Vertical Dynamic Interaction Research Articles

Dynamic responses of locomotive axle box bearings under varying operating conditions

Axle box bearings (ABBs) are a vital component of a railway locomotive, ensuring their reliable performance requires an in-depth understanding of their dynamic behaviour. This work theoretically derives a locomotive–track longitudinal–vertical coupled dynamics model that takes detailed dynamics of the ABBs into account. The model considers a number of complex interactions, such as non-linear contact and friction forces within the ABB, wheel–rail interactions, and track irregularities. The motions of ABB are coupled with the wheelset and bogie frame, allowing detailed investigation of the ABB dynamic behaviour during operation. The correctness of the proposed model is verified in both the time and frequency domains by comparing with field test. The dynamic characteristics of ABBs under varying operating conditions were comprehensively investigated. The results demonstrate that track irregularities have a non-negligible influence on the dynamic forces of the locomotive ABB. Furthermore, the axle load transfer occurs during loading conditions, which leads to significant differences in vertical and longitudinal dynamic interactions within the ABBs across wheelsets. It is clear that the operating conditions of the locomotive should be suitably considered in the dynamic assessment, operation, and maintenance of ABBs.

Vertical dynamic interaction and group efficiency factor for floating pile group in layered soil

AbstractThis paper theoretically estimates the dynamic pile–soil interaction and the group efficiency factor for pile group in layered soil through energy‐based method. The vertical dynamic interaction of partially embedded single piles and their surrounding layered soil is analytically deduced from Hamilton's principle. Combined with a series of numerical simulations, the soil attenuation factor from energy method is modified for adapting to the wave speed variation in various layers. Then, the pile‐to‐pile interaction factor is directly solved with the help of the transfer‐matrix method. The dynamic governing equation of pile group with an elevated rigid cap is established by superposing the pile‐to‐pile interaction factors. Finally, the dynamic impedance of pile group is obtained and derived into a group efficiency factor. Compared with the plane strain method, this present method can produce a more suitable soil attenuation factor and a dynamic interaction factor at low frequency range, which is exploited for practical engineering design. The effects of subsoil layer, subsoil layer, and unembedded pile segment on group efficiency factor are investigated. The results show that the real part of group efficiency factor decreases at high frequency range for a small pile spacing, which may be detriment to the pile group capacity. Besides that, the combined effects of unembedded segment and weak surface soil on group efficiency factor are highlighted.

Simulation study on the influence of wheel irregularity on the vertical dynamics of wheel–rail interaction for high-speed railway track using bond graph

Dynamic interaction between train wheel and high-speed slab is an important issue when evaluating the contact force at the wheel–rail interface under the influence of an irregularity either on the train wheel or on the rail. In this paper, an influence of amplitude of irregularity along with the vehicle speed on the dynamics of wheel–rail interaction for high-speed railway tracks is being analyzed. For this purpose, single-wheel high-speed railway (HSR) track interaction models are developed using the bond graph modeling technique. The HSR track model consists of two layers of beam, i.e., rail and concrete slab. Both the rail and slab track is modeled using the Euler–Bernoulli beam theory. The Hertzian contact theory at the wheel–rail interface has been considered for this analysis. The vertical dynamic interaction between a train wheel and a high-speed slab track is compiled in the time domain using a bond graph approach coupled with a technique known as modal superposition. Irregularity present on the wheel is characterized as smooth, moderate, and severe depending upon the variation of irregularity amplitude. An expeditious increase of maximum contact force has been observed between the speed range of 200 and 250 km/h. Beyond the speed of 250 km/h, there is a gradual increment of contact force up to its peak value. When the train speed is beyond 288 km/h, there is a gradual decrease in maximum contact force. This kind of several other useful dynamic responses in terms of wheel acceleration and wheel–rail overlap are also evaluated.

