Vertical random vibration analysis of vehicle–track coupled system using Green's function method

A vertical vehicle–track coupled dynamic model, consisting of a high-speed train on a continuously supported rail, is established in the frequency-domain. The solution is obtained efficiently by use of the Green's function method, which can determine the vibration response over a wide range of frequency without any limitations due to modal truncation. Moreover, real track irregularity spectra can be used conveniently as input. The effect of the flexibility of both track and car body on the entire vehicle–track coupled dynamic response is investigated. A multi-body model of a vehicle with either rigid or flexible car body is defined running on three kinds of track: a rigid rail, a track stiffness model and a Timoshenko beam model. The results show that neglecting the track flexibility leads to an overestimation of both the contact force and the whole vehicle vibration response. The car body flexibility affects the ride quality of the vehicle and the coupling through the track and can be significant in certain frequency ranges. Finally, the effect of railpad and ballast stiffness on the vehicle–track coupled vibration is analysed, indicating that the stiffness of the railpad has an influence on the system in a higher frequency range than the ballast.     

轨道不平顺波长和幅值对高速动车组动力学性能的影响分析

为了更有针对性地控制高速铁路轨道不平顺,保证某型高速动车组在线路上安全、平稳运行,通过动力学仿真,分析轨道不平顺波长和幅值对车辆动力学性能的影响规律,确定轨道养护维修时需重点关注的不平顺波长范围.针对某型高速动车组,将轨道不平顺简化为不同波长、不同幅值的谐波型高低、轨向、水平和扭曲不平顺,基于车辆-轨道耦合动力学模型对不平顺激扰下的车辆动力学性能进行仿真分析.分析表明,中、短波长的不平顺对轮轨动力相互作用和行车安全性有显著影响,对运行平稳性也有一定影响;中、长波长的不平顺主要影响运行平稳性.不平顺幅值对车辆动力学性能指标的影响规律基本呈线性变化,敏感波长条件下的不平顺幅值变化对系统动力响应的影响尤为显著.分析结果确定了某型高速动车组的敏感波长范围及其幅值影响规律,明确了养护维修重点,将有助于提高养护维修效率,提升高速动车组运行品质.      

High-speed trains subject to abrupt braking

The dynamic response of high-speed train subject to braking is investigated using the moving element method. Possible sliding of wheels over the rails is accounted for. The train is modelled as a 15-DOF system comprising of a car body, two bogies and four wheels interconnected by spring-damping units. The rail is modelled as a Euler–Bernoulli beam resting on a two-parameter elastic damped foundation. The interaction between the moving train and track-foundation is accounted for through the normal and tangential wheel–rail contact forces. The effects of braking torque, wheel–rail contact condition, initial train speed and severity of railhead roughness on the dynamic response of the high-speed train are investigated. For a given initial train speed and track irregularity, the study revealed that there is an optimal braking torque that would result in the smallest braking distance with no occurrence of wheel sliding, representing a good compromise between train instability and safety.     

Effect of temperature- and frequency-dependent dynamic properties of rail pads on high-speed vehicle–track coupled vibrations

