Influence of yaw stiffness on the nonlinear dynamics of railway wheelset

A numerical simulation of the dynamic behavior of a railway wheelset is presented. The contact forces between the wheel and the rail are estimated using Johnson and Vermeulen theory of creepages. Nonlinear governing equations of motion of wheelset on a straight track are solved using fourth-order Runge–Kutta method. Both symmetric and asymmetric oscillations and chaotic motion are observed. The influence of yaw stiffness and axial velocity on the response of wheelset is studied. Broadband chaotic motion is developed at various velocity levels. The results are presented in the form of time evolution, phase plots, Poincare maps and bifurcation diagrams. The Lyapunov exponent is calculated and its variation with time is presented. Intermittency is observed. There is a shift in the bifurcation diagram by increasing the yaw stiffness. It indicates that chaotic behavior could be delayed with increasing yaw stiffness.     

Influence of the rail pad on a vehicle travelling on a straight track

In the following text, the influence of the stiffness of rail fastening systems of high‐speed railway lines on the dynamic stability of a vehicle travelling along a straight track is investigated. The dynamic behaviour of the rail head in lateral direction is incorporated in the model by means of mechanical models derivedfrom a preceding frequency analysis of the track. For the wheel‐rail interface, linear contact mechanics and kinematic relations are applied. The investigation leads to the conclusion that on high‐speed lines the critical velocity of vehicles is reduced as a result of the increased elasticity of the rail pads. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient decoupling technique applied to the numerical time integration of advanced interaction models for railway dynamics

Railway interaction is characterised by the coupling between the train and the track introduced through the wheel/rail contact. The introduction of the flexibility in the wheelset and the track through the finite element (FE) method in the last four decades has permitted to study high-frequency phenomena such as rolling noise and squeal, whose origin lies in the strongly non-steady state and non-linear behaviour of the contact forces that arise from the small contact area. In order to address models with a large number of degrees of freedom, innovative Eulerian-modal models for wheelsets with rotation and cyclic tracks have been developed in recent years. The aim of this paper is to extend the resulting formulation to an uncoupled linear matrix equation of motion that allows solving each equation independently for each time step, considerably reducing the associated computational cost. The decoupling integration method proposed is compared in terms of computational performance with Newmark and Runge-Kutta schemes, commonly used in vehicle dynamics, for simulations with the leading wheelset negotiating a tangent track and accounting the rail roughness.     

Rail-bridge coupling element of unequal lengths for analysing train-track-bridge interaction systems

This paper presents a rail-bridge coupling element of unequal lengths, in which the length of a bridge element is longer than that of a rail element, to investigate the dynamic problem of train-track-bridge interaction systems. The equation of motion in matrix form is given for a train-track-bridge interaction system with the proposed element. The first two numerical examples with two types of bridge models are chosen to illustrate the application of the proposed element. The results show that, for the same length of rail element, (1) the dynamic responses of train, track and bridge obtained by the proposed element are almost identical to those obtained by the rail-bridge coupling element of equal length, and (2) compared with the rail-bridge coupling element of equal length, the proposed element can help to save computer time. Furthermore, the influence of the length of rail element on the dynamic responses of rail is significant. However, the influence of the length of rail element on the dynamic responses of bridge is insignificant. Therefore, the proposed element with a shorter rail element and a longer bridge element may be adopted to study the dynamic responses of a train-track-bridge interaction system. The last numerical example is to investigate the effects of two types of track models on the dynamic responses of vehicle, rail and bridge. The results show that: (1) there are differences of the dynamic responses of vehicle, rail and bridge based on the single-layer and double-layer track models, (2) the maximum differences increase with the increase of the mass of sleeper, (3) the double-layer track model is more accurate.     

轨道结构随机场模型与车辆-轨道耦合随机动力分析

将轨道结构视为一个参数随机系统,提出并建立了轨道结构的随机场模型.利用车辆-轨道耦合动力学的基本方法,将轨道系统有限单元模型与多刚体车辆模型相结合,建立了考虑铁路线路参数空-时随机变化的车辆-轨道动力计算模型.算例表明:所提出的方法较为可靠且高效;线路参数随机性对车辆-轨道系统的动力响应有明显的影响,随线路参数离散程度的增加,可能造成行车不安全、轨道损伤加剧等一些问题.     

