High Speed Railway Track Dynamics

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Track Dynamics Research Contents and Related Standards

   Xiaoyan Lei

   Abstract

   With the rise in train speeds, axle loads, and traffic density along with the broader engineering applications of new vehicles and track structures, the vehicle-track interaction has grown increasingly complex. Consequently, train safety and stability are more significantly influenced by elevated dynamic stresses. High-Speed Railway Track Dynamics provides a robust foundation for studying the intricate wheel-rail relationship and interaction mechanisms, offering critical insights for optimizing vehicle and track structure design. This chapter reviews recent advancements in track dynamics modeling and analytical methods while introducing key aspects of track dynamics analysis. Additionally, it evaluates relevant standards and limits, including safety and stability criteria for high-speed trains, track maintenance and management standards for high-speed railways, environmental noise and vibration regulations for rail transit, and vibration thresholds for historic building structures.

3. Chapter 2. Analytic Method for Dynamic Analysis of the Track Structure

   Xiaoyan Lei

   Abstract

   It is well established that higher train speeds can amplify dynamic forces exerted on the track and ground surface. This effect becomes particularly pronounced under high-speed operations. Research indicates that when train speeds approach or exceed certain critical thresholds, surface waves generated by high-speed trains can induce severe vibrations in the track structure. To investigate the underlying mechanisms of these vibrations, this chapter employs an analytical approach for dynamic analysis of the track structure. A continuous elastic foundation beam model is developed to simulate the track system, examining the characteristics of ground waves induced by high-speed trains and their contribution to track vibrations. Additionally, the influence of track stiffness on critical speed and vibration response is systematically analyzed using this analytical method.

4. Chapter 3. Fourier Transform Method for Dynamic Analysis of the Track Structure

   Xiaoyan Lei

   Abstract

   With the advantages of computational efficiency and analytical flexibility, the Fourier transform method is widely used in the dynamic analysis of various engineering structures. In this chapter, the Fourier transform method is applied to the dynamic analysis of track structures under moving loads. By establishing single-layer, double-layer, and three-layer continuous elastic beam models for the track structure and deriving the vibration equations in the frequency domain, the algorithm for solving the dynamic response of the track structure using the Fourier transform method is presented. Based on the proposed model and algorithm, the source code, “Slab Track Dynamics Calculation Program STDYN_1.0,” developed using MATLAB mathematical tools, is provided in Appendix B. As application examples, two case studies are presented: the dynamic analysis of high-speed railway slab track structures and mixed passenger-freight railway track structures.

5. Chapter 4. Analysis of Vibration Behavior of the Elevated Track Structure

   Xiaoyan Lei

   Abstract

   Elevated track structures are widely employed in high-speed railways and urban rail transit systems due to their space efficiency, minimal land occupation, controlled settlement, and cost-effectiveness in both construction and operation. These advantages make them an effective solution for mitigating urban traffic congestion. However, a notable drawback is their vibration and noise emissions, which can impact the surrounding environment along the rail corridor. In this chapter, two distinct models, an analytical model and a three-dimensional finite element model (3D FEM), are developed to analyze the dynamic behavior of the elevated track structures. Using these models, the study examines the velocity admittance distribution of box girders and U-beams, and the vibration attenuation characteristics of the elevated track structure as a function of distance. Finally, a comparative analysis is conducted to evaluate the adaptability and accuracy of the two modeling approaches.

6. Chapter 5. Track Irregularity Power Spectrum and Numerical Simulation

   Xiaoyan Lei

   Abstract

   Track irregularity is the main excitation source of the vehicle-track coupling system vibration. Track irregularities can be divided into track profile irregularity, track cross-level irregularity, track alignment irregularity, and track gauge irregularity. In vehicle-track coupling dynamics, track random irregularities are typically characterized using power spectral density (PSD) functions, which play a crucial role in vibration analysis. This chapter provides an overview of the PSD functions adopted by various countries, including the United States, Germany, China, and Japan, for describing track irregularities. Additionally, it introduces a numerical method for generating track random irregularity samples based on these track irregularity PSD functions.

