A coupled model for train-track-bridge stochastic analysis with consideration of spatial variation and temporal evolution

Due to random characteristics of system parameters and excitations, the dynamic assessment and prediction for the train-track-bridge interaction systems become rather complex issues needing to be addressed, especially considering the longitudinal inhomogeneity and uncertainty of dynamic properties in physics and correspondingly their temporal evolutions. In this paper, a temporal-spatial coupled model is developed to fully deal with the deterministically/non-deterministically computational and analytical matters in the train-track-bridge interactions with a novelty, where a train-track-bridge interaction model is newly developed by effectively coupling the three-dimensional nonlinear wheel-rail contact model and the finite element theory, moreover, the Monte-Carlo method (MCM) and Karhunen–Loève expansion (KLE) are effectively united to model the random field of track-bridge systems, and a spectral evolution method accompanied by a track irregularity probabilistic model are introduced to select the most representative track irregularity sets and to characterize their random evolutions in temporal dimension. In terms of random vibration analysis, the high-efficiency and effectiveness of this developed model is validated by comparing to a robust method, i.e., MCM. Apart from validations, multi-applications of the temporal-spatial coupled model from aspects of deterministic computation, random vibration, resonant analysis and long-term dynamic prediction, etc., have been fully presented to illustrate the universality of the proposed model.     

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.     

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.     

A probabilistic model for track random irregularities in vehicle/track coupled dynamics

Track irregularities generally viewed as weak stationary random processes are perhaps the most important excitations to the vehicle/track coupled system. To better clarify the random vibration characteristics and probabilistic relationships between track random irregularities and dynamic behaviors of vehicle/track systems, it is a necessity to consider the full properties of track irregularities on amplitude, frequency and probability in vehicle/track interactions. The purpose of this paper is to develop a probabilistic model to select representative and realistic track irregularity sets from numerous data with higher efficiency and accuracy. To establish the vehicle/track interaction model, the finite element method and vehicle/track coupled dynamics are adopted and effectively combined, which can be used to reveal the interaction mechanisms between the moving vehicles and the guiding tracks. Moreover, the probabilistic transmission relationships between track irregularities and system responses are addressed by introducing a probability density evolution method. Through detailed comparisons with the experimental measurements and other advanced models, this proposed model is proved to be fairly effective and highly efficient.     

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.     

Comportamiento dinámico de viaductos cortos considerando la interacción vehículo-vía-estructura-suelo

The dynamic vehicle-track-bridge-soil interaction is studied on high speed lines. The analysis is carried out using a general and fully three dimensional multi-body-finite element-boundary element model, formulated in the time domain to predict vibrations due to the train passage over the bridge. The vehicle is modelled as a multi-body system, the track and the bridge are modelled using finite elements and the soil is considered as a homogeneous half-space by the boundary element method. Usually, moving force model and moving mass model are employed to study the dynamic response of bridges. In this work, the multi-body system allows one to consider the quasi-static and dynamic excitation mechanisms. Soil-structure interaction is taken into account on the dynamic structure behaviour on simply-supported short span bridges. The influence of soil-structure interaction is analysed in both resonant and non-resonant regimes.     

Longitudinal track-bridge interaction due to loading sequences

For continuously welded rails the longitudinal track–bridge interaction plays an important role. The coupling between track and bridge consists of a ballasted interface or a slab track with rail–pads. European codes describe this coupling by means of a bilinear force–displacement relation with making a difference wether the track is loaded or not. This contribution is focussed on an incremental treatment of the long–term loading from seasonal temperature–change followed by several cycles unloaded–loaded–unloaded due to passing trains. As one cycle happens in a rather short time, the mass accelerations are involved, too. The load–character of the coupling–variation will be explained and a comparison between a statical and a dynamical treatment will be presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

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.     

介质与结构三维动力相互作用分析的半解析边界元—半解析有限元结合法

本文将半解析边界元一半解析有限无结合法用于介质与结构的动力相互作用研究:用半解析边界元法分析具有复杂地表面的半无限介质,用半解析有限元法分析具有任意截面形状的柱体结构,利用介质与结构交界面上的位移相容条件和力平衡条件,将介质与结构联系起来。联立京解上述半解析边界元方程和半解析有限元方程,对应每一时间步进,可同时求出介质与结构交界面上的位移、速度、加速度和相互作用力以及地表面的运动情况.与目前广泛研究的边界元—有限元结合法相比,本方法在介质与结构二个个区域各降低了一维空间,因而离散单元数和计算工作量大幅度减少,人工输入数据非常简单.文中还考虑了地下结构的长跨比效应、厚度效应和介质效应.     

