. Prepare the initial matrices of the train, track and bridge model; this step is the same for the LCM and the SCM.

Step 2. Construct the integrated time-dependent equation of motion of the train–track subsystem.

Step 3. Update the train–track subsystem according to the positions of the vehicles.

Step 4. Calculate the force vector $F_{rb}$ in equation (24) according to the bridge motion obtained in the previous iteration loop (or the last time step) to solve for the train–track subsystem.

Step 5. Calculate the force vector $F_{b}$ in equation (25) according to the rail motion obtained in Step 4 to solve for the bridge subsystem.

Step 6. Calculate the errors between the updated rail–bridge interaction forces after Step 5 and those found the previous iteration loop for the convergence check. If the convergence check is not satisfied, then return to Step 4; otherwise, proceed to the next time step or stop the program if the calculation is complete.

The computation procedure for the HSA is shown in Figure 2. To illustrate the differences among the computation strategies of the HSA, the LCM and the SCM, the computation procedures for the LCM and SCM are shown in Figure 3. More details regarding the LCM and SCM can be found from Liu et al. (2014) and Neves et al. (2014).

Figure 2.

Flowcharts of computation procedure for the hybrid solution algorithm.

Figure 3.

Flowcharts of the computation procedures: (a) loosely coupled method and (b) strongly coupled method.

When the LCM is used to separate the CTTBS into a train subsystem and a track–bridge subsystem at the locations of the wheel–rail contact points, a small time step of less than 1E−4 s is required to maintain the stability of the iterative solution of the equation of motion. Alternatively, the SCM could be used; however, in the SCM, the complexity of the matrices is increased by coupling the train and bridge, which reduces the computation efficiency of solving the equations of motion. In contrast to the LCM and SCM, the HM separates the CTTBS into a train–track subsystem and a bridge subsystem at the interfaces between the track and the bridge. The equilibrium conditions in the HM are enforced at the track–bridge interfaces instead of at the wheel–rail contact points. Because the high-frequency components of the track–bridge forces are small, the convergent results can be obtained more easily. Furthermore, the matrices of the bridge subsystem need not be updated at each time step; only the matrices of the train–track subsystem change at each time step, thereby significantly reducing the computational cost at each time step.

Case study and discussion

Case description of a CTTBS

A cable-stayed bridge located on the new Chinese high-speed line between Shanghai and Kunming is considered as numerical example to verify the accuracy and computational efficiency of the proposed HSA (see Figure 4(a)). The bridge is a single-tower double-cable-plane bridge consisting of three spans with an overall length of 224 m (112 m + 80 m + 30 m). The strength grade of the concrete is C50, and the density is 26 kN/m³. The geometric characteristics of the channel section girder are shown in Figure 4(b), the girder width is 10.8 m and the girder heights in spans 1 and 2 are 3.7 and 3.5 m, respectively. The cables are made of parallel steel wire with a diameter of 7 mm and a tensile strength of 1670 MPa. The secondary dead load (including the ballast, sleepers and rails) is 67.9 kN/m³. The bridge’s total mass is approximately 11,800 ton.

Figure 4.

Cable-stayed bridge: (a) elevation and (b) schematic diagram of a girder section.

FE modelling

The 3D FE model of the cable-stayed bridge, which is shown in Figure 5, including the rail, the ballast track, the girders and their connections, is established using ANSYS. The entire model consists of the cable–stayed bridge and two simply supported girders at either end of the cable-stayed bridge. The rails, sleepers, girders, piers and tower are all modelled using beam elements (see Figure 5). Distributed uniaxial tension–compression spring–dashpot units are used to connect the rails and sleepers in both the lateral and vertical directions to simulate the stiffness and damping of the rail pads and fasteners (see Figure 5). Uniaxial spring–dashpot units are also used to reproduce the stiffness and damping of the ballast in the track. The non-structural mass of the ballast bed is added to the self-weight of the girders. Table 1 summarizes the track system’s main physical properties. Although a stay cable actually has a curved form between two anchorage points because of the sagging effect caused by the self-weight of the cable, a straight truss element is used to approximate such a curved cable; its equivalent modulus, $E_{eq}$ , is as given by Ernst (1965).

Figure 5.

The 3D FE model of the cable-stayed bridge and a detail view of the track.

Table 1.

Properties of the track system.

