Vehicle-bridge System Research Articles

Research on Simplified Calculation Method of Coupled Vibration of Vehicle-Bridge System

A modified moving loads’ model is proposed for the vehicle-bridge coupling vibration simulation. Taking the vehicle-bridge interaction model (VBI) as the reference, the accuracy and applicability of the three calculation models, namely, moving loads’ model, moving mass model, and spring-damper-mass model, are compared using the frequently-used railway simply-supported beam with a span of 32 meters as the research object. Influencing factors such as vehicle speed, mass ratio of vehicle and beam, and primary spring stiffness on the dynamic response of the vehicle-bridge system are discussed in detail. The results show that the moving load model has the best performance on the stability of the deviation rate, but its calculation results are smaller than the other two methods as well as the VBI. The values of the deviation rate for the moving mass model and the spring-damper-mass model are large, and the stability of those are insufficient in the range of 80%∼120% of the first resonance velocity. Except for that, the results of the two models are in good agreement with the VBI model. According to above analysis, a modified moving loads’ model with two amplification coefficients, namely, 1.10 for the range of 90%∼105% of the first resonance velocity and 1.05 for other velocities, are proposed, which has higher calculation efficiency and accuracy.

Ill-Posedness Determination of Moving Force Identification and Parameters Selection for Regularization Methods

Moving force identification (MFI) from dynamic responses of bridges is a typical inverse problem with ill-posedness. Under the efforts of researchers, some regularization methods have been presented to solve the ill-posed problem, but there still lacks an effective index to reveal the ill-posedness of the vehicle–bridge dynamic system such that it can be utilized as a guidance for the regularization parameter selection. In this paper, an ill-posedness indicator (IPI) defined as the ratio of the Fourier coefficient to the singular value is adopted to reveal the ill-posedness in the MFI problem. Simulation results show that the larger the IPI value is, the more obvious the ill-posedness of the vehicle–bridge system equation, namely, the intrinsic factor of ill-posedness in MFI is attributed to very large IPI value. The maximum IPI value increases with the increasing noise level, which leads directly to the ill-posedness of the vehicle–bridge system equation. In addition, a relative percentage error (RPE) is used to select the optimal regularization parameters, while evaluating the ill-posedness existing in the MFI. Using the proposed IPI value, the influence of ill-posedness on identified results is evaluated in this study, which can assist qualitatively and quantitatively in selecting optimal regularization parameters and proper regularization methods.

A Dynamic Partitioning Method to solve the vehicle-bridge interaction problem

This paper presents a Dynamic Partitioning Method (DPM) to solve the vehicle-bridge interaction (VBI) problem via a set of exclusively second-order ordinary differential equations (ODEs). The partitioning of the coupled VBI problem follows a localized Lagrange multipliers approach that introduces auxiliary contact bodies between the vehicle’s wheels and the sustaining bridge. The introduction of contact bodies, instead of merely static points, allows the assignment of proper mass, damping and stiffness properties to the involved constrains. These properties are estimated in a systematic manner, based on a consistent application of Newton’s law of motion to mechanical systems subjected to bilateral constraints. In turn, this leads to a dynamic representation of motion constraints and associated Lagrange multipliers. Subsequently, both equations of motion and constraint equations yield a set of ODEs. This ODE formulation avoids constraint drifts and instabilities associated with differential–algebraic equations, typically adopted to solve constrained mechanical problems. Numerical applications show that, when combined with appropriate numerical analysis schemes, DPM can considerably decrease the computational cost of the analysis, especially for large vehicle-bridge systems. Thus, compared to existing methods to treat VBI, DPM is both accurate and cost-efficient.