车桥耦合系统随机振动的时域显式解法

在桥面和轨道随机不平顺作用下,车桥耦合系统振动是一个典型的非平稳随机振动问题.笔者分别建立表征物理演变机制的车辆系统和桥梁系统的动力响应显式表达式,然后利用车桥之间的运动相容条件,建立车桥之间接触力关于桥面不平顺的显式表达式.在此基础上,即可直接利用统计矩运算法则,获得车桥接触力的统计矩演化规律,并进一步计算车辆系统和桥梁系统关键响应的演变统计矩.此外,也可以基于车桥接触力关于桥面不平顺的显式表达式,高效地进行随机模拟（即Monte Carlo模拟, MCS）,以获得车桥耦合系统关键响应的演变统计矩及其他统计信息.在上述过程中,由于实现了车桥耦合系统物理演变机制和概率演化规律的相对分离,在响应统计矩计算中,无需反复求解车桥耦合系统的运动微分方程,且可以仅针对车桥接触力及其他所关注的关键响应开展降维计算,大幅提高了车桥耦合系统随机振动的计算效率.数值算例表明,所提出的方法具有理想的计算精度和计算效率.     

Study on numerical simulation methods of the railway vehicle-track dynamic interaction

A very efficient numerical simulation method of the railway vehicle–track dynamic interaction is described. When a vehicle runs at high speed on the railway track, contact forces between a wheel and a rail vary dynamically due to the profile irregularities existing on the surface of the rail. A large variation of contact forces causes undesired deteriorations of a track and its substructures. Therefore these dynamic contact forces are of main concern of the railway engineers. However it is very difficult to measure such dynamic contact forces directly. So it is important to develop an appropriate numerical simulation model and identify structural factors having a large influence on the variation of contact forces. When a contact force is expressed by the linearized Hertzian contact spring model, the equation of motions of the system is expressed as a second–order linear time–variant differential equation which has a time–dependent stiffness coefficient. Applying a well–known Newmark direct integration method, a numerical simulation is reduced to solving iteratively a time–variant, large–scale sparse, symmetric positive–definite linear system. In this study, by defining a special vector named a contact point one, it is shown that this time–variant stiffness coefficient can be expressed simply as a product of the contact point vector and its transpose and so the Sherman–Morrison–Woodbury formula applied for updating the inverse of the coefficient matrix. As a result, the execution of numerical simulation can be carried out very efficiently. A comparison of the computational time is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

车桥系统的耦合振动

通过用正弦波形模拟桥面的不平和考虑移动车辆-桥梁间的相互作用,在Euler-Bernoulli梁理论的基础上建立了一种车桥系统的耦合振动模型．利用模态分析法和Runge-Kutta法对模型进行数值求解,获得了车桥系统耦合振动的动态响应和共振曲线．发现车桥耦合振动的共振曲线中存在两个共振区域,一个反映主共振而另一个反映次共振．讨论了桥面不平、桥梁振型和车辆间的相互作用对系统振动的影响．数值结果表明,这些参数对系统振动的影响很大,桥面不平和振型对车桥系统耦合振动的影响不能忽略,设计车速应该远离临界车速．     

A beam segment element for dynamic analysis of large aqueducts

Large aqueduct structure is a complex structure that is commonly used in hydraulic engineering, especially in large-scale water conveying projects. The analysis of dynamic response for an aqueduct structure is necessary if the aqueduct is built in an earthquake area. Traditional 3D finite element analysis is time consuming and the existing simplified response method cannot take into account all the effects, such as the bending-torsion coupling effect and the constrained torsion, of the deformations of the thin wall structure of the aqueduct body. For this special structure, a simple and yet accurate model for dynamic analysis is needed. In this paper, a beam segment element is developed and used for the calculation of dynamic response for aqueduct structures. With the frame of the aqueduct being modeled using beam element, the proposed model can calculate the dynamic response of the whole aqueduct structures. Results are compared with that of a general purpose finite element analysis software using 3D finite element model. Good agreement is achieved between the two models. However, the proposed model needs less elements and much less computing time.     

A dynamic programming method for matching rail-track monitoring data

A commonly used method of monitoring the condition of rail track is to run an inspection vehicle over the track at intervals of about 3 months. Measurements of several geometric properties of the track are automatically recorded about every 900 mm, resulting in long sequences of data (signals) arising from runs of up to 100 km. Condition monitoring is done by comparing the results of a current run with those of a previously recorded reference run. Before this can be done, the two signals need to be aligned so that corresponding distance measurements in each signal actually refer to the same point on the track. A procedure for matching the two signals is presented, which has at its heart a dynamic programming method. The procedure is demonstrated on data from rail tracks in Australia.     