Dynamic properties of an individual wheelset drive of a railway vehicle

Spatial vibrations of an individual wheelset drive supported by rubber silent blocks in a two–axle bogie's frame of a railway vehicle are studied. The method, used for a creation of a mathematical model, is based on a decomposition of the wheelset drive into subsystems — a drive assembled from an engine and a gear transmission, a flexible composite hollow shaft and a flexible wheelset. Subsystems are modeled in local configuration spaces and they are mutually connected by deformable couplings. An adhesive characteristic in a wheel–rail contact is described using the Kalker's theory. The nonlinear mathematical model respects spatial vibrations of the drive's components and the flexible wheelset, a deformable rail ballast and it is used for investigating transient dynamic responses on different types of excitations. Experimentally investigated spatial bogie's frame displacements or displacements derived from a total model of the bogie caused by spatial irregularities of the rail track are main sources of excitations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Track–Monitoring of Wheel–Rail–Systems

Inspection and maintenance of railway networks is a complex and expensive task. Special measurement vehicles are used to record the geometrical properties of railway lines within required time intervals. Due to the extent of measurement data the quality of railway track is evaluated considering only a few parameters. Although safety and comfort of wheel–rail–systems depend on the dynamical behavior, current inspection vehicles are not equipped to measure dynamic properties. In this paper, we will discuss a novel approach to evaluate the quality of railway tracks based on wheel–rail dynamics: Wheelset dynamics of subway trains are analyzed by Karhunen–Loève Transformation to extract the principal dynamics from the collected measurement data. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A stochastic dynamic model of train-track-bridge coupled system based on probability density evolution method

An innovative stochastic dynamic model of a 3D train-track-bridge coupled system (TTBS) with refined wheel/rail interaction is established for a high-speed railway based on the random theory of probability density evolution method (PDEM). The multi-coupling effect of excitations can be simultaneously input into the new model, e.g. random track irregularity, random vehicle loads, stochastic system parameters, et al. Moreover, a new approach, named “Number theoretic method of multi-target probability functions” (NTM-mp), is developed to obtain the discrete point sets of multidimensional random parameters in hypercube space, aims to solve the point design of system uncertainty. The stochastic harmonic function (SHF) is applied to generate representative random track irregularity samples. The results of TTBS got by PDEM are verified with several typical case studies for its efficiency and reliability, which are the deterministic results in the representative publication, the Monte Carlo method (MCM) results, and the field testing results on the high-speed railway. At last, a typical case study of TTBS on a high-speed railway is presented for numerical analysis. Discussions and significant conclusions on the random dynamic responses are presented.     

Transversal vibration analysis of the upper span of nonlinear closed-loop track systems

In order to analyze the effect of track pre-tension on the transversal vibration of the upper track, a multibody dynamics model of a tracked vehicle needs to be established. In view of the complex structure of tracked vehicles, the computational efficiency of conventional methods cannot meet engineering design requirements. In this paper, the Riccati Transfer Matrix Method for nonlinear closed-loop multibody systems is proposed and applied to the modeling and simulation of tracked vehicle systems. It avoids the overall dynamic equations of the system and achieves efficient computation with a smaller matrix scale. By selecting accelerations and forces as elements of the state vectors instead of coordinates and forces, linearization is avoided. The track chain is cut in a revolute joint. Its effect on the closed-loop system is represented by a set of constraint equations and internal constraint forces. An improved Riccati transformation for closed-loop systems is proposed to connect the internal force and acceleration parts in the state vector with the system constraint internal forces. The track pre-tension can be adjusted by moving the position of the idler relative to the chassis. The root mean square of vertical acceleration of mid-span with respect to the chassis is selected to evaluate the transversal vibration of the upper track. Simulation of a tracked vehicle driving on different road conditions is carried out with different idler displacements. Finally, a reasonable adjustment strategy for the track tensioner is proposed corresponding to different running conditions.     