Interaction Dynamic Response of a High-Speed Train Moving Over Curved Bridges with Deficient or Surplus Superelevation

This paper proposes a three-dimensional dynamic model for high-speed railway trains moving over curved bridges considering the transition curves, circular curves, and superelevation. Key features of this study are to consider the nonlinear geometrical relationships and creep relationships between the wheels and rail, for which the interactive iterative numerical algorithms are developed based on the equations of vertical displacement and rolling of wheelset, and the torsional resonance conditions of the vehicle–bridge system are verified. The results show that the torsional vibration will cause amplification on vertical dynamic response of the beam on the outside edge of the curve. The deficient/surplus superelevation plays an important role in the lateral and torsional angular displacements of the bridge, and the peak of the torsional resonance response can be reduced by adjusting the practical superelevation of the curve. The variations of wheel–load reduction rate and derailment coefficient in the curve section are positively correlated to the deficient/surplus superelevation. The curve radius is the key factor affecting the wear and fatigue of wheel–rail, and when the curve radius is greater than 7000 m, the wear and fatigue can be significantly reduced. Running at a deficient superelevation level can also reduce the wear and fatigue.

Time-varying characteristics analysis of vehicle-bridge interaction system based on modified S-transform reassignment technique

The health monitoring of bridge under moving vehicles demands a high-efficient time-varying characteristic analysis method to extract the frequency fluctuations caused by vehicle-bridge dynamic interaction effect. Characteristics identification of structures becomes a challenging task especially when time-varying vibration modes are involved. For this purpose, an instantaneous frequency identification technique based on modified S-transform reassignment is presented in this paper. Firstly, the general approximation between structural instantaneous frequency and S-transform is deduced. The Gaussian window is employed in S-transform and a novel first-order frequency function with two parameters is added into the Gaussian window function to improve the time–frequency resolution. The added function makes the Gaussian window scalable-sliding and the optimal values of the parameters are selected based on time–frequency concentration criterion. Considering the Rényi entropy as the measurement of criterion, the information performance extracted from S-transform spectrogram is evaluated. Then the time–frequency reassignment technique is applied to improve the readability of S-transform spectrogram to assure the accuracy of the extracted instantaneous characteristics, which enhances the time–frequency concentration efficiently. The proposed method is verified by numerical simulation, the results obtained by the spectrograms derived from short-time Fourier transform, wavelet transform, modified S-transform and reassigned modified S-transform are compared. Afterwards, the time-varying frequency components related to the vehicle-bridge interaction are extracted from the response signal. Numerical simulations involving different parameter analysis and laboratory experiments are conducted to validate the flexibility of the technique in vehicle-bridge system. The effect of vehicle weight, speed and road surface roughness on the identified results are discussed. Both numerical and experimental results have demonstrated that a higher time–frequency concentration in the spectrogram can be observed of the proposed modified S-transform reassignment method than other time–frequency methods, and subsequently a higher accuracy of instantaneous frequency identification is achieved which provides a good tool for time-varying characteristic analysis in bridge health monitoring.

Mechanism Study of Vehicle–Bridge Dynamic Interaction under Near-Fault Ground Motions

The effect of vehicle on the seismic responses of highway bridge has usually been neglected under the assumption that the possibility of vehicles crossing highway bridge or viaduct with a concurrent earthquake is extremely low. Furthermore, there is no in phase movement of vehicle and bridge was observed in past earthquake events. However, in the heavily trafficked urban area this is probably not the case. The mechanism of vehicle–bridge dynamic interaction still remains unclear. Therefore, an analytical vehicle–bridge model with contact elements is built to study the dynamic interaction and the seismic responses of vehicle–bridge system when subjected to near-fault ground motions. In this vehicle–bridge model, vehicle and highway bridge are coupled through Coulomb friction forces in the horizontal directions and through Hertz impact force in the vertical direction, respectively. The horizontal Coulomb friction forces varying with the variation of vertical Hertz impact force under earthquake excitations. Analysis results reveal that vehicle sometimes has a significant influence on the vertical seismic displacement response of bridge and mass ratio of vehicle to bridge is one of the primary factors governing the effect of vehicle on the seismic responses of bridge. Influence of vehicle frequency and damping ratio on the seismic responses of bridge are both related to mass ratio and become significant with the increasing of mass ratio. However, for the responses of the vehicle itself, influence of vehicle frequency and damping ratio is significant only when the mass ratio is small. Due to the weak coupling of vehicle and bridge in the horizontal directions, horizontal displacements of the vehicle can reach several meters, hence possibly endanger the traveling safety of vehicle on highway bridge.