Dynamic response of vehicle-track interaction at the bolted rail joint in the presence of the polygonal wheel

Rail joints are the weakest link in the railway track structure due to the presence of discontinuities in the track stiffness and wheel-rail contact geometry. This paper presents the dynamic response of vehicle-track interaction at the bolted rail joint (BRJ) in the presence of polygonal wheel wear. A three-dimensional coupled vehicle-track dynamic model was established using the finite element method (FEM) and the multi-body system (MBS) approach, in which the flexibilities of the wheelset and track are taken into account. The proposed dynamic model is validated by the comparisons of field measurement and calculated vertical axle-box acceleration in time and frequency domains. The effects of the flexible wheelset in the presence of polygonal wear on the vehicle-track dynamic interaction at the BRJs are illustrated through comparisons with those obtained using without polygonal wheel model. The results show that the polygonal wheel wear and the rail joint irregularities can induce high-magnitude vertical wheel-rail contact force, and exacerbate the vibration responses of the axle box, the wheelset, and the rails. The simulation results also indicate that the wheelset bending modes, the P2 resonance due to rail joint excitation, and the track vibration modes have significant effects on the vertical vehicle-track dynamic interaction at the supported and suspended BRJs.

Comprehensive efficient vertical and lateral track dynamic model to study the evolution of rail corrugation in sharp curves

• A highly accurate and fast model is developed for the study of rail corrugation. • Experimental receptances are precisely fitted using RFP and genetic algorithms. • Lateral system dynamics is added to the model preserving short computational times. • The model has been validated with experimental measurements. This paper addresses a time/space domain model, extremely efficient in terms of computation, which combines the vertical and lateral dynamics of the track to predict undulatory wear depending on track and vehicle parameters. As it is known, this is a source of noise and vibration that can provoke very high levels. The model is obtained from the transformation of a model in the frequency domain to the time/space domain by means of the rational fraction polynomials method. Multiobjective genetic algorithms are used to minimize the error in fitting receptances and to ensure the stability of the system. This model includes the vertical wheel-rail dynamic interaction with a Hertzian spring, and the FASTSIM algorithm is used to study tangential contact. The model is validated with the experimental corrugation measurements taken on a metro line.

Vertical dynamic interaction factors for offshore thin-walled pipe piles

With the development of offshore wind power in deep water, jacket foundations are increasingly used. These piles are always thin-walled pipe piles, whose dynamic interactions are different from onshore pile groups in several ways; for example, scour may develop around the piles, which may reduce a single pile’s impedance, the dynamic interactions between piles and the impedances of pile groups. Jacket foundations are usually exposed to marine dynamic loads and are likely to experience vertical and rocking vibrations. Both kinds of vibrations are related to the vertical dynamic interaction between piles, which are the focus of this study. The effects of soil rigidity, soil permeability, pile spacing, excitation frequency, scour type and scour depth on dynamic vertical interaction factors are studied in this paper by a rigorous semi-analytical method. Numerical results for parameter analysis are illustrated, and corresponding simplified closed-form equations to predict dynamic interactions are also deduced. The dynamic interaction factors are then used to study tripod piles and tetrapod piles. Dynamic interaction may increase or decrease the pile impedances, mostly depending on the pile spacing ratio and non-dimensional frequency, and closed-form equations are obtained to forecast the amplification or reduction.

Wheel–rail impact loads and axle bending stress simulated for generic distributions and shapes of discrete wheel tread damage