7. Chapter 6. Vertical Vibration Model for the Track Structure and the Vehicle

   Xiaoyan Lei

   Abstract

   Numerical method, in particular the finite element method, is a powerful tool to analyze the dynamic response of the vehicle-track coupling system, and the key to solve the complicated dynamic interactions between the vehicle and the track is to construct a reasonable model for the track structure and the vehicle. As the basis for establishing the dynamic analysis model of various vehicle-track coupling systems, the fundamental concepts of the dynamic finite element method (FEM) and the numerical algorithm for 2D beam elements are introduced. A computationally efficient and easily programmable generalized beam element model for the track structure is presented. Beginning with a simple single-wheel vehicle model incorporating a primary suspension system, the analysis progressively describes half-vehicle and full-vehicle models equipped with both primary and secondary suspension systems. The corresponding dynamics equations for each vehicle model are derived.

8. Chapter 7. A Cross-Iteration Algorithm for Vehicle–Track Nonlinear Coupling Vibration Analysis

   Xiaoyan Lei

   Abstract

   The dynamic finite element equation of the vehicle-track nonlinear coupling system is a large, high-order and nonlinear coupled second-order differential equation set, the efficiency of the conventional algorithm for solving the large and complicated differential equation set is not satisfied. In this chapter, the cross-iterative algorithm for solving the dynamic finite element equation of nonlinear vehicle-track coupling system is described, and the algorithm verification and convergence analysis are carried out by given examples. In simulation analysis, the whole system is divided into two subsystems, i.e., the vehicle subsystem considered as a vehicle element with a primary and secondary suspension system, and the track or the track-bridge subsystem regarded as a three elastic beam model. Coupling of the two systems is achieved by equilibrium conditions for wheel-rail nonlinear contact forces and geometrical compatibility conditions. In order to accelerate the iterative convergence, a relaxation technique is introduced to modify the wheel-rail contact forces. The cross-iteration algorithm has the shining advantages of higher efficiency, better precision, and simple programming. Based on the proposed model and the algorithm, the source code, “Train-track-continuous bridge coupling system dynamics calculation program VTBDYN_1.0,” developed using MATLAB mathematical tools, is provided in Appendix C.

9. Chapter 8. Moving Element Model and Its Algorithm

   Xiaoyan Lei

   Abstract

   The dynamics equation of the vehicle-track coupling system based on energy principle is a large-scale coupled equation set. Within this system, the vehicle and the track structure are interdependent, coupled through the wheel-rail contact force. This chapter establishes three progressively complex models: (1) a moving element model for a single wheel, (2) a moving element model for a single wheel incorporating a primary suspension system, and (3) a moving element model for a single wheel with both primary and secondary suspension systems. Based on these models, the algorithm for simultaneously solving the dynamic responses of the train and track is described, and the corresponding explicit formulas are presented. As an application example, a dynamic analysis of the train-track-bridge coupling system is conducted, yielding several conclusions. The source code, “Dynamics Calculation Program WTBDYN_1.0 for the Moving Wheelset with Primary and Secondary Suspension-Track-Continuous Bridge Coupling System,” developed using MATLAB mathematical tools, is provided in Appendix D.

10. Chapter 9. Model and Algorithm for Track Element and Vehicle Element

    Xiaoyan Lei

    Abstract

    With the rapid development of high-speed railways, trains traveling at high speeds induce significant dynamic stress on the track structure and strong vehicle vibrations, directly impacting operational safety and passenger comfort. In this chapter, a track element model, a slab track-bridge element model, and a 15-node, 26-degree-of-freedom vehicle element model are proposed. Using the finite element method and Lagrange’s equation, the stiffness, mass, and damping matrices for both the track and vehicle elements are derived. The train-track-subgrade (or bridge) coupling system is discretized into finite vehicle and track elements, with the track-subgrade (or bridge) system modeled as a series of track elements and the train represented by vehicle elements. During computation, the global stiffness, mass, and damping matrices of the track-subgrade (or bridge) system are generated only once. In subsequent time-step calculations, only the vehicle element matrices need to be assembled into the global matrices of the track-subgrade (or bridge) system. This approach significantly improves computational efficiency.