A model for investigating vehicle-bridge interaction under high moving speed

The speed of rail vehicles become higher and higher over two decades, and China has unveiled a prototype highspeed train in October 2020 that has been able to reach 400 km/h. At such high speeds, wheel-rail force items that had previously been ignored in common computational model should be reevaluated and reconsidered. Aiming at this problem, a new model for investigating the vehicle-bridge interaction at high moving speed is proposed. Comparing with the common model, the new model was more accurate and applicable, because it additionally considers the second-order pseudo-inertia forces effect and its modeling equilibrium position was based on the initial deformed curve of bridge, which could include the influences of temperature, pre-camber, shrinkage and creep deformation, and pier uneven settlement, etc. Taking 5 km/h as the speed interval, the dynamic responses of the classical vehicle-bridge system in the speed range of 5 km/h to 400 km/h are studied. The results show that ignoring the second-order pseudo-inertia force will underestimate the dynamic response of vehicle-bridge system and make the high-speed railway bridge structure design unsafe.

Damage localization and quantification for beam bridges based on frequency variation of parked vehicle-bridge systems

Frequency variations are normally employed for damage detection as frequencies possess the advantages of the high accuracy and high anti-noise capability. However, the validity and applicability of such methods are restricted due to the limited order of frequencies. A damage localization and quantification approach is proposed for beam bridges by using the frequency variation of parked vehicle-bridge systems. Firstly, the phenomenon of frequency variation of parked vehicle-bridge systems is briefly illustrated. Secondly, a damage location index (DLI) is defined by the frequency change rate (FCR) of the vehicle-bridge systems with the vehicle parked at different positions before and after damage. Then the relationship between the frequency-parameter change rate (FPCR) and damage severity is established via finite element model (FEM) and further employed to formulate a damage equation(s) for severity estimation. The dimension of coefficient matrix in damage equations is reduced to large extent as the damage locations are determined by DLI in advance, which is meaningful in enhancing the effectiveness of severity estimation. Results of numerical and experimental examples with different damage scenarios indicate that the proposed approach can localize and quantify damages with high accuracy and efficiency by using frequency information only. The effects of measurement noise and FEM error are discussed to examine the robustness of the proposed approach. The cases when damage occurs in neighbor elements are analyzed to investigate the feasibility and robustness of the proposed approach.

Dynamic interaction analysis of suspended monorail vehicle and bridge subject to crosswinds

This paper develops a fully coupled wind-vehicle-bridge (WVB) interaction model to evaluate the dynamic performance and riding comfort of the suspended monorail vehicle-bridge system subject to turbulent crosswinds. The dynamic interactions between the vehicle and the bridge, the wind and the bridge, and the wind and the vehicle are considered in the model. The suspended monorail vehicle is modeled as a mass-spring-damper system and the motion equations of the multi-body system are derived. Through generating nonuniformity wind velocity fields consisting of various wind average velocities, an accurate approach to simulate the actual aerodynamic wind forces acting on the bridge and on moving vehicles is proposed. Further, a multi-span bridge is chosen as an example to illustrate the dynamic behaviors of the proposed WVB system. Results indicate that the turbulent crosswinds could have a great influence on the dynamic performance of the suspended monorail vehicle-bridge system, and it should be taken into account in the simulation of aerodynamic wind forces to guarantee the accurateness of vehicle vibration analysis. Besides, passengers could experience uncomfortable when vehicles pass through the bridge under strong turbulent crosswinds.