Wheel–rail impact loads generated by discrete wheel tread irregularities may result in high dynamic bending stresses in the wheelset axle, leading to a decrease in component life and an elevated risk for fatigue failure. In this paper, a versatile and cost-efficient method to simulate the vertical dynamic interaction between a wheelset and railway track, accounting for generic distributions and shapes of wheel tread damage, is presented. The wheelset (comprising two wheels, axle and any attached equipment for braking and power transmission) and track with two discretely supported rails are described by three-dimensional finite element (FE) models. The coupling between the two wheel‒rail contacts (one on each wheel) via the wheelset axle and via the sleepers is considered. The simulation of dynamic vehicle–track interaction is carried out in the time domain using a convolution integral approach, while the non-linear wheel–rail normal contact is solved using Kalker's variational method. Wheelset designs that are non-symmetric with respect to the centre of the axle, track support conditions that are non-symmetric with respect to the centre of the track, as well as non-symmetric distributions of tread damage on the two wheels (or irregularities on the two rails) can be studied. Time-variant stresses are computed for the locations in the wheelset axle which are prone to fatigue. Based on Green's functions for stress established using the wheelset FE model, this is achieved in a post-processing step. An extensive parametric study has been performed where wheel–rail impact loads and axle stresses have been computed for different distributions and sizes of tread damage as well as for different train speeds.

Analytical Model for Vertical Dynamic Interaction between Circular Raft and Saturated Soils

Abstract The raft–soil dynamic interaction problem is of great significance in the fields of marine geotechnical engineering and traffic engineering. This paper develops an analytical model to stud...

Vertical dynamic interactions of poroelastic soils and embedded piles considering the effects of pile-soil radial deformations

The paper presents an analytical solution for the vertical dynamic interaction analysis of a poroelastic soil layer and an embedded pile with the consideration of pile-soil radial deformations. The soil is treated a three-dimensional porous continuum and described by the Boer’s poroelastic model, while the pile is treated as a two-dimensional rod with both radial and vertical deformations of which the equation of motion is derived by the Hamilton’s variational principle. Without the introduction of potential functions, first take the volumetric strain of soil skeleton and pore fluid pressure as intermediate variables to deal with the equations of motion for the soil and then use the separation of variables to solve the equations of motion for the soil and the pile. By imposing the boundary and continuity conditions of the pile-soil system, the dynamic impedance in frequency domain and the velocity response in time domain of the pile top are obtained. The present solution is then verified by comparing with the corresponding finite element model computation results and the existing solutions. The effects of the pile-soil parameters on the dynamic characteristic of the pile-soil system are also analyzed. Some significant conclusions are drawn, which can provide useful reference for related engineering practice.

Dynamic Performance of the LMS Maglev Train–Track–Bridge System Under Uneven Settlement for Two Typical Bridges

To investigate the dynamic performance of the low-to-medium-speed (LMS) maglev train and bridge system under uneven ground settlement, a refined vertical dynamic interaction model of the LMS maglev train–track–bridge system with uneven settlement is proposed. Firstly, the numerical model is verified based on the field test. Secondly, the dynamic performances of the system induced by uneven settlements are numerically analyzed. Furthermore, numerical studies are carried out to investigate the effect of various uneven settlement types, to compare the performances of the two typical bridges, and to assess the contribution of the F-rail in the presence of uneven settlement. The results show that uneven settlement has a significant enlargement effect on the dynamic responses of the car body and levitation module, but a very weak influence on the bridge. Both the patterns of uneven settlement and bridge types significantly affect the dynamic response of the maglev train to various levels. The numerical model excluding the track structure will overestimate the dynamic responses of the levitation module. It is suggested that the dynamic interaction model for the maglev train–track–bridge system be selected to simulate the influence of uneven settlement for better accuracy.

Study on Vertical Vibration of Partially Exposed Friction Pipe Pile Groups Based on Fictitious Soil Pipe Pile Model