11. Chapter 10. Dynamic Analysis of the Vehicle–Track Coupling System with Finite Elements in a Moving Frame of Reference

    Xiaoyan Lei

    Abstract

    The dynamic analysis models of the vehicle-track coupling systems discussed in previous chapters all face the challenge of truncated boundary effects on the solutions. To address this issue, this chapter presents a finite element method in a moving frame of reference (FEMFR) for the dynamic analysis of the vehicle-track coupling systems. A three-layer continuous beam model is developed for the slab track, and the mass, damping, and stiffness matrices of the slab track element in a moving frame of reference are derived. The vehicle is modeled as a 26-degree-of-freedom (DOF) element. This approach effectively eliminates the influence of truncated boundaries on the computational results while improving both accuracy and efficiency. The main advantage of this method is that, unlike conventional finite element method (FEM), the moving vehicle always acts at a fixed point in the numerical model. This eliminates the need to track the contact point relative to individual elements. Furthermore, the vehicle remains within the computational domain throughout the simulation, and never runs out of the truncated model. As an application example, dynamic analysis of high-speed train and slab track coupling system is carried out.

12. Chapter 11. Model for Vertical Dynamic Analysis of the Vehicle–Track–Subgrade–Ground Coupling System

    Xiaoyan Lei

    Abstract

    Chapters 6, 7, 8, 9, and 10 detailed several models and their algorithms for vertical dynamic analysis of the vehicle-track coupling system. These analyses excluded subgrade and ground effects, instead simplifying the subgrade as viscoelastic damping elements. However, an accurate assessment of the vehicle-induced dynamic actions on track structures must account for interactions among the vehicle, track, subgrade, and ground. In this chapter, both the slab track-embankment-ground system model and the ballast track-embankment-ground system model under moving loads are established, and the flexibility matrices for the moving vehicle at the wheelset and for the track structure at the wheel–rail contact point are derived respectively. By incorporating moving axle loads and track irregularity, an numerical algorithm for the coupled vibration analysis of the moving vehicle-track-subgrade-ground system is presented. As an application case, dynamic response analysis of a high-speed train-track-subgrade-ground coupling system is carried out.

13. Chapter 12. Analysis of Dynamic Behavior of the Train, Ballast Track, and Subgrade Coupling System

    Xiaoyan Lei

    Abstract

    There are a large number of bridges, grade crossings, and rigid culverts involved in railway transportation, which result in plenty of track transitions. When high-speed trains pass through regions with abrupt changes in vertical stiffness, significant additional wheel–rail contact forces are generated. In this chapter, a dynamic analysis model of the train–ballasted track–subgrade coupled system is developed based on the vehicle-element and track-element modeling algorithms. To address the recurring issue of track faults caused by train-induced impacts at roadbed–bridge transitions in high-speed railways, parameter studies are conducted to examine the effects of train speed, track foundation stiffness, and transition irregularity on the dynamic responses of both the vehicle and the track structure in transition zones.