A Non-Iterative Integration Scheme Enriching the Solution to the Coupled Maglev Vehicle–Bridge System

A non-iterative integration scheme is presented in this study to enrich the solutions to the coupled equations of the maglev vehicle–bridge system. The proposed integration scheme is composed of two integration methods aiming at providing the solutions to equation of motion and state-space equation. First, the equation of motion of the simply supported girder bridge is transformed by the modal superposition method. Then the state-space equation is used to describe the motions of both the vehicle and the suspension control system, with the associated matrices assembled using the fully computerized approach. By adopting this integration scheme, only pure vector calculations arise in the solution process, regardless of the existence of time-dependent displacement and velocity on the right-hand sides of the two coupled equations. The proposed integration method is of the second-order accuracy with and without damping. Being equipped with adequate numerical dissipation and dispersion, the method also possesses the characteristic of little computing errors, as can be achieved through the use of different pairs of parameters. Finally, numerical simulations have been conducted to assess the influence of different feedback gains, three types of bridges with different lengths, and guideway irregularity on the maglev vehicle–bridge system.

Efficient non-stationary random vibration analysis of vehicle-bridge system based on an improved explicit time-domain method

Based on the recently developed explicit time-domain method (ETDM), this paper presents a new efficient computational framework for non-stationary random vibration analysis of the vehicle-bridge system (VBS) subjected to random track irregularity excitation. In this framework, the time-varying VBS equation is discretized using the time-stepping integration method, leading to an explicit expression for the stochastic system response in terms of the track irregularity excitation. Based on this expression, the response moments of the VBS can be easily evaluated in a direct way; the response probability density can also be obtained by using the Gaussian characteristic of the irregularity excitation. To further improve the computational efficiency, an efficient spectral representation-based formulation is proposed to calculate the excitation covariance matrix required in the analysis, which avoids tedious Fourier integrals required in the traditional ETDM. In the numerical application, the accuracy and efficiency of the proposed method are validated by the solutions obtained using the Monte Carlo and pseudo-excitation methods; the effects of several parameters on the stochastic responses of the VBS are investigated.

Theoretical and experimental study on vibration modal for simply supported bridge

For the highway bridges with small or medium span, a vehicle will stay on the bridge after the excitation. So the measured values of the vibration characteristics of the bridge are related to the vehiclebridge system. In order to analyse the natural vibration characteristics of simply supported bridge and the loaded frequency of vehicle-bridge system, theoretical analysis was carried out firstly, then the simply supported beam calculation model was built based on finite element method, and the results were compared with the measured values. The finite element model of a mass block-simply supported beam system was built by regarding the vehicles as a concentrated mass block attached. Both the vibration modals of the bridge system and the loaded frequencies of the vehicle-bridge system under diffident vehicle positions were obtained by software analysing. As a result, the variation characteristics of the loaded frequency for this simply supported bridge were extracted and discussed through the comparison of the results from both finite element method and the measured.

Simultaneous identification of the frequencies and track irregularities of high-speed railway bridges from vehicle vibration data

Bridge frequencies and track irregularities are both the focuses of railway bridge condition assessment, which are coupled with each other in a vehicle-bridge system. Available algorithms face a great challenge when applied to simultaneously identify the natural frequencies and track irregularities of railway bridges using on-board measurement data. This paper proposes a novel algorithm for simultaneously identifying the frequencies and track irregularities of high-speed railway bridges using vehicle dynamic responses for the first time. An extended state-space model with unknown input condensation is established for time-dependent vehicle-bridge systems. We subsequently propose a new extended Kalman filter algorithm with an adaptive procedure for accelerating the convergence of estimation, which can simultaneously identify the frequencies and track irregularities of a railway bridge when a vehicle is running on it. The effectiveness of the proposed algorithm has been illustrated via numerical simulations of two real high-speed railway bridges. The proposed algorithm provides a low-cost and high-efficient approach for identifying the natural frequencies and track irregularities of high-speed railway bridges.

Impact coefficient analysis of track beams due to moving suspended monorail vehicles

To study the impact coefficient of track beams due to moving suspended monorail vehicles under the bridge deck, this paper presents a vehicle-bridge dynamic model based on the rigid-flexible coupling method. Finite element method is used to establish the bridge model, and the vehicle is simulated as a multi-body dynamic model The mechanical characteristics of running tire are precisely modelled and the dynamic equations for the suspended monorail vehicle-bridge coupling system are derived. The proposed approach employs a double-line 7-span bridge with 210 m length as a numerical example, the impact performance of two track beams are studied and the results are verified by comparing with the theoretical solutions. The influences of critical system parameters, such as the train formation and double-line trains loads, on the dynamic responses of vehicles and the bridge are explored. The results show that it is feasible to study the impact effect with the proposed method. The separation phenomenon of rubber running tires jumping up on track surface occurs when vehicle speed is high, which may lead to negative impact coefficients. The impact coefficients of the track beam vary much with different locations, which can even be negative values at locations nearing the pier area.