Regarding soil below the pipe pile as fictitious soil pipe pile, the vertical dynamic model of partially exposed friction pipe pile is established. Winkler spring-damper model is used to simulate the vertical dynamic interaction between soil around the pile and pipe pile, and between pile core soil and pipe pile by ignoring the radial displacement and circumferential displacement of soil. Considering the displacement boundary conditions of the soil around pile and pile core soil, the effects of the soil around pile and pile core soil on the pipe pile are obtained, and the complex stiffness of vertical vibration of a partially exposed friction pipe pile is obtained by using the initial parameter method and the transfer matrix method. The vertical dynamic interaction factor of partially exposed friction pipe pile-pipe pile is obtained by considering the dynamic interaction between pipe pile and pipe pile. Based on the principle of pile groups superposition and the vertical dynamic interaction factor of pipe pipe-pipe pile, the vertical vibration of partially exposed friction pipe pile groups is studied, and the vertical dynamic impedance of pipe pile groups is also obtained, and the influences of relevant parameters on the vertical dynamic impedance of partially exposed friction pipe pile groups are investigated by numerical examples of 3 × 3 pipe pile groups. The results show that the dynamic stiffness and damping of end-bearing friction pipe pile groups are larger than friction pipe pile groups, and the change of peak values of curves varying with frequency are gentler. The restraint effect of bedrock under pipe pile can increase the vertical dynamic impedance of pipe pile groups. Increasing the length of pipe pile is beneficial to increase the vertical dynamic impedance of pipe pile groups. The larger the pipe pile spacing, the sharper the peak values of curves of the dynamic stiffness and damping curves of pipe pile groups varying with frequency. The influence of pipe pile geometry on vertical vibration of pipe pile groups is greater than that of physical properties such as elastic modulus of pipe pile and shear modulus of soil.

Simulation of vertical dynamic vehicle–track interaction using a three-dimensional slab track model

When improving track design, a better understanding of the track’s damage modes is needed, and the railway industry is then dependent on the availability of accurate simulations of the dynamic vehicle–track interaction. In the present study, the vertical dynamic interaction between a travelling railway vehicle and a slab track is simulated in the time domain by using an extended state-space vector approach. A three-dimensional slab track model is launched where the rails are modelled using Rayleigh–Timoshenko beam elements and the concrete panel and roadbed are represented by using either shell or solid finite elements. From the parameterised track model, which is developed in Abaqus using Python scripts, the system matrices are exported to Matlab where the simulation of the dynamic vehicle–track interaction is performed. A complex-valued modal superposition technique is employed, which reduces the computational cost of the simulation. In a post-processing step, calculated wheel–rail contact forces from the dynamic analysis are used as input to the Abaqus track model where various track responses are evaluated. In particular, the time history of principal stresses is determined at critical locations in the concrete panel. Also the influence of the speed of the vehicle on the wheel–rail contact forces, and the influence of a transverse culvert below the track (modelled as a local increase of the foundation bedding modulus) on the track stiffness variation at the rail level, are investigated. A mesh convergence study for a range of track responses has been conducted including investigations of when to use linear or quadratic elements and shell or solid elements. Finally, the presented three-dimensional models have been compared to an alternative two-dimensional model to determine in what situations a two-dimensional model is sufficient.

Dynamic interaction analysis of bridges induced by a low-to-medium–speed maglev train

The structure of low-to-medium–speed maglev trains significantly differs from that of traditional wheel/rail trains, leading to significant differences between the coupling vibration mechanism of the train and bridge systems. To determine the vertical dynamic interaction of the low-to-medium–speed maglev train–bridge system, a dynamic interaction model was established and studied, based on a proportional–integral–derivative active suspension control system and modal superposition method. The simulation model was validated through bridge dynamic field tests on the Changsha low-to-medium–speed maglev commercial line. The vertical dynamic characteristics of the system were analyzed for bridges with different girder heights. Subsequently, the mechanism of the vertical resonance of the bridge induced by the maglev train was analyzed carefully. The results show that reducing the bridge rigidity increases the electromagnetic levitation force, thereby increasing the dynamic response. The low-speed resonance in the bridge is caused by the circulation loading frequency of the adjacent electromagnetic force, whereas the normal-speed resonance is induced by the self-frequency of the electromagnetic levitation force.