The soft under baseplate pad of WJ-8 rail fastener frequently used in China’s high-speed railways was taken as the study subject, and a laboratory test was performed to measure its temperature and frequency-dependent dynamic performance at 0.3?Hz and at ?60°C to 20°C with intervals of 2.5°C. Its higher frequency-dependent results at different temperatures were then further predicted based on the time–temperature superposition (TTS) and Williams–Landel–Ferry (WLF) formula. The fractional derivative Kelvin–Voigt (FDKV) model was used to represent the temperature- and frequency-dependent dynamic properties of the tested rail pad. By means of the FDKV model for rail pads and vehicle–track coupled dynamic theory, high-speed vehicle–track coupled vibrations due to temperature- and frequency-dependent dynamic properties of rail pads was investigated. Finally, further combining with the measured frequency-dependent dynamic performance of vehicle’s rubber primary suspension, the high-speed vehicle–track coupled vibration responses were discussed. It is found that the storage stiffness and loss factor of the tested rail pad are sensitive to low temperatures or high frequencies. The proposed FDKV model for the frequency-dependent storage stiffness and loss factors of the tested rail pad can basically meet the fitting precision, especially at ordinary temperatures. The numerical simulation results indicate that the vertical vibration levels of high-speed vehicle–track coupled systems calculated with the FDKV model for rail pads in time domain are higher than those calculated with the ordinary Kelvin–Voigt (KV) model for rail pads. Additionally, the temperature- and frequency-dependent dynamic properties of the tested rail pads would alter the vertical vibration acceleration levels (VALs) of the car body and bogie in 1/3 octave frequencies above 31.5?Hz, especially enlarge the vertical VALs of the wheel set and rail in 1/3 octave frequencies of 31.5–100?Hz and above 315?Hz, which are the dominant frequencies of ground vibration acceleration and rolling noise (or bridge noise) caused by high-speed railways respectively. Since the fractional derivative value of the adopted rubber primary suspension, unlike the tested rail pad, is very close to 1, its frequency-dependent dynamic performance has little effect on high-speed vehicle–track coupled vibration responses.     

Interaction study of train–bridge–track systems using Maxwell model

The investigation of problems related to the interaction of train, bridge and track systems has been accelerated by the emergence of high-speed trains. Such studies are required, not only for the endurance issues regarding bridge and tracks, but to assure trains’ functionality and performance. The suspension mechanism of train systems is of prime importance in defining the functionality of high-speed trains, and accurate mathematical models of the mechanism are imperative. This paper introduces a numerical technique for an interaction study of train–bridge–track systems based on Maxwell (three-element type) modeling of the suspension mechanisms of vehicles. Track irregularity in sinusoidal form is also integrated into the mathematical model. Although the proposed technique is simple in formulation, it offers phenomenal accuracy in representing the interaction of train, track and bridge systems. In a numerical example, the dynamic behavior of a train–bridge system has been studied. Results of this analysis provide a valuable insight into the contributing roles of different parameters in this subject.     

Interaction study of train-bridge-track systems using Maxwell model

The investigation of problems related to the interaction of train, bridge and track systems has been accelerated by the emergence of high-speed trains. Such studies are required, not only for the endurance issues regarding bridge and tracks, but to assure trains' functionality and performance. The suspension mechanism of train systems is of prime importance in defining the functionality of high-speed trains, and accurate mathematical models of the mechanism are imperative. This paper introduces a numerical technique for an interaction study of train-bridge-track systems based on Maxwell (three-element type) modeling of the suspension mechanisms of vehicles. Track irregularity in sinusoidal form is also integrated into the mathematical model. Although the proposed technique is simple in formulation, it offers phenomenal accuracy in representing the interaction of train, track and bridge systems. In a numerical example, the dynamic behavior of a train-bridge system has been studied. Results of this analysis provide a valuable insight into the contributing roles of different parameters in this subject.     

Experimental and numerical investigation of the effect of rail corrugation on the behaviour of rail fastenings

This paper presents the results of a detailed investigation of the effects of rail corrugation on the dynamic behaviour of metro rail fastenings, obtained from extensive experiments conducted on site and from simulations of train–track dynamics. The results of tests conducted with a metro train operating on corrugated tracks are presented and discussed first. A three-dimensional (3D) model of the metro train and a slab track was developed using multi-body dynamics modelling and the finite element method to simulate the effect of rail corrugation on the dynamic behaviour of rail fastenings. In the model, the metro train is modelled as a multi-rigid body system, and the slab track is modelled as a discrete elastic support system consisting of two Timoshenko beams for the rails, a 3D solid finite element (FE) model for the slabs, periodic discrete viscoelastic elements for the rail fastenings that connect the rails to the slabs, and uniformly viscoelastic elements for the subgrade beneath the slabs. The proposed train–track model was used to investigate the effects of rail corrugation on the dynamic behaviour of the metro track system and fastenings. An FE model for the rail fastenings was also developed and was used to calculate the stresses in the clips, some of which rupture under the excitation of rail corrugation. The results of the field experiments and dynamics simulations provide an insight into the root causes of the fracture of the clips, and several remedies are suggested for mitigating strong vibrations and failure of metro rail fastening systems.     