Property	Notation	Unit	Value
Rail mass per unit length	$m_{r}$	kg	60
Sleeper mass	$m_{s}$	kg	340
Vertical stiffness of a rail pad	$k_{pv}$	$N / m$	6.0 × 10⁷
Vertical damping of a rail pad	$c_{pv}$	$N s / m$	7.5 × 10⁴
Lateral stiffness of a rail pad	$k_{pl}$	$N / m$	3.0 × 10⁷
Lateral damping of a rail pad	$c_{pl}$	$N s / m$	6.0 × 10⁴
Longitudinal stiffness of a rail pad	$k_{pd}$	$N / m$	3.0 × 10⁷
Longitudinal damping of a rail pad	$c_{pd}$	$N s / m$	2.0 × 10³
Torsion stiffness of a rail pad	$k_{pt}$	$N m / rad$	1.0 × 10¹⁰
Torsion damping of a rail pad	$c_{pt}$	$N m s / rad$	1.0 × 10⁵
Density of the ballast layer	$ρ_{b}$	$kg / m^{3}$	1.7 × 10³
Vertical stiffness of the ballast layer per unit length	$k_{bv}$	$N / m$	5.0 × 10⁸
Vertical damping of the ballast layer per unit length	$c_{bv}$	$N s / m$	2.0 × 10⁵
Horizontal stiffness of the ballast layer per unit length	$k_{bh}$	$N / m$	1.56 × 10⁸
Horizontal damping of the ballast layer per unit length	$c_{bh}$	$N s / m$	8.0 × 10⁴
Shear stiffness of the ballast layer per unit length	$k_{bs}$	$N / m$	7.84 × 10⁷
Shear damping of the ballast layer per unit length	$c_{bs}$	$N s / m$	8.0 × 10⁴

Modelling the track and bridge together in ANSYS allows easy confirmation of the supporting nodes of the rail; the mass, stiffness and damping matrices for these models are then separated before the dynamic analysis using a MATLAB program with the form shown in section ‘Equation of motion for the CTTBS’. The equations of motion of the track and bridge are expressed using the DSM, as described in sections ‘Matrices for the rail’ and ‘Matrices for the bridge’. The bridge subsystem has about 120,000 DOFs, whereas the track subsystem has about 2000 DOFs.

In this study, the ICE-3 high-speed train is adopted for the dynamic analysis of the CTTBS. The train consists of a locomotive followed by six carriages and another locomotive. The detailed properties of the ICE-3 train are shown in Du et al. (2012). The train subsystem has 80 DOFs.

Results and analysis

The complete history of the train travelling over the bridge at a speed of $250 km / h$ is analysed with HSA, SCM and LCM, separately. Rayleigh damping is adopted, and the damping ratio of the bridge is assumed to be 0.02. Track irregularity data generated from the German low-disturb track spectra are adopted, as shown in Figure 6. The total numerical simulation time is 8 s.

Figure 6.

Vertical track irregularity sample.

In this article, we focused on the result (the bridge response, the vehicle response and the wheel–rail contact forces) even adopting a large time step, whose accuracy would be acceptable. However, such a time step (0.001 s) may induce convergency problems for LCM. After a few trail calculations, it was found that if the time step was larger than or equal to 0.0003 s, the results produced using LCM would not converge. Therefore, the time step of 0.0002 s was used for the LCM and the time step of 0.001 s was used for the SCM and HSA.

Comparisons of the displacement and acceleration are shown in Figures 7 and 8 for point A (see Figure 4), respectively. There were excellent agreements among the three methods, with a maximum error of less than 5%. Similarly, good comparisons were obtained for the vertical acceleration of the first car body (Figure 9) and for the contact force between the first wheelset and the rail (Figure 10). Zoomed-in plots are provided for acceleration and the contact force (see Figures 8 and 10, respectively) to clear illustration of the differences. Again, satisfactory agreements were found for the different methods.

Figure 7.

Vertical displacement at point A.

Figure 8.

Vertical acceleration at point A.

Figure 9.

Vertical acceleration of the first car body.

Figure 10.

Wheel–rail contact force.