车辆-轨道垂向耦合系统求解过程的改进算法

考虑到轨道结构长度随系统响应持时的增加而增长，提出了一种改进的车辆 轨道垂向耦合系统的动力响应求解算法.该算法事先选定某一定长度的轨道结构，并获得该轨道结构的质量矩阵、阻尼矩阵和刚度矩阵；通过在求解过程中不断地对车辆子系统定位，判断是否需要对车辆子系统的位置和轨道结构的响应矩阵进行调整，以此来达到仅增加系统响应持时而不增加轨道结构长度的目的.算例表明:该改进加快算法是精确、高效的，不仅可以真实地模拟车辆在轨道上的前进运行状态，而且可以保证轨道子系统的轨道单元数量不随系统响应持时的增加而增长，这为快速求解车辆 轨道垂向耦合系统提供了一种有效的计算方法.     

On the nonlinear dynamics of a high-speed railway vehicle with nonsmooth elements

A comprehensive lateral mathematical model of a high-speed railway vehicle with 17 degrees of freedom is built to study its nonlinear dynamics on a straight line. The hunting stability is explored by using eigenvalue analysis, bifurcation diagrams and the first Lyapunov coefficient. Due to the non-smoothness in the wheel/rail contact table and the secondary suspension elements such as anti-yaw dampers, bump stops, etc., an event-driven strategy is applied in the integration process of the ODEs. As an example, the nonlinear dynamics of a Chinese Railway High-speed (CRH) vehicle (named CRH2) is investigated. The result shows that the stable equilibrium of the high-speed railway vehicle loses its stability through a supercritical Hopf bifurcation with the increase of the forward speed, and the stable periodic solution loses its stability through a grazing bifurcation with the decrease of the forward speed. The influence of primary system parameters on the hunting stability of the high-speed railway vehicle is also investigated.     

Simulation of ice under mechanical and thermal load

The contact physics of the wheel-rail contact of a railway vehicle under presence of water and ice at low temperatures is still not completely understood. For the investigation of the particular process in the contact zone a simulation is required, which is able to calculate the normal and tangential contact, the temperature field and the fluid-structural interaction between wheel and rail at low temperatures under presence of snow and ice. For that purpose the behaviour of ice under wheel-rail contact conditions is an important part. In this paper the thermal dynamic model of TSHIJOV [1], [3] for an adiabatic ice probe is updated by the new IAPWS equations of state for water [5] and ice phase Ih [4]. In a first approximation an ice specimen is loaded by specific wheel-rail contact pressure distributions calculated by the half-space formulation to clarify if phase transitions of ice can exist. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The variability of dynamic responses of beams resting on elastic foundation subjected to vehicle with random system parameters

This paper investigates the variability of dynamic responses of a beam resting on an elastic foundation, which is subjected to a vehicle with uncertain parameters, such as random mass, stiffness, damping of the vehicle and random fields of mass density, and the elastic modulus of the beam and stiffness of elastic foundation. The vehicle is modeled as a two-degree-of-freedom spring-damper-mass system. The equations of motion of the beam was constructed using a finite element method. The mass and elastic properties of the beam, and the stiffness of foundation are assumed to be Gaussian random fields and were simulated by the spectral represent method. Masses, stiffness of the spring, and the damping coefficient of the vehicle are assumed as Gaussian random variables. The numerical analyses were performed using the finite element method (FEM) in conjunction with the Monte Carlo simulation (MCS). The variability of dynamic responses of the beam were investigated with various cases of random parameters. For each sample, the equations of motions were solved with the Wilson-q integral method to find dynamic responses. The influence of random system parameters and their correlation on the response variability is discussed in detail.     

A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions

In this article, we study an explicit scheme for the solution of sine‐Gordon equation when the space discretization is carried out by an overlapping multidomain pseudo‐spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth‐order Runge–Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo‐spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix–vector multiplications. Therefore, we propose a multidomain pseudo‐spectral method for the numerical simulation of the sine‐Gordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments show that the multidomain scheme has an excellent long‐time numerical behavior for the sine‐Gordon equation in one and two dimensions. Copyright © 2013 John Wiley & Sons, Ltd.     