Probabilistic assessment of railway vehicle-curved track systems considering track random irregularities

The randomness of track irregularities directly leads to the random vibration of the vehicle–track systems. To assess the dynamic performance of a railway system in more comprehensive and practical ways, a framework for probabilistic assessment of vehicle-curved track systems is developed by effectively integrating a vehicle–track coupled model (VTCM), a track irregularity probabilistic model (TIPM) with a probability density evolution method (PDEM). In VTCM, the railway vehicle and the curved track are coupled by the nonlinear wheel–rail interaction forces, and through TIPM, the ergodic properties of random track irregularities on amplitudes, wavelengths and probabilities can be properly considered in the dynamic calculations. Lastly, PDEM, a newly developed method for solving probabilistic transmissions between stochastic excitations and deterministic dynamic responses, is introduced to this probabilistic assessment model. Numerical examples validate the correctness and practicability of the proposed models. In this paper, the results of probabilistic assessment are presented to illustrate the dynamic behaviours of a high-speed railway vehicle subject to curved tracks with various radii, and to demonstrate the importance of considering the actual status of wheel–rail contacts and curve negotiation effects in vehicle-curved track interactions.     

Stochastic vibration of the vehicle–bridge system subject to non-uniform ground motions

A study of a train moving along a cable-stayed bridge is performed by considering both the stationary track irregularity and a non-stationary earthquake. A detailed bridge model with 3972 degrees of freedom is established while the train model consists of two locomotives and eight carriages. The equations of motion of the coupled system are obtained by using the displacement continuous condition at the contact, with track irregularities. The earthquake is assumed to occur once the train has entered the bridge. The pseudo-excitation method is used to find the random responses of the coupled system, and the results indicate that the effect of the earthquake is much greater than that of the track irregularities. The paper discusses the influence of the intensity of the earthquake, the wave propagation velocity, the speed of the train, and the dynamic interaction between the vehicles and the bridge.     

Investigation of the effects of sleeper-passing impacts on the high-speed train

The sleeper-passing impact has always been considered negligible in normal conditions, while the experimental data obtained from a High-speed train in a cold weather expressed significant sleeper-passing impacts on the axle box, bogie frame and car body. Therefore, in this study, a vertical coupled vehicle/track dynamic model was developed to investigate the sleeper-passing impacts and its effects on the dynamic performance of the high-speed train. In the model, the dynamic model of vehicle is established with 10 degrees of freedom. The track model is formulated with two rails supported on the discrete supports through the finite element method. The contact forces between the wheel and rail are estimated using the non-linear Hertz contact theory. The parametric studies are conducted to analyse effects of both the vehicle speeds and the discrete support stiffness on the sleeper-passing impacts. The results show that the sleeper-passing impacts become extremely significant with the increased support stiffness of track, especially when the frequencies of sleeper-passing impacts approach to the resonance frequencies of wheel/track system. The damping of primary suspension can effectively lower the magnitude of impacts in the resonance speed ranges, but has little effect on other speed ranges. Finally, a more comprehensively coupled vehicle/track dynamic model integrating with a flexible wheel set is developed to discuss the sleeper-passing-induced flexible vibration of wheel set.     

