Stochastic Analysis on the Resonance of Railway Trains Moving over a Series of Simply Supported Bridges

Abstract

Resonance problems encountered in vehicle-bridge interaction (VBI) have attracted widespread concern over the past decades. Due to system random characteristics, the prediction of resonant speeds and responses will become more complicated. To this end, this study presents stochastic analysis on the resonance of railway trains moving over a series of simply supported bridges with consideration of the randomness of system parameters. A train-slab track-bridge (TSB) vertically coupled dynamics model is established following the basic principle of vehicle-track-coupled dynamics. The railway train is composed of multiple vehicles, and each of them is built by seven rigid parts assigned with a total of 10 degrees of freedom. The rail, track slab, and bridge are considered as Euler–Bernoulli beams, and the vibration equations of which are established by the modal superposition method (MSM). Except for the nonlinear wheel-rail interaction based on the Hertz contact theory, the other coupling relations between each subsystem are assumed to be linear elastic. The number theory method is employed to obtain the representative sample point sets of the random parameters, and the flow trajectories of probabilities for the TSB dynamics system are captured by a probability density evolution method (PDEM). Numerical results indicate that the maximum bridge and vehicle responses are mainly dominated by the primary train-induced resonant speed; the last vehicle of a train will be more seriously excited when the bridges are set in resonance by the train; the resonant speeds and responses are rather sensitive to the system randomness, and the possible maximum amplitudes predicted by the PDEM are significantly underestimated by the traditional deterministic method; optimized parameters of the TSB system are preliminary obtained based on the representative point sets and imposed screening conditions.

Simplification of earthquake induced vehicle bridge interaction contact simulation for urban viaducts

“…If compare Equation (8) and Equation (12), it is clearly seen that the vertical force F 2z changes from the 3/2 order of φ to the linear relationship of φ. This is one approach to simplify the contact model, and Equation ( 12) is named as modified piecewise tangent model herein.…”

“…A conventional VBI system can be described as a dynamic system where motor vehicle and bridge are interacted through contact element as shown in Figure 1. As for contact element, in the existing VBI research, the contact model can be normally divided into three kinds: (i) linear elastic spring model 5,6 without considering the separation of vehicle and bridge, which may lead to tension in the contact model in extreme cases, which is inconsistent with the actual situation; (ii) nonlinear Hertz model, 8,10–14 although it can simulate the nonlinear behavior like separation and collision of vehicles and bridges, it will increase the complexity of the coupling equation and reduce the efficiency of the solution; and (iii) piecewise model. For example, 16–18 when the contact element is in tension, the stiffness is 0, and only when the contact element is compressed, the stiffness is k t , which is used to simulate the separation and collision between vehicles and bridges.…”

“…(ii) nonlinear Hertz model, 8,[10][11][12][13][14] although it can simulate the nonlinear behavior like separation and collision of vehicles and bridges, it will increase the complexity of the coupling equation and reduce the efficiency of the solution; and (iii) piecewise model. For example, [16][17][18] when the contact element is in tension, the stiffness is 0, and only when the contact element is compressed, the stiffness is k t , which is used to simulate the separation and collision between vehicles and bridges.…”

“…In order to simulate the detachment between vehicle and bridge, the endpoint of φ in Equation ( 11) needs to be modified as shown in Equation (12):…”

“…In fact, Hertz model is often used in collision analysis between structures 9 and in contact simulation between wheel and rail of rail transit bridges. [10][11][12][13][14] Normally, the Hertz model links two objects, the contact force and displacement between the two objects are expressed as nonlinear functions related to material and geometric characteristics of contact. 15 Through its nonlinear stiffness, Hertz model can simulate the vertical nonlinear contact force of tire and analyze the detachment or collision between vehicle and bridge induced by earthquake excitation.…”