Contact simulation for the calculation of the kinematic excitation of vibrations from surface roughness of wheel and rail

A simulation method has been developed to predict the sound emission of the railway wheelsets due to the excitation of wheelset vibrations by surface roughness of wheel and rail. This model treats the contact in a linearized manner and is therefore not capable of modelling the local effects of the roughness in the wheel-rail contact zone. These effects are very important since the contact acts as a low pass filter which is calculated within a contact simulation. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

二系悬挂条件下的车-路垂向耦合系统的动力模型

在车辆的走行过程中,上部与下部是相互作用和影响的．因此,轨道交通问题实际上就是线路下部结构和车辆系统的体系匹配问题．在一系悬挂条件下的车-路系统耦合动力分析模型基础上,考虑了包含转向架在内的车辆的实际构成和轨下基础包括路基和地基的参振特性,利用轨道维护标准模拟行走不平顺激励,通过位移相容条件,从理论上研究车辆-轨道-路基体系的动力相互作用,建立了二系悬挂条件下的车辆-轨道-路基系统的垂向动力分析模型．为高速铁路路基的动力特性分析和设计提供参考．     

A generalized Duffing equation with the Coulomb’s friction law and Signorini–type contact conditions

This work provides mathematical and numerical analyses for a spring–mass system, in which Signorini–type contact conditions and Coulomb’s friction law with thermal effects are taken into consideration. The motion of a mass attached to a viscoelastic (Kelvin–Voigt type) nonlinear spring is described by a generalized Duffing equation. Signorini contact conditions are understood as extended complementarity conditions (CCs), where convolution is incorporated, allowing to consider thermal aspects of an obstacle. We prove the existence of global weak solutions for the highly nonlinear differential equation system with all the conditions, based on the regularized differential equation and the normal compliance condition with the standard mollifier. In addition, we investigate what side effects produce higher singularities of contact forces in dynamic contact problems, which is also supported by numerical evidences. Numerical schemes are proposed and then several groups of data are selected for the display of our numerical simulations.     

Rolling Contact Problem with the Generalized Coulomb Friction

This paper deals with the numerical solution of the wheel - rail rolling contact problems. The unilateral dynamic contact problem between a rigid wheel and a viscoelastic rail lying on a rigid foundation is considered. The contact with the generalized Coulomb friction law occurs at a portion of the boundary of the contacting bodies. The Coulomb friction model where the friction coefficient is assumed to be Lipschitz continuous function of the sliding velocity is assumed. Moreover Archard's law of wear in the contact zone is assumed. This contact problem is governed by the evolutionary variational inequality of the second order. Finite difference and finite element methods are used to discretize this dynamic contact problem. Numerical examples are provided. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A novel dynamics model for railway ballastless track with medium-thick slabs

Motivated by the requirements for elaborated slab ballastless track dynamics analysis in practical engineering application, a novel dynamic model for the railway ballastless tracks with medium-thick slabs is proposed in this work based on the Reissner–Mindlin plate theory, and it is implemented into the coupled dynamics analysis of a vehicle and the ballastless track. First, an efficient and easily programmable computational algorithm is adopted to solve the transverse deflection of the Reissner–Mindlin plate, in which the displacements and shear strains are chosen as the independent variables and subsequently constructed by spline functions, resulting in no shear-locking effect. The involved partial differential equations are transformed into ordinary ones by using the energy variation principle. Further, a mathematical model for the ballastless track dynamics analysis is established, which can consider the effects of the shear deformation and moment of inertia involved in the medium-thick track slab. Experimental verification and comparative analysis with other models demonstrate the accuracy and efficiency of the proposed model. Finally, a spatially coupled dynamics model of a vehicle and the ballastless track is developed, and it is efficiently solved by using the hybrid explicit-implicit time integration method. Compared with the widely used modelling the track slab by elastic thin plate, the reliability and advantages of the proposed vehicle-slab track coupled dynamics model are demonstrated.