Train–track–bridge dynamic interaction: a state-of-the-art review

ABSTRACT

Train–track–bridge dynamic interaction is a fundamental concern in the field of railway engineering, which plays an extremely important role in the optimal design of railway bridges, especially in high-speed railways and heavy-haul railways. This paper systematically presents a state-of-the-art review of train–track–bridge dynamic interaction. The evolution process of train–bridge dynamic interaction model is described briefly, from the simplest moving constant force model to the sophisticated train–track–bridge dynamic interaction model (TTBDIM). The modelling methodology of the key elements in the TTBDIM is systematically reviewed, including the train, the track, the bridge, the wheel–rail contact, the track–bridge interaction, the system excitation and the solution algorithm. The significance of detailed track modelling in the whole system is highlighted. The experimental research and filed test focusing on modelling validation, safety assessment and long-term performance investigation of the train–track–bridge system are briefly presented. The practical applications of train–track–bridge dynamic interaction theory are comprehensively discussed in terms of the system dynamic performance evaluation, the system safety assessment and train-induced environmental vibration and noise prediction. The guidance is provided on further improvement of the train–track–bridge dynamic interaction model and the challenging research topics in the future.     

Dynamic response of the train–track–bridge system subjected to derailment impacts

Derailments on bridges, although not frequent, when occurs due to a complex dynamic interaction of the train–track–bridge structural system, are very severe. Furthermore, the forced vibration induced by the post-derailment impacts can toss out the derailed wagons from the bridge deck with severe consequences to the traffic underneath and the safety of the occupants of the wagons. This paper presents a study of the train–track–bridge interaction during a heavy freight train crossing a concrete box girder bridge from a normal operation to a derailed state. A numerical model that considers the bridge vibration, train–track interaction and the train post-derailment behaviour is formulated based on a coupled finite-element – multi-body dynamics (FE-MBD) theory. The model is applied to predict the post-derailment behaviour of a freight train composed of one locomotive and several wagons, as well as the dynamic response of a straight single-span simply supported bridge containing ballast track subjected to derailment impacts. For this purpose, a typical derailment scenario of a heavy freight train passing over a severe track geometry defect is introduced. The dynamic derailment behaviour of the heavy freight train and the dynamic responses of the rail bridge are illustrated through numerical examples. The results exhibit the potential for tossing out of the derailed trains from the unstable increase in the yaw angle signature and a lower rate of increase of the bridge deck bending moment compared to the increase in the static axle load of the derailed wheelset.     

Effects of rail thermal stress on the dynamic response of vehicle and track

A new method is proposed to obtain the dynamic responses of the vehicle–track coupling system under the conditions of rail thermal stress changes in high-speed railways. Exact models are established with different rail longitudinal forces, in which multibody dynamic models are used for vehicles and the direct stiffness method for structures. In order to provide a general, simple and flexible formulation to express longitudinal stress distribution, the accurate model of long slab track consists of many small units with parameters which can be initialised separately. The exact analytical equation of track frequency and modal function was obtained by the transition matrix method, which can be used in calculating the dynamic response of wheel–rail coupling model. The proposed model is verified through comparisons with other classical solutions. Under the influence of train velocities and track irregularities, the specific vibration performances that frequency shifted and amplitude peak enhanced with thermal force are demonstrated through examples. The results show that the response analyses of vehicle and track have great application potentiality for fast estimation of the rail longitudinal stress.     

Ride comfort of high-speed trains travelling over railway bridges

The ride comfort of high-speed trains passing over railway bridges is studied in this paper. A parametric study is carried out using a time domain model. The effects of some design parameters are investigated such as damping and stiffness of the suspension system and also ballast stiffness. The influence of the track irregularity and train speed on two comfort indicators, namely Sperling's comfort index and the maximum acceleration level are also studied. Two types of railway bridges, a simple girder and an elastically supported bridge are considered.