Effects of wind-barrier parameters on dynamic responses of wind-road vehicle–bridge system

Accidents, such as deflections, side slips, and even rolls, can easily occur when road vehicles moving on the bridge deck at high speed are subjected to sudden changes owing to strong crosswinds. Wind barriers are an effective way to ensure the driving safety of vehicles on the bridge but with potential negative influences on the aerodynamic performance. Therefore, it is critical to evaluate the shielding effect of wind barriers on road vehicles by assessing the overall impact on the vehicle–bridge system. To optimize the wind-barrier scheme, a series of wind tunnel tests is conducted to evaluate the effects of barrier parameters on the aerodynamic characteristics of the wind-vehicle-bridge system. In this study, the aerodynamic force coefficients of the wind–vehicle–bridge system installed with wind barriers with four different heights and three ventilation ratios were investigated. With wind barriers installed on the bridge deck, dynamic responses of the wind–vehicle–bridge system were simulated with the proposed prediction procedures as those used in an earlier study. Results show that wind barriers on the bridge deck have significant influences on the aerodynamic force coefficients of the bridge, especially on the side-force coefficients, and on the dynamic responses of the wind–vehicle–bridge system. Additionally, the increase of the barrier height and the decrease of the barrier ventilation ratio considerably increase and decrease the lateral and torsional displacements of the bridge and the vehicles, respectively, but with only minor influence on the vertical displacements of both bridges and vehicles. Therefore, it is necessary to include the wind barrier in both experimental and numerical simulations of the wind-road vehicle-bridge system to explore the effect of different barrier parameters on the wind-vehicle-bridge system.

Influence of bridge parameters on monorail vehicle–bridge system— A research with multi-rigid body and multi-flexible body coupling theory and Park method

The parameters of the monorail bridge, which have a significant effect on the vehicle, are analyzed in this paper. With Park method, a complicated solver of the second order with a variable step size, equations insolvable to traditional methods can be solved and a more complicated dynamic model could be built. Different from some traditional simulation of vehicle–bridge coupling system, the bridge is divided into the girder and piers. The finite element model of each part is created separately in Ansys and coupled with the vehicle system in UM, which helps the analysis of the interaction forces between flexible components and better simulation accuracy can be obtained. FIALA model is used to calculate the wheel force. These methods make the measurement more precise. The study shows that all the variables except for the stiffener thickness have little effect on the vibration of the vehicle due to the good vibration reduction performance of the vehicle system. The resonance of the girder could be avoided by changing the thickness of the stiffeners and web plates. In addition, high stiffness and damping of the connection point of the girder and pier are helpful in alleviating the vibration of the bridge.

Fatigue life estimation of continuous girder bridges based on the sequence of loading

A need exists to determine the fatigue life of multi-span continuous bridges in the design stage. It is difficult to conduct a full-scale fatigue test of a bridge, hence the simulation of bridge-vehicle interaction dynamics is important to estimate the fatigue life of bridges. The vehicles entering the bridge are random in nature. In the present study, the arrival time of the vehicles is considered a random variable following Poisson process to simulate the actual traffic. The bridge-vehicle coupled system equations have been derived wherein the vehicle arrival rate appears as random variable casting difficulty in obtaining a direct solution. The solution approach has been devised using Orthogonal Polynomial Expansion method. Response statistics so obtained for a multi-span continuous bridge has been used in fatigue life estimation involving crack initiation phase and crack propagation phase modifying conventional linear damage rule. The sequence of loading is also considered in the modified fatigue formulation. The effect of the vehicle speed, road surface irregularity and the arrival rate of the vehicle on fatigue life estimation of the bridge have been studied.

