Vertical Vibration Analysis Of Train-track/bridge System

Our country keeps on raising the speed of the trains.With the increase of the train speed,the dynamic impacts on the track structure or the bridge structure increase as well as the dynamic responses of the trains.Though the vertical vibration of the train-track/bridge system has been investigated in the extensive literature,the following issues remain such as the formulae calculating the shear force and the bending moment at any cross-section of a rail,the finite element analysis of the dynamic responses of the high-speed train-slab track-bridge system,the safety for the wheel-rail separation,and the stress/deflection impact factor of the bridge.Aiming at these issues,the following series of studies have been conducted and some conclusions drawn.1.Based on the equilibrium condition of force of beam element,some finite-element formulae are presented to calculate the shear force and the bending moment at any cross-section of a rail on either continuously or discretely viscoelastic foundation subjected to a moving train.Furthermore,the proposed formulae can be easily degenerated into either formulae calculating the shear force and the bending moment of a simply supported or a continuous Bernoulli-Euler beam subjected to a moving train,or formulae calculating the shear force and the bending moment of a Bernoulli-Euler beam on Winlder foundation under static loads.The correctness of the proposed formulae has been verified by the analytical solutions to some numerical examples.The solutions obtained by the proposed formulae have been compared with those calculated directly from the derivatives of the element displacements,and the numerical results have shown that the proposed formulae not only raise the efficiency but also improve the accuracy of calculation.2.A train,the ballast track on each approach embankment of bridges,the slab track on bridges,and a series of bridges have been considered as an entire system,in which the rail,the slabs and the bridges have been divided into a finite number of beam elements,respectively.The equation of motion for this system has been formulated by means of the principle of total potential energy with stationary value in elastic system dynamics and the "set-in-right-position" rule for formulating matrices, with both the principle and the rule being presented by academician Qingyuan Zeng. The effects of the two types of definite track irregularities,that is,the irregularity caused by the invalid rail fastenings and the single harmonic irregularity on track surface,on the dynamic responses of this system have been investigated.The dynamic responses of the trains,the slab tracks and the bridges caused by a high-speed train passing through the track with profile random irregularities have also been studied.The numerical results have shown that(ⅰ)in order to guarantee the safe running of high-speed trains or that the rails being acted on by sustainable forces, the fastenings must be kept in normal working condition,and the maximum depth of track unevenness has to be kept within limits;(ⅱ)the stress impact factors,not the deflection impact factors should be applied in the design of simply-supported beam bridges;(ⅲ)the parameters for the 24m-span bridge beams and for the tracks on the Qinhuangdao-Shenyang passenger transport line can be applied in the high-speed railway design to guarantee the normal running of high-speed trains.3.Based on the rigid wheel-rail contact model,the equation of motion of a two-axle vehicle-bridge system with one or two wheelsets of vehicle separating from the bridge has been derived.The whole process of the wheel-rail contact,separation, and re-contact has been simulated.The numerical results have shown that:when the wheel separates from the rail,on the condition of equal wavelength l_αof the track irregularity,the bigger the maximum depth of track unevenness is and the higher the train speed becomes,the higher the wheel will separate from the rail,the bigger the wheel-track impact force,the longer the duration of the second wheel-rail separation, and the shorter the duration between the two separations.4.A matrix equation is derived for the vertical motion of a system that is composed of a simply supported through truss bridge subjected to a moving train consisting of a locomotive with two-stage suspension and a set of freight cars with central single-stage suspension.Based on the spectral density power function of the random profile irregularity on the three main railway lines in China,both the deflection impact factors at the nodes of the down chord member and the stress impact factors of each member of a simply supported 64m-span through steel truss bridge subjected to a speed-raising freight train or passenger train are calculated, respectively.The numerical results have shown that:(ⅰ)for a bridge under a speed-raising passenger train at a constant speed of 160,180 or 200 km/h,its deflection impact factors and stress impact factors are smaller than those corresponding factors for a bridge under a speed-raising freight train at a constant speed of 60,70,80 or 90 km/h;(ⅱ)in the truss bridge design,the stress impact factor,not the deflection impact factor,should be used.5.Two types of equations of motion,one based on the finite element method and the other on the modal analysis method,for the vertical vibration of the train,track and bridge interaction system are derived with two-layer track model.The numerical results obtained by the finite element method are compared with those by the modal analysis method.The numerical results have shown:(ⅰ)the finite element method can yield results of high accuracy for the responses of either the rail or the bridge;(ⅱ)the modal analysis method can yield results of high accuracy for the responses of the bridge;(ⅲ)when the same degrees of freedom of rail have been adopted by the two methods,compared with the finite element method,the modal analysis method yields the results for the responses of the rail not only with lower efficiency but with lower accuracy,in particular and the results for the bending moment of the rail are not accurate enough.