Timoshenko beam theory is used for modelling the rail and bridge and two layers of parallel damped springs in conjunction with a layer of mass are used to model the rail-pads, sleepers and ballast. A randomly irregular vertical track profile is modelled, characterized by its power spectral density (PSD). The ‘roughness’ is generated for three classes of tracks. Nonlinear Hertz theory is used for modelling the wheel-rail contact. The influences of some nonlinear parameters in a carriage-track-bridge system, such as the load-stiffening characteristics of the rail-pad and the ballast and that of rubber elements in the primary and secondary suspension systems, on the comfort indicators are also studied. Based on Galerkin's method of solution, a new analytical approach is developed for the combination between the rigid and flexural mode shapes, which could be used not only for elastically supported bridges but also other beam-type structures.     

Extended applications of track irregularity probabilistic model and vehicle–slab track coupled model on dynamics of railway systems

Track irregularities are inevitably in a process of stochastic evolution due to the uncertainty and continuity of wheel–rail interactions. For depicting the dynamic behaviours of vehicle–track coupling system caused by track random irregularities thoroughly, it is a necessity to develop a track irregularity probabilistic model to simulate rail surface irregularities with ergodic properties on amplitudes, wavelengths and probabilities, and to build a three-dimensional vehicle–track coupled model by properly considering the wheel–rail nonlinear contact mechanisms. In the present study, the vehicle–track coupled model is programmed by combining finite element method with wheel–rail coupling model firstly. Then, in light of the capability of power spectral density (PSD) in characterising amplitudes and wavelengths of stationary random signals, a track irregularity probabilistic model is presented to reveal and simulate the whole characteristics of track irregularity PSD. Finally, extended applications from three aspects, that is, extreme analysis, reliability analysis and response relationships between dynamic indices, are conducted to the evaluation and application of the proposed models.     

High-frequency random vibration analysis of a high-speed vehicle–track system with the frequency-dependent dynamic properties of rail pads using a hybrid SEM–SM method

A hybrid Spectral Element Method (SEM)–Symplectic Method(SM) method for high-efficiency computation of the high-frequency random vibrations of a high-speed vehicle–track system with the frequency-dependent dynamic properties of rail pads is presented. First, the Williams-Landel-Ferry (WLF) formula and Fractional Derivative Zener (FDZ) model were, respectively, applied for prediction and representation of the frequency-dependent dynamic properties of Vossloh 300 rail pads frequently used in China's high-speed railway. Then, the proposed hybrid SEM–SM method was used to investigate the influence of the frequency-dependent dynamic performance of Vossloh 300 rail pads on the high-frequency random vibrations of high-speed vehicle–track systems at various train speeds or different levels of rail surface roughness. The experimental results indicate that the storage stiffness and loss factors of Vossloh 300 rail pad increase with the decrease in dynamic loads or the increase in preloads within 0.1–10,000?Hz at 20°C, and basically linearly increase with frequency in a logarithmic coordinate system. The results computed by the hybrid SEM–SM method demonstrate that the frequency-dependent viscous damping of Vossloh 300 rail pads, compared with its constant viscous damping and frequency-dependent stiffness, has a much more conspicuous influence on the medium-frequency (i.e. 20–63?Hz) random vibrations of car bodies and rail fasteners, and on the mid- (i.e. 20–63?Hz) and high-frequency (i.e. 630–1250?Hz) random vibrations of bogies, wheels and rails, especially with the increase in train speeds or the deterioration of rail surface roughness. The two sensitive frequency bands can also be validated by frequency response function (FRF) analysis of the proposed infinite rail–fastener model. The mid and high frequencies influenced by the frequency-dependent viscous damping of rail pads are exactly the dominant frequencies of ground vibration acceleration and wheel rolling noise caused by high-speed railways, respectively. Even though the existing time-domain (or frequency-domain) finite track models associated with the time-domain (or frequency-domain) fractional derivative viscoelastic (FDV) models of rail pads can also be used to reach the same conclusions, the hybrid SEM–SM method in which only one element is required to compute the high-order vibration modes of infinite rail is more appropriate for high-efficiency analysis of the high-frequency random vibrations of high-speed vehicle–track systems.     

