Journal of Nonlinear Dynamics

Volume 2016 (2016), Article ID 8356160, 17 pages

http://dx.doi.org/10.1155/2016/8356160

## Dynamic Analysis of the High Speed Train and Slab Track Nonlinear Coupling System with the Cross Iteration Algorithm

Engineering Research Center of Railway Environment Vibration and Noise, Ministry of Education, East China Jiaotong University, Nanchang 330013, China

Received 28 October 2015; Revised 13 January 2016; Accepted 28 January 2016

Academic Editor: Usama El Shamy

Copyright © 2016 Xiaoyan Lei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A model for dynamic analysis of the vehicle-track nonlinear coupling system is established by the finite element method. The whole system is divided into two subsystems: the vehicle subsystem and the track subsystem. Coupling of the two subsystems is achieved by equilibrium conditions for wheel-to-rail nonlinear contact forces and geometrical compatibility conditions. To solve the nonlinear dynamics equations for the vehicle-track coupling system, a cross iteration algorithm and a relaxation technique are presented. Examples of vibration analysis of the vehicle and slab track coupling system induced by China’s high speed train CRH3 are given. In the computation, the influences of linear and nonlinear wheel-to-rail contact models and different train speeds are considered. It is found that the cross iteration algorithm and the relaxation technique have the following advantages: simple programming; fast convergence; shorter computation time; and greater accuracy. The analyzed dynamic responses for the vehicle and the track with the wheel-to-rail linear contact model are greater than those with the wheel-to-rail nonlinear contact model, where the increasing range of the displacement and the acceleration is about 10%, and the increasing range of the wheel-to-rail contact force is less than 5%.

#### 1. Introduction

Since the opening of the world’s first high speed railway in 1964 (with its advantages of high speed and convenience, safety and comfort, environmental-friendliness and low energy consumption, larger carrying capacity, and the availability of all-day transportation), high speed railway has shown strong competitiveness amongst other modes of transport. According to the statistics by the International Union of Railways, as of November 1, 2013, there are a total of 11,605 km of high speed railways in operation, 4883 km in construction, and another 12,570 km of high speed railways planned to be built in other countries of the world. In China, there are 11,028 km of high speed railways in operation and another 12,000 km under construction, making up half the total amount of the world’s total high speed railways, with the country owning the longest and the largest scale of high speed railways both in operation and construction. With the rapid development of high speed railways, ballastless slab track has been widely used throughout the world [1, 2]. In China, over 80% high speed railways are ballastless slab track. Compared with the traditional ballast track, it has the advantages of high stability, long service life, high geometric regularity, uniform track stiffness, low maintenance, and good durability.