Influence of the track structure on the vertical dynamic interaction analysis of the low-to-medium-speed maglev train-bridge system

The low-to-medium-speed maglev train is stably suspended near the rated suspension gap. The suspension force acts directly on the track and is transmitted to the bridge. The maglev track structure is novel, and the influence mechanism of the track structure on the coupled vibration of the maglev train-bridge system is unknown. Therefore, in this study, we propose vertical dynamic interaction models of the low-to-medium-speed maglev train-bridge system and the low-to-medium-speed maglev train-track-bridge system to analyse the influence mechanism of the maglev track structure on the vertical dynamic interaction of the low-to-medium-speed maglev train-bridge system. The vibration characteristics of the F-rail and the influence mechanism of the track structure on the dynamic responses of the bridge are discussed in detail. The study verifies that the local deformation of the F-rail is self-evident and cannot be ignored. In addition, the influence of the F-rail on the dynamic interaction of the maglev train-bridge system is mainly reflected in two aspects: first, the vibration of the bridge in the high-frequency band increases due to the high frequency and intensive local vibration of the F-rail itself. Second, the vibrations of the bridge and the F-rail in the low-frequency band increase due to the periodic irregularities caused by the local deformation of the F-rail. In this study, we consider the vertical dynamic interaction model of the low-to-medium-speed maglev train-track-bridge system.

High efficient dynamic analysis of vehicle–track–subgrade vertical interaction based on Green function method

ABSTRACTA new method is proposed to efficiently simulate vehicle-track-subgrade vertical dynamic interactions based on the Green function method. In this simulation, the vehicle-track-subgrade system is divided into two subsystems, the vehicle-rail subsystem and the slab-subgrade subsystem, which are coupled through fastener connections. In the vehicle-rail subsystem, the vehicle is modelled as a multi-rigid-body system with 10 degrees of freedom, and the rail is simplified as a Bernoulli-Euler beam supported by rail pads. Dynamic equations of the vehicle-rail subsystem are described by differential equations. In the slab-subgrade subsystem, a three-dimensional finite element model is established to obtain the Green functions of the system induced by a unit impulse excitation. Based on the derived Green functions, dynamic equations of the slab-subgrade subsystem are formulated by the Duhamel integral. The interaction between the two subsystems is transferred by the fastener forces, and the dynamic responses of the whole system are solved through explicit numerical methods. The proposed method is subsequently validated through a detailed numerical comparison with the result obtained by the finite element method. Finally, the method is applied to investigate the dynamic property of the subgrade system under high-speed train operation.

Analysis of Vertical Dynamic Cross Interaction between Adjacent Footings for Railway Tracks through Soil

An efficient semi-analytical method is presented for the vertical analysis of adjacent railway tracks considering the dynamic cross interaction (DCI) effect through the semi-infinite soil medium. The interfaces between soil and tracks are discretized into a series of strip elements to solve the unknown contact pressure influenced by DCI effect. The Green function for each element under uniform harmonic excitation is derived and calculated by the piecewise integration and Cauchy principal value integral. The vertical impedance matrix is finally obtained by the superposition method. The accuracy and the effectiveness in high frequency range are verified by the convergence study, as well as the comparison study with the rigorous results from the mix-boundary-value method. The influence of the soil properties and the tracks distance on the DCI effect are discussed in detail in the parameter studies. It is shown that the DCI effect increases with the decrease of the distance ratio S/L.

Prediction of rail non-uniform wear – Influence of track random irregularity

A numerical model for the prediction of rail non-uniform wear evolution is proposed. The vehicle-track coupled dynamics model, Kalker's variational method and the material wear model developed by the University of Sheffield are combined to acquire the distribution of rail wear at each contact patch. Rail wear distributions for particular cross-sections at regular intervals are calculated by a new sampling and accumulation strategy. Moving average smoothing and spline interpolation are employed to obtain a relatively smooth rail surface. Based on the proposed model, the non-uniform wear evolution of a 100 m-long rail and its effect on wheel-rail dynamic interaction are investigated based on a Chinese high-speed vehicle (CRH2 EMU) that runs on the straight-line section of a ballastless track (CRTS I). The simulation results indicate that the wear band is almost parallel to the rolling direction and shows an extended trend during the wear evolution. The distribution of the rail wear area along the wheel rolling direction shows agreement with the wear number. The average wear area after the passage of one vehicle remains constant even though the wear is continuously updated. Non-uniform rail wear has a minimal effect on the wheel-rail vertical dynamic interaction but has a significant influence on the wheel-rail lateral interaction.