Vertical Vibration Analysis of Vehicle/Imperfect Track Systems

Summary This paper studies the vertical vibration of a vehicle traveling on an imperfect track system. The car body and sleepers are modeled as Timoshenko beams with finite length, and the rail is assumed as an infinite Timoshenko beam with discrete supports. Imperfection of the track system comes from a sleeper lost partial support by the ballast. Since deflection of the rail is limited within a certain interval where the vehicle is passing over, the infinite domain problem can be transformed into a finite domain problem with moving boundary. In this work, the equations of motion of the car body, rail and sleepers are discretized first by the finite element method. The discretized equations of motion for the vehicle and track systems are then assembled, respectively. Finally, the Newmark method is applied to obtain the response of the vehicle and track systems at each time step. The effect of the vehicle speed on the response of the vehicle and track systems is investigated.     

Transverse Vibration of Train-Bridge and Train-Track Time Varying System and the Theory of Random Energy Analysis for Train Derailment

Summary In this paper, the fundamental problems in the calculation of transverse vibration of train-bridge and train-track time-varying system (hereinafter referred to as the system) are expounded. That is, (1) Proper solution to transverse vibration of the system cannot be obtained by establishing separate transverse vibration equation groups for the car and the bridge (or track); (2) The exciting source of transverse vibration of the system has not been made definite; (3) It is difficult to carry out the random analysis of vibration of the time varying system as the theory of the random vibration analysis for the time varying system has not been established. Our thinking and methods to solve these problems are introduced. On the above-mentioned basis, the theory of random energy analysis for train derailment is presented. The main contents of this theory are as follows: method of random energy analysis of transverse vibration of the system; geometric criterion of derailment; mechanism of derailment caused by the combined track irregularity and energy increment criterion for derailment evaluation; calculation of the entire derailing process; method of improvement of track parameters for preventing straight line speed-raise freight trains from derailment and of the calculation of the safety coefficient against derailment. This theory is used to calculate a case of freight train derailment, which corresponds to an actually occurred accident. A train in another derailment test is judged to be not derailed and its maximum vibration responses are calculated. The field test results are well responded.     

Application of the structural articulation method to dynamic impact loading of railway bridges – a case study

This article demonstrates a practical application of the structural articulation method. An existing prototype railway bridge was selected to compare our new method with the industry codes of practice. The response history and dynamic increment of the bridge were investigated through a variety of methods: lump sum mass analysis (LSMA) and suspension system analysis (SSA) for a single-axle force, and SSA for multi-axle forces. We considered both a local irregularity and a global sinusoidal irregularity. The dynamic impact load induced by either form of track irregularity increases approximately linearly with the vehicle speed up to a certain point, then tends to decrease gradually. This behaviour reveals that the dynamic impact load induced by track irregularities is dominated by the resonance of the bridge. If a bridge must support multiple axles, or if an especially accurate dynamic impact factor is required for safety reasons, then multi-axle SSA is recommended because this approach is the most accurate and likely to produce a weaker response than single-axle analysis. The random irregularity is generated by the stochastic track irregularity process. It is found that the dynamic impact load induced by the random irregularity is negligible, compared with the deterministic irregularity.     

Vibration control in train–bridge–track systems

To study the problems associated with vibration control of train–bridge–track systems a mathematical model with the capability of representing supplementary vibrational control devices is proposed. The train system is assumed as rigid bodies supported on double-deck suspension mechanism with semi-active features. The bridge system is modeled using the modal approach. Vibration control for bridge responses is provided by tuned mass dampers. A non-classical incremental Eigen analysis is proposed to trace the system characteristics across the time. In an example, the capability of the proposed model in investigating the vibration control prospects of a bridge–train system is shown. The results indicate the effectiveness of active suspension mechanism in reducing train's body movements, particularly the pitching angle and the vertical accelerations. Accordingly, the results also verify the potential of TMD devices in reducing the bridge responses at resonance motions.