A Modified Bridge System method to characterize and decouple vehicle–bridge interaction

This paper examines the effect of traversing vehicles on the vibration of the supporting bridge. It shows that the effect of vehicle–bridge interaction (VBI) on the mechanical system of the bridge breaks down into three constituent mechanisms: (a) an additional damping, (b) an additional stiffness, and (c) a modified loading term. The study characterizes these three mechanisms with explicit analytical expressions in the case of a single-degree of freedom (SDOF) vehicle–SDOF bridge system. The analysis reveals that the impedance ratio—defined as the ratio of the vehicle’s damping to the bridge’s mechanical impedance—is a dominant coupling parameter of VBI, in addition to the (well-known) vehicle-to-bridge stiffness ratio. Subsequently, the study proposes an original approach, namely the Modified Bridge System (MBS) method, to decouple the VBI problem when the stiffness ratio is small. The MBS approach solves a modified bridge system, independent of the traversing vehicle. Numerical applications on simply supported railway bridges show that the MBS method outperforms other decoupling approaches, such as the moving load approximation. At the same time, compared to the solution of the coupled vehicle–bridge system, the proposed MBS method is equally accurate but computationally more efficient. The paper also outlines limitations of the proposed approach related to the single-mode representation of both vehicle and bridge.

Fault Diagnosis for a PDE Suspended-Bridge Model With Kalman Filter and Statistical Decision Making

This article proposes a fault diagnosis method for the partial differential equation (PDE) model that describes the interaction between a suspended-bridge and a moving vehicle. Using semidiscretization and the finite differences approach, and considering a grid of N equidistant points on the bridge's longitudinal axis, the PDE that describes the spatiotemporal dynamics of the bridge is decomposed into a set of N ordinary differential equations (ODEs). Thus, a state-space model of the bridge and vehicle interaction is obtained comprising N+2 ordinary differential equations. Equivalently, one can obtain a matrix description of this state-space model of dimension 2N+4. Moreover, by denoting transformed control inputs, this state-space model is finally written in the canonical Brunovsky form which is known to be observable. For the latter state-space representation of the bridge-vehicle system in the canonical form, state estimation is performed with the use of Kalman filtering. Next, by comparing the outputs of the Kalman Filter with the real-time outputs of the bridge-vehicle system, the residuals sequence is generated. It is demonstrated that the sum of the squares of the residuals vectors weighted by the inverse of their covariance matrix stands for a stochastic variable, which follows the χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> distribution. Thus, by exploiting the 96% or the 98% confidence intervals of the χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> distribution one can define fault thresholds which show in an almost infallible manner, whether the bridge retains its structural integrity or whether it has been subjected to a fault.

Random Characteristics of Vehicle-Bridge System Vibration by an Optimized Pseudo Excitation Method

The pseudo excitation method (PEM) is improved for its efficiency by incorporating the self-adaptive Gauss integration (SGI) technology as a new combining integration. The PEM can transform the random rail irregularities into some pseudo harmonic excitation, which is a mature approach to deal with the random excitation for vehicle–bridge systems. The SGI was used to distinguish the significant from the insignificant parts of an integral section for the random excitation frequency on the stochastic response of the system, thereby reducing the computational effort required for the random vibration analysis of the system. Also, the SGI can intelligently handle the recognized integral section, by subdividing the important sections into several necessary frequency points, making rough decomposition, and allowing the unimportant regions to be eliminated. Based on selected frequency points, the deterministic pseudo harmonic excitations were generated, and then the standard deviation (SD) of the time history for the system was calculated by the PEM. The vehicle subsystem was simulated as a 23-degree of freedom model, and the bridge subsystem as a three-dimensional finite element model. The time-varying power spectral density (PSD) plots of the system were presented. Besides, the cumulative distribution function (CDF) of the response was calculated using Poisson’s crossing assumption. The random characteristics for the vehicle–bridge vibrations for different speeds and rail irregularities were calculated.