Vertical impedance of a pile in layered saturated viscoelastic half-space considering radial inhomogeneity

A simplified analytical approach is presented to investigated the vertical impedance of a pile embedded in layered saturated viscoelastic half-space considering radial inhomogeneity. The vertical dynamic interaction between the soil and pile is simulated by the Beam-on-Dynamic-Winkler-Foundation (BDWF) model. To characterize the variation of the soil properties in the disturbed zone around the pile, an improved complex stiffness transfer model is developed to determine the Winkler moduli based on the wave propagation theory of saturated porous medium. Vertical impedance of the pile head is obtained by solving the differential equation for axial vibration of the pile, combined with transfer-matrix formulations to deal with the layered property of the soil along the pile shaft. The validity and accuracy of the analytical solutions are demonstrated through the comparison examples for the cases of different radial variations of soil properties. Parametric studies are performed to investigate the influences of the Darcy permeability coefficient, width ratio and shear modulus ratio of the disturbed zone on the vertical impedance of the pile. It is revealed that the permeability of the pore fluid has an influence on the impedance of the pile when the Darcy permeability coefficient is large than 1 × 10−3. In addition, the different disturbance degrees of the soil layers along the pile shaft are suggested to take into consideration in analysis of the vertical impedance of the pile embedded in layered saturated viscoelastic half-space.

Metro train-track-tunnel-soil vertical dynamic interactions – semi-analytical approach

ABSTRACTAn accurate evaluation of the metro train-track-tunnel-soil dynamic interactions is very important in engineering practice. Therefore, an innovative train-track-tunnel-soil vertical coupled model was established to calculate the train-induced dynamic response of the system based on the integration of the substructure method and the semi-analytical approach. The modelling of each subsystem was presented, including the multi-rigid model of the train subsystem, the beam model of the track subsystem and the semi-analytical model of the tunnel-soil subsystem. The wheel-rail contact theory, the fastener force model and the displacement compatibility conditions were used to determine the dynamic interactive relationship between the subsystems. An explicit integral approach was used in the proposed model to analyse the complex system in the time domain which was capable of modelling highly nonlinear loading behaviours of the train-track-tunnel-soil system. The proposed model was validated by comparing with the vehicle-track coupled dynamics software VICT, the Pipe-in-Pipe model and the measured data from the onsite test in the tunnel of Ningbo Metro line 2. An illustrative example of the model was then presented to display the dynamic responses of the tunnel system under the passage of a metro train.

Simulation of vertical dynamic vehicle–track interaction using a two-dimensional slab track model

ABSTRACTThe vertical dynamic interaction between a railway vehicle and a slab track is simulated in the time domain using an extended state-space vector approach in combination with a complex-valued modal superposition technique for the linear, time-invariant and two-dimensional track model. Wheel–rail contact forces, bending moments in the concrete panel and load distributions on the supporting foundation are evaluated. Two generic slab track models including one or two layers of concrete slabs are presented. The upper layer containing the discrete slab panels is described by decoupled beams of finite length, while the lower layer is a continuous beam. Both the rail and concrete layers are modelled using Rayleigh–Timoshenko beam theory. Rail receptances for the two slab track models are compared with the receptance of a traditional ballasted track. The described procedure is demonstrated by two application examples involving: (i) the periodic response due to the rail seat passing frequency as influenced by the vehicle speed and a foundation stiffness gradient and (ii) the transient response due to a local rail irregularity (dipped welded joint).