Table 2 summarizes the elapsed times for the three methods. All calculations were performed using a personal computer equipped with Intel^® Core™ i7-4790K quad core processors running at 4.00 GHz. Unlike in the LCM, the coefficient matrices of the train–track subsystem in the HSA must be updated at every time step, which slightly extends the calculation time for a single step in the HSA (0.047 s in this example) compared to the LCM (0.039 s in this example). Thus, HSA is not superior in terms of computation efficiency with the same time step. However, HSA method can adopt larger time step (as in this article, a time step of 0.001 s is adopted) with lower accuracy only for the rail acceleration and still acceptable accuracy for the bridge responses, vehicle responses and the wheel–rail contact force, as shown in Figures 7 to 10. As a result, the elapsed time for the LCM (1546 s in this example) is significantly longer than that for the HSA (379 s in this example). Compared with the SCM, the dimensionality of the train–track subsystem coefficient matrices in the HSA is significantly lower, thereby improving the computational efficiency. The calculation time for a single step in the SCM is 0.121s and that in the HSA is 0.047s, in this example.

Table 2.

Calculation times for the three methods.

Method	Elapsed time (s)	Time step (s)	Total integration steps	Calculation time per step (s)
LCM	1546	0.0002	40,000	0.039
SCM	965	0.001	8000	0.121
HSA	379	0.001	8000	0.047

LCM: loosely coupled method; SCM: strongly coupled method; HSA: hybrid solution algorithm.

Conclusion

This article established an HM to perform dynamic analysis of a CTTBS. In the HM, the CTTBS is separated into a train–track subsystem and a bridge subsystem. These subsystems are coupled by enforcing the compatibility of the forces at the contact points between the track and the bridge. The proposed HSA which is based on the HM combines the SCM and the LCM. To demonstrate the proposed HM and the corresponding HSA, numerical analyses of the interactions of a specific CTTBS were conducted using the HM, SCM and LCM for a high-speed train travelling across a cable-stayed bridge. The numerical results demonstrated that the results obtained from the three methods are in excellent agreement with each other.

Compared with the SCM and LCM, the characteristics of the HM and the corresponding HSA are as follows:

The equations of motion of the train–track subsystem and the bridge subsystem demonstrate that only the mass, damping and stiffness matrices of the train–track subsystem are time dependent and need to be updated. Conversely, the bridge subsystem matrices are independent of time and do not require updating. The time consumed at each time step of the HSA is relatively low compared with that for the SCM.

Coupling the train–track subsystem and the bridge subsystem is achieved by imposing equilibrium conditions that relate the responses of the track to the corresponding responses of the bridge. In comparison to the LCM, the HSA can achieve better computation stability, although it is not superior in terms of computation efficiency with the same time step.

The train is idealized as a combination of a number of rigid bodies connected by a series of springs and dampers, whereas the track and bridge are modelled using the conventional FE method to account for all significant deformations. The matrices of the train–track subsystem are automatically constructed using the proposed method and are not affected by the bridge subsystem. This reduces the difficulty of establishing the equations of motion of the CTTBS.

If the accurate rail acceleration is not considered, the proposed HSA is an efficient method for studying the interaction problem of a train travelling across a complex bridge, such as a long-span plate-truss cable-stayed bridge.

The integration time step is set to be 0.001 s for the HSA in this article; it is enough to analyse the dynamic response of the bridge and train but not to obtain the accurate rail acceleration, and shortening the integration time to obtain accurate rail response will reduce the computation efficiency of HSA. However, the HSA separates the train–track–bridge system into train–track subsystem which includes the high-frequency domain (wheel–rail contact) and the bridge subsystem whose natural frequency is relatively small (about 10 Hz; Gialleonardo et al., 2012). This kind of domain decomposition method offers an opportunity to apply the multi-time-step algorithm, in which the train–track subsystem will use fine time step to obtain the accurate rail vibrations and the bridge subsystem will use coarse time step to improve the computation efficiency. The multi-time-time step method based on the HSA will be presented in a forthcoming publication.

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 study was financially supported by the National Natural Science Foundation of China (grant nos 51378511, 51678576), the Open Project Program of the State Key Laboratory of Traction Power of China (grant no. TPL1601) and the Fundamental Research Funds for the Central Universities of Central South University (grant no. 2016ZZTS421).

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A hybrid solution for studying vibrations of coupled train–track–bridge system

Abstract

Keywords

Introduction

Train, track and bridge models

Train model

Track and bridge models

Equation of motion for the CTTBS

Displacement vectors

Matrices for the vehicles

Matrices for the rail

Matrices for the bridge

Matrices of the vehicle–rail interaction

Load vectors of the vehicles, rail and bridge

Hybrid solution algorithm

Case study and discussion

Case description of a CTTBS

FE modelling

Results and analysis

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References