14. Chapter 13. Analysis of Dynamic Behavior of the Train-Slab Track-Subgrade Coupling System

    Xiaoyan Lei

    Abstract

    In this chapter, the dynamic behavior of the train-slab track-subgrade coupling system is analyzed by using the vehicle and track element model and algorithm. Parameter analyses are conducted on the stiffness and damping of the rail fastening system, the CA mortar layer, and the subgrade. The results indicate that, among the six parameters considered, the stiffness of the rail fastening system and the subgrade have the most significant influence on the dynamic response of both the train and the track structure. Increasing the stiffness of the rail pad and fasteners helps mitigate wheel-rail interaction, whereas reducing their stiffness aids in attenuating vibrations in the track foundation. Additionally, lower subgrade stiffness may result in considerable dynamic responses within the track structure. Therefore, selecting appropriate stiffness values for the rail pad and fasteners, along with increasing the subgrade stiffness, is essential to ensure the safe and stable operation of the track system. The findings from the parameter analysis provide valuable guidance for optimizing the structural parameters of the slab track and supporting the design of track structures.

15. Chapter 14. Analysis of Dynamic Behavior of the Transition Section Between Ballast Track and Ballastless Track

    Xiaoyan Lei

    Abstract

    Due to the substantial stiffness difference between the ballasted and ballastless tracks, additional wheel-rail contact forces are generated when a high-speed train transitions from a ballasted to a ballastless track. Studies on transition zones have shown that under long-term train loading, a range of issues emerge in these sections, resulting in damage to the track structure. Based on typical structural configurations and design parameters of ballast-ballastless track transition sections, this chapter utilizes the vehicle element and track element models as well as the computational methodology introduced in Chap. 9 to analyze the influence of train speed and track stiffness on the dynamic response of both the train and the track structure. Parameter studies are carried out to assess the transition performance between ballasted and ballastless tracks. The analysis seeks to offer theoretical support for the design and stiffness selection in transition zones.

16. Chapter 15. Analysis of Medium and High-Frequency Vibration for Track Structure

    Xiaoyan Lei

    Abstract

    The ballastless track structure exhibits wide-frequency vibration characteristics under high-speed train loading. While the traditional finite element method is effective for low-frequency structural vibration analysis, it is less suitable for medium- and high-frequency vibration analysis. In this chapter, a frequency-domain vibration model of the vehicle–track system is established based on Spectral Element Method (SEM). The track structure is modeled as a three-layer system using Euler and Timoshenko beams, and the vehicle is represented as a full multi-body model. Based on the exact solutions of the beam wave equation, the dynamic stiffness matrix of spectral elements for the slab track structure is derived. The corresponding spectral element equations are then formulated by applying the boundary conditions. Track irregularities are simulated as virtual excitations via the virtual excitation method. By solving the coupled vehicle–track system equations formulated through the spectral element method, the vibration response of both the vehicle and the track structure in the frequency domain is obtained. The advantages of the spectral element method lie in its modeling flexibility, algorithmic generality, and ease of implementation. It achieves high computational accuracy and efficiency with a small number of elements across a broad range of analysis frequencies.

17. Chapter 16. Dynamic Analysis of High-Speed Train-Track Spatial Nonlinear Coupling System

    Xiaoyan Lei

    Abstract

    The speed of 400 km/h represents a milestone that humanity has been striving for in wheel-rail high-speed railway systems, and the research is extremely challenging. In this study, a sophisticated rigid-flexible coupled dynamic model of a vehicle–track spatial nonlinear coupling system is developed. The model incorporates the elastic deformation and three-dimensional vibrations of the rail, track slab, and concrete base, while the vehicle is simplified as a rigid body with 31 degrees of freedom. The “Trace Method” and the “Minimum Distance Method” are employed to identify the spatial contact points between the wheel and rail, followed by a quasi-elastic contact correction. Based on this, the geometric relationship of the wheel-rail spatial contact is established. Given the complexity of the dynamic behavior in the vehicle–track coupling system and the computational cost associated with locating the spatial contact points, an improved cross-iteration algorithm is developed and a numerical approach is proposed. By integrating the Trace Method into the cross-iteration process, this strategy significantly improves both computational efficiency and model accuracy. As an application case, the adaptability of existing Chinese high-speed railways to operate at 400 km/h under track random irregularity excitation is investigated, and the dynamic performance of the coupled high-speed train–track system is evaluated.

18. Backmatter