It is well known that the transportation function of a railway system is achieved by wheel-to-rail interaction, and the design and the manufacture of locomotives and rolling stock, as well as the design and the construction of the line itself, are required for understanding the dynamic characteristics of the vehicle and track coupling system. To ensure the train moves safely and smoothly, a good dynamic performance of the railway line is required. The development of a mathematical model and the simulation technique for analysis of the dynamic behavior of ballastless slab track will be helpful to achieve improved component design and maintenance schedules. These models are used to understand the interactions of the track and vehicle components and to measure the vibration characteristics of the track structure. Some literature has focused on recent developments in slab track and slab track design [3, 4]. However, only a few references have been made to the dynamic behavior of a ballastless slab track under moving vehicles. Zhai et al. developed a vehicle/slab track interaction model based on the theory of vehicle-track coupling dynamics and analyzed the dynamic properties of slab tracks used in high speed railways. The effects of the elasticity and damping of the CA-layer under the slab on system dynamics were also investigated [5]. Xiang et al. established a dynamic analysis model of the lateral finite strip and slab segment element based on the structural characteristics of the ballastless track, such as the Bögl slab track [6]. Both the motor car and trailers of the high speed train were modeled as a multibody system with two suspensions. The vertical displacements of the rail and the slab were obtained by the traditional static model and by the dynamic analysis model of the lateral finite strip and slab segment element, respectively. The calculated results demonstrate that the vertical static and dynamic maximum displacements of the rail and the slab are close and the values are in general range. Next, Xiang et al. put forward a new spatial vibration model of track segment element of the slab track according to structural characteristic of a slab track [7]. The spatial vibration equation set of the high speed train and slab track system was then established on the basis of the rule of “set-in-right-position” for formulating system matrices. The equation set was solved by the Wilson- direct integration method. The theory was verified by the high speed running experiment carried out on the slab track of the Qinghuang dao-Shenyang passenger transport line. Dong et al. carried out experimental validation of a numerical model for prediction of the dynamic response of ballastless subgrade of high speed railways [8]. 3D consistent viscous-spring artificial boundary elements are introduced in the ABAQUS software, the effects of the distance between multisource loading points and the viscous-spring artificial boundary on the simulation results under train loading conditions are discussed, and 3D finite element ballastless track/subgrade models have been developed. Comparison between the field measured data of CRH2 at the National Railway Test Site and the ABAQUS simulation results shows good agreement. The results indicate that the dynamic stress of the subgrade increases with the speed. Lei and Zhang presented a model for dynamic analysis of vehicle-track-subgrade coupling system to the CRTS II (China Railway Track System) ballastless slab track system [9]. Based on the model, a new type of ballastless slab track element is developed, and the associated stiffness matrix, mass matrix, and damping matrix for the element is deduced. This element includes rail, rail pad and fastener, prefab slab, cement-asphalt mortar, hydraulically bonded layer, and subgrade. By means of the Lagrange equation, the finite element equation for analysis of dynamic behavior of the ballastless slab track is formulated. As application examples, parameter studies on the track vibration of the ballastless slab track structure, such as stiffness and damping resulting from the rail pad and fastener, CA mortar, and subgrade, are investigated. In order to investigate the dynamic behavior of the train and slab track coupling system, Lei and Wang studied a new approach with finite elements in a moving frame of reference based on conceptions of vehicle element and track element [10]. By discretizing the slab track subsystem into track elements that flow with the moving vehicle, the proposed method eliminates the need for keeping track of the vehicle position with respect to the track model. The governing equations are formulated in a coordinate system moving at a constant velocity, and the associated stiffness matrix, mass matrix, and damping matrix for the track element in a moving frame of reference are derived. The vehicle element is introduced to model a car with primary and secondary suspension systems, which has 26 degrees of freedom, where 10 degrees of freedom are used to describe the vertical movement of the car, and 16 degrees of freedom are associated with rail displacements. The method is shown to work for varying train speed and track parameters and has several advantages over the conventional finite element method in a fixed system of reference. (It is known that track irregularity is the principal exciting source inducing vibrations of the train and the track.) Kang et al. proposed a calculation method, fitting function and fitting spectrum of power spectral density (PSD) of track irregularity for high speed railways [11]. To improve the calculation accuracy of PSD of track irregularity, a linear interpolation and a wavelet analysis are given and used to eliminate the outliers and trends in the track irregularities. Based on the method and the data from the high speed comprehensive inspection train, the track irregularity PSD fitting equation was obtained, as well as the frequency multiplication energy table, to reflect the impact of track periodical structure of high speed railways. PSD of the ballastless track irregularities of high speed railway includes fitting spectrum and frequency multiplication energy table, which provides the basis for maintenance and optimization design for high speed railways. By means of a parallel algorithm based on the pseudo excitation method (PEM), Zhang et al. investigated the nonstationary random response of a vertically coupled vehicle-slab track system subjected to random excitation induced by track irregularity [12]. The vehicle is simplified as a multibody system with 10 degrees of freedom and the slab track is represented by a three-layer Bernoulli-Euler beam model which includes the rail, slab, and roadbed. Linear wheel-rail contact provides interaction between the vehicle and slab track models. In the above contributions, studies are mainly focused on the vehicle-track linear coupling dynamic problems and have their own characteristics, whereas the analysis of dynamic response for the vehicle-track nonlinear coupling system is rarely given attention, as is the corresponding algorithm. Varandas et al. presented a methodology to predict the settlement of railway track in transition zones due to train loading [13]. The methodology is based on dynamic calculations using a (non-linear) train-track interaction model and an incremental settlement model. Nguyen et al. carried out comparison of dynamic effects of high speed traffic load on ballasted track using a simplified two-dimensional and full three-dimensional model [14]. In these models, the vehicle and track are coupled via a nonlinear Hertz contact mechanism. The method of Lagrange multipliers is used for the contact constraint enforcement between the wheel and rail. Due to the contact nonlinearities, the numerical simulations are performed in the time domain, using a direct integration method for the transient problem. Yang and Fonder presented an iterative scheme to analyze the dynamic response of a bridge-vehicle system [15]. The method consists in dividing the whole system into two subsystems at the interface of the bridge and vehicles, and these two subsystems are solved separately. Their compatibility at the interface is achieved by an iterative procedure with underrelaxation or with Aitken acceleration. The proposed method is more efficient in nonlinear dynamic responses because, in this case, the iterations are necessary whether the system is solved as a whole or not. In each iteration step, judgment must be made to satisfy equilibrium conditions for wheel-rail nonlinear contact forces and geometrical compatibility conditions, and the select of the time step will be limited when dealing with problem of the larger amplitude of the track random irregularity. A fundamental model is established for analyzing the train-track-bridge dynamic interactions by Zhai and Xia, in which the vehicle subsystem is coupled with the track subsystem through a spatially interacted wheel-rail model [16, 17]; and the track subsystem is coupled with the bridge subsystem by a track-bridge dynamic interaction model. An explicit-implicit integration scheme is adopted to numerically solve the equations of motion of the large nonlinear dynamic system in the time domain. Computer simulation software, that is, the train-track-bridge interaction simulation software (TTBSIM), is developed to predict the vertical and lateral dynamic responses of the train-track-bridge coupled system, and the effectiveness of the TTBSIM simulation for dynamic evaluation of complex bridge structures in high speed railways is demonstrated [16, 17]. Neves et al. pointed out that the equations of motion of the structure and vehicles are complemented with additional compatibility equations that relate nodal displacements of the vehicles to the displacements of the corresponding points on the surface of the structure, with no sliding or separation [18, 19]. In order to avoid the system matrix being time dependent at each time step and deal with the nonlinear contact problem, a model for dynamic analysis of the vehicle and slab track nonlinear coupling system is proposed by finite element method in this paper. The whole system is divided into two subsystems: the vehicle subsystem (considered as a complete locomotive or rolling stock unit with a primary and secondary suspension system) and the track subsystem (regarded as three-layer elastic beam model). Coupling of the two systems is achieved by equilibrium conditions for wheel-to-rail nonlinear contact forces and geometrical compatibility conditions. A cross iteration algorithm is presented to solve the dynamics equations of the vehicle-track nonlinear coupling system. In order to accelerate the iteration convergence rate, a relaxation technique is introduced to modify the wheel-to-rail contact forces. By contrasting with a reference example, the correctness of the algorithm is verified. The example of vehicle and track vibration induced by China high speed train CRH3 moving on the slab track is given, in which the influences of the linear and nonlinear wheel-to-rail contact model and the different train speeds are considered. The results demonstrate that the cross iteration algorithm has the advantages of simple programming, fast convergence rate, less computation time and high accuracy.