Three-dimensional vehicle-ballasted track-subgrade interaction: Model construction and numerical analysis

In this work a three-dimensional vehicle-ballasted track-subgrade interaction model is developed, where the vehicle is modeled as a multi-rigid-body system, the track-subgrade interaction is modelled by finite element method (FEM) with the rail modelled as beam elements, the sleeper and the subgrade layers modelled as solid elements, and the vehicle and the track-subgrade system are coupled by unified matrix formulations and solved simultaneously and time-dependently. As the modelling advancement, the versatility for FEM construction of track-subgrade systems has been promoted, where the elemental scale can be arbitrarily chosen without consideration of the node-to-node matching principle as the conventional methods; besides the vehicle and the track-subgrade systems have been coupled effectively by wheel-rail contact models. The applicability of various wheel-rail contact models has been discussed, and to reveal the effectiveness of this model in solving engineering problems such as the soil elasticity unevenness and the contact break of “hanging sleepers”, numerical examples have also been presented with referencable conclusions.     

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)     

跨海特大型桥梁风-浪耦合作用的随机振动分析

考虑风-浪耦合场中风和波浪特征参数的相关性,建立了基于有限元法与边界元法联合分析的特大型桥梁风-浪耦合作用运动方程.其中,作用在大型深水基础上的波浪力采用势流理论和边界元法进行计算,并建立有限单元与边界元单元组的映射关系,将边界单元组上的波浪力映射到结构有限单元上;作用在桥梁上的气动力通过有限元法进行计算,包括由脉动风激发的非定常抖振力和由气弹相互作用产生的自激力.在此基础上,基于随机振动分析的高效算法——虚拟激励法,建立了计算桥梁风-浪耦合作用响应的分析方法.最后,针对某跨海超大跨桥梁方案进行研究,结果表明:与风致响应相比,风-浪耦合作用下桥梁深水基础内力显著增大,其中波浪激发的侧向剪力占主导地位,波浪激发的侧向弯矩在海床附近与风致响应基本相当,但在海床以下更大;斜风-波浪耦合作用下的主梁内力响应和深水基础内力响应比正交风-波浪耦合作用下的结果更大.因此,在跨海桥梁设计中,必须考虑风-浪耦合作用效应.     

Comprehensive optimization of urban rail transit timetable by minimizing total travel times under time-dependent passenger demand and congested conditions

The comprehensive optimization of the timetables of urban rail transit systems under more realistic conditions is essential for their practical operation. Currently, most time-dependent timetabling models do not adequately consider train capacity and variable operation parameters. To bridge this gap, this study mainly investigates the timetable design problem of the urban rail transit system so as to adapt to time-dependent passenger demand under congested conditions by considering the variable number of trains, train running time, and train dwell time. Two nonlinear non-convex programming models are formulated to design timetables with the objective of minimizing the total passenger travel time (TTT) under the constraints of train operations, and passenger boarding and alighting processes. The difference between the two models is that one is a train-capacity unconstrained model and the other is a train-capacity constrained model. The proposed models are examined through real-world cases solved by the adaptive large neighborhood search algorithm. The results show that the first model can minimize passenger TTT under dynamic passenger demand, whereas the second can comprehensively optimize passenger TTT and meantime keep the train load factor within a reasonable level. Accordingly, it is concluded that the proposed models are more realistic.     

The interactive vibration behavior in a suspension bridge system under moving vehicle loads and vertical seismic excitations

In this paper, the vibration behavior of a suspension bridge due to moving vehicle loads with vertical support motions caused by earthquake is studied. The suspension bridge system is presented here by two coupled nonlinear cable–beam equations aiming to describe both the dynamic characteristics for the supporting cable and the roadbed, respectively. The dynamic effect of traffic vehicles are modeled as a row of equidistant moving forces, while the earthquake movement is simulated as the vertical oscillation of boundary supports. The governing integro-differential equations are transferred into a set of ordinary differential equations, which can be solved analytically in the present study. Furthermore, the world’s largest designed suspended bridge – Messina Bridge – is examined (central span of length 3.3 km) and the modified Kobe earthquake records is applied to the calculations in order to validate the present study and the proposed methodology. As a result, the deformation of the cable produces more oscillations than that of the beam since the material property of the cable is more flexible. It is shown that the interaction of both the moving loads and the seismic forces can substantially amplify the response of long-span suspension bridge system especially in the vicinity of the end supports.