列车—轨道（桥梁）系统竖向振动分析
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摘要
我国列车不断提速。随着列车速度的不断提高,轨道、桥梁结构受到的动力冲击作用越来越大,列车自身的动力响应也在增大。国内外对列车—轨道(桥梁)竖向振动的研究很多,但在下列方面仍存在一些问题:例如:钢轨截面剪力和弯矩的计算公式,高速列车—板式轨道—桥梁系统动力响应的有限元分析,车轮跳离钢轨的安全性,桥梁的挠度和应力冲击系数等。本文针对上述问题,开展了一系列的工作,取得了如下成果:
1.首次基于单元动力的平衡条件,提出了计算移动列车作用下连续或离散粘弹性基础钢轨任意截面剪力和弯矩的有限元公式。所提出的公式不仅能退化为计算移动列车作用下简支或连续Bernoulli-Euler梁任意截面剪力和弯矩的有限元公式,也能退化为计算静止荷载作用下Winkler基础Bernoulli-Euler梁任意截面剪力和弯矩的有限元公式。用几个算例验证了所提出公式的正确性,并与使用直接对位移求导方法的计算结果进行了对比。数值结果表明:与直接对位移求导的方法相比,运用本文所提出的有限元公式可以显著提高计算结果的精度及计算效率。
2.视列车、桥梁两端有碴轨道、桥上板式轨道以及多座桥梁为一个整体系统,将轨道和桥梁用有限元离散,利用曾庆元院士提出的弹性系统动力学总势能不变值原理及形成矩阵的“对号入座”法则,首次建立了有限元形式的系统振动矩阵方程。分析了两种确定性轨道不平顺(钢轨扣件失效不平顺、钢轨表面存在单一谐波型不平顺)对系统动力响应的影响;研究了轨道高低随机不平顺激扰下高速列车与桥上板式轨道系统的动力响应。数值结果表明:(1)为了高速列车安全运行及轨道处于良好的受力状态,应保证扣件处于正常的工作状态,严格限制短波长不平顺的波深;(2)简支梁桥设计宜用应力冲击系数,而不宜用挠度冲击系数;(3)秦沈客运专线24m跨度桥梁以及轨道参数能保证高速列车正常运行,故能用于高速铁路设计之中。
3.基于轮轨刚性接触模型,首次导出了考虑轮对跳离钢轨的2轴车辆—桥梁振动矩阵方程,数值模拟了车轮从正常接触钢轨、跳离钢轨到再次接触钢轨的动态过程。数值结果显示:当车轮跳离钢轨时,对于同样的不平顺波长l_a,不平顺波深(?)越大、车速越高,车轮跳离得越高,轮轨冲击力越大,第2次轮轨分离的持续时间越长,轮轨两次分离之间的时间间隔越短。
4.将下承式简支桁架桥、桥上的双层悬挂机车及其牵引的多辆具有一系中央悬挂的货车视为一个系统,首次导出了该系统的竖向振动矩阵方程。利用我国三大干线轨道高低随机不平顺功率谱密度函数,计算了提速货物列车和旅客列车通过64m跨度下承式简支钢桁架桥时下弦杆节点的挠度冲击系数和每根杆件的应力冲击系数。数值结果显示:(1)提速旅客列车以速度160、180、200 km/h运行时的桥梁挠度冲击系数和杆件应力冲击系数均小于货物列车以速度60、70、80、90 km/h运行时相应的冲击系数;(2)在桁梁设计中宜用应力冲击系数,而不宜用挠度冲击系数。
5.分别导出了有限元法和模态分析法的双层轨道结构模型的列车—有碴轨道—桥梁系统竖向振动的矩阵方程。对比了两种方法的计算结果和计算效率,数值结果表明:(1)用有限元法计算钢轨响应和桥梁响应,均能获得较高精度的计算结果;(2)用模态分析法计算桥梁响应时,可以获得较高精度的计算结果;(3)当采用相同的自由度计算钢轨响应时,与有限元法相比,模态分析法的计算精度低(特别是弯矩的计算精度低),并且计算效率低。
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.
1.首次基于单元动力的平衡条件,提出了计算移动列车作用下连续或离散粘弹性基础钢轨任意截面剪力和弯矩的有限元公式。所提出的公式不仅能退化为计算移动列车作用下简支或连续Bernoulli-Euler梁任意截面剪力和弯矩的有限元公式,也能退化为计算静止荷载作用下Winkler基础Bernoulli-Euler梁任意截面剪力和弯矩的有限元公式。用几个算例验证了所提出公式的正确性,并与使用直接对位移求导方法的计算结果进行了对比。数值结果表明:与直接对位移求导的方法相比,运用本文所提出的有限元公式可以显著提高计算结果的精度及计算效率。
2.视列车、桥梁两端有碴轨道、桥上板式轨道以及多座桥梁为一个整体系统,将轨道和桥梁用有限元离散,利用曾庆元院士提出的弹性系统动力学总势能不变值原理及形成矩阵的“对号入座”法则,首次建立了有限元形式的系统振动矩阵方程。分析了两种确定性轨道不平顺(钢轨扣件失效不平顺、钢轨表面存在单一谐波型不平顺)对系统动力响应的影响;研究了轨道高低随机不平顺激扰下高速列车与桥上板式轨道系统的动力响应。数值结果表明:(1)为了高速列车安全运行及轨道处于良好的受力状态,应保证扣件处于正常的工作状态,严格限制短波长不平顺的波深;(2)简支梁桥设计宜用应力冲击系数,而不宜用挠度冲击系数;(3)秦沈客运专线24m跨度桥梁以及轨道参数能保证高速列车正常运行,故能用于高速铁路设计之中。
3.基于轮轨刚性接触模型,首次导出了考虑轮对跳离钢轨的2轴车辆—桥梁振动矩阵方程,数值模拟了车轮从正常接触钢轨、跳离钢轨到再次接触钢轨的动态过程。数值结果显示:当车轮跳离钢轨时,对于同样的不平顺波长l_a,不平顺波深(?)越大、车速越高,车轮跳离得越高,轮轨冲击力越大,第2次轮轨分离的持续时间越长,轮轨两次分离之间的时间间隔越短。
4.将下承式简支桁架桥、桥上的双层悬挂机车及其牵引的多辆具有一系中央悬挂的货车视为一个系统,首次导出了该系统的竖向振动矩阵方程。利用我国三大干线轨道高低随机不平顺功率谱密度函数,计算了提速货物列车和旅客列车通过64m跨度下承式简支钢桁架桥时下弦杆节点的挠度冲击系数和每根杆件的应力冲击系数。数值结果显示:(1)提速旅客列车以速度160、180、200 km/h运行时的桥梁挠度冲击系数和杆件应力冲击系数均小于货物列车以速度60、70、80、90 km/h运行时相应的冲击系数;(2)在桁梁设计中宜用应力冲击系数,而不宜用挠度冲击系数。
5.分别导出了有限元法和模态分析法的双层轨道结构模型的列车—有碴轨道—桥梁系统竖向振动的矩阵方程。对比了两种方法的计算结果和计算效率,数值结果表明:(1)用有限元法计算钢轨响应和桥梁响应,均能获得较高精度的计算结果;(2)用模态分析法计算桥梁响应时,可以获得较高精度的计算结果;(3)当采用相同的自由度计算钢轨响应时,与有限元法相比,模态分析法的计算精度低(特别是弯矩的计算精度低),并且计算效率低。
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.
引文
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