#### 2. Fundamental Assumptions

The following assumptions are made in establishing the model to analyze the dynamic behavior of the ballastless slab track of high speed railways:(1)Only vertical dynamic loads are considered in the model.(2)Since the vehicle and the slab track are symmetrical about the center-line of the track, only half of the coupling system is used for ease of calculation.(3)The upper structure in the vehicle and slab track coupling system is a complete locomotive or rolling stock unit with a primary and secondary suspension system, in which vertical and pitch motion for both vehicle and bogie are considered.(4)The lower structure in the coupling system is a CRTS II slab track where rails are considered as beams with finite length resting on a discrete pad. The elastic and damping behavior for the rail pad and the fastener of the track structure are represented with stiffness and damping coefficients and .(5)The concrete slab is simplified as a beam, and only vertical dynamic responses are considered. The elastic and damping behavior resulting from cement-asphalt mortar (CA mortar) of the track structure are and .(6)The hydraulically bonded layer (HBL) is simplified as a beam, and only vertical dynamic responses are considered. The elastic and damping behavior resulting from subgrade of the track structure are and .(7)A nonlinear relationship between two elastic contact cylinders perpendicular to each other is used in coupling the vehicle and the track.

#### 3. Vehicle Subsystem

The vehicle subsystem is a complete locomotive or rolling stock unit with a primary and secondary suspension system, as shown in Figure 1. In the model, and are mass and pitch inertia for the car body; and are mass and pitch inertia for the bogie; , and , stand for stiffness and damping coefficients for the primary and secondary suspension systems of the vehicle, respectively; and are vertical displacement and angular displacement of pitch motion for the car body; and are the vertical displacement and angular displacement of pitch motion for the th bogie; is the vertical displacement for the th wheel; is the mass of the th wheel and is the wheel-to-rail contact force for the th wheel. The nodal displacement vector for this element can be defined asBy means of the Lagrange equation, the dynamic equation of vehicle for the coupling system can be obtained aswhere , , and represent the mass, damping, and stiffness matrixes for the upper structure and the explicit expressions can be described as and can be found in the literature [9].