城市轨道交通环境振动源函数反演的论证研究
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摘要
作为国家自然科学基金重点项目“城市轨道交通引起的环境振动及传播规律”（50538030）的最后一部分研究成果的总结，本文在课题组取得突破进展的基础上，进一步论证一个尚待深入研究的阶段性结论，改善从轨道交通环境振动观测数据中去除本底振动的处理方法，探讨频率-波数谱的应用，论证项目的主要成果在表达振动衰减中局部放大的能力、探讨其机理，完善项目组发展的频率-波数域动力格林函数法计算列车-轨道-场地系统的三维动力响应、反演轮轨谱参数的方法。
城市轨道交通引起环境振动的卓越频带在10Hz-80Hz，近距离处主要在60Hz-80Hz。在文献[4]提出广泛应用的有限元方法计算这一高频振动在土层中传播有局限性的基础上，本文通过具体的算例论证时域-空间域方法模拟高频波传播的适用性。设定单覆盖层半空间模型表面作用2Hz、10Hz、30Hz和60Hz四种频率的单位简谐激励，采用波动有限元加透射边界计算1米、2米、5米和10米四种尺寸网格模型在距离振源5米-20米处的竖向振动位移幅值。通过与频率-波数域动力格林函数法的结果比较，指出了在许多条件下时间-空间域方法的计算误差比较明显；对于给定频率的激励，网格越小，有限元计算结果误差越小；对于给定的网格尺寸，频率越低，有限元计算结果误差越小；目前广泛采用的、基于一维离散网格模型分析稳定性研究提出的最大离散网格尺寸限定准则不能够确保三维计算结果在60Hz-80Hz频段的可靠性。归纳得到初步的定量结论为最大的离散网格不能超过感兴趣的最小波长的1/25。同时指出了计算的稳定性与离散的时间步长有关，对于一般的城市轨道交通环境振动分析，网格尺寸取1米时，时间步长需要取到0.0001秒，致使计算量很大，计算时间非常长，普通的个人计算机往往需要一周才能计算一个简单例子。
去除环境振动观测记录中的本底振动是提高信噪比，揭示轨道交通环境振动特征的一个重要环节。本文分析现有的从轨道交通环境振动观测记录中去除本底振动方法的不足，进一步考虑本底振动与观测振动互相关性，提出了一个去除本底振动的新方法——自互功率谱法，推导了全套计算公式。选定一组列车达到前足够久的观测记录和列车通过时的观测记录为本底振动和轨道交通“环境振动”，再从两者叠加得到的“观测记录”中去除本底振动，与原来选定的“环境振动”比较，通过实际算例论证了方法的可靠性。结果表明，本文提出的自互功率谱法能够更有效地去除观测数据中的本底干扰，获得更贴近实际的轨道交通环境振动的功率谱、时程、振动级VAL以及加权振级VLz。尤其是在本底振动占优势的低频段，比现有方法有明显的改善。
台阵观测的优势在于各观测点振动的自相关及互相关特性在一定程度上体现了城市轨道交通振动场的空间结构。为了探讨振动源对这一空间结构的影响，对一台阵6个点的环境振动观测数据进行总体功率谱分析及分时段功率谱分析，采用最大似然法计算了列车通过台阵一侧不同时段振动场的F-K谱，逐一时段分析了F-K谱的特征变化。据不同时段F-K谱的特征，分析指出了相应时刻列车相对台阵的不同方位（与列车在轨道上的位置一一对应），展现了借助台阵振动场的F-K谱判断列车运行过程的能力，并推算出列车运行的速度。结果表明，只要时段划分得足够小，就可以直接从上述F-K谱中提取轨道交通激励源的移动速度。
接着，本文采用频率-波数域动力格林函数计算地表激励下单一覆盖层-半空间表面的竖向位移，论证了表达轨道交通环境振动现场观测中多次发现的地面振动衰减局部放大现象的能力。通过七个半空间与覆盖层的模量比1、5、6、7、8、9和10，四个覆盖层厚度3、5、7和9米，7个激励频率16Hz、48Hz、56Hz、64Hz、72Hz、80Hz及108Hz，三个覆盖层阻尼比，0.025、0.05和0.1等诸多因素对局部放大影响的分析，指出了局部放大受埋藏的主要速度界面控制，局部放大区的位置和放大的强度与该界面两侧的模量比（波阻抗比）、覆盖层厚度、激励的频率及覆盖土层的阻尼比等因素有关。一般来说，速度界面的波阻抗越大，局部放大越强烈；给定其他条件，使直达的表面波与半空间表面折射波越接近同时到达的覆盖层厚度使局部放大越为明显，使两者波峰越靠近的激励频率会引起更为明显的局部放大；覆盖层阻尼比越小，局部放大越强烈。结合简单的波传播路径分析，给出了城市轨道交通环境振动衰减局部放大机理的一个解释，亦从一个侧面验证了本文采用的频率-波数域的动力格林函数方法计算轨道交通环境振动的可靠性。
为了适应地下水埋藏较浅场地环境振动分析的需要，基于饱和多孔介质动力分析的门福录模型，采用Haskell-Thomson传递矩阵法，在直角坐标系下推导了饱和层状场地频率-波数域动力格林函数。通过饱和土层参数取ρ2=0.0001kg/m~3，f=1，Ew=0.0001Pa，b=0.0001Pa·s/m~2，使其十分接近单相介质来退化，计算4种场地模型的地表竖向位移，与相应单相介质模型计算的结果比较，论证了本文推导的计算公式和编程的可靠性。计算了8个频率10Hz、20Hz、30Hz、40Hz、50Hz、60Hz、70Hz和80Hz单位简谐点源激励下单一饱和土覆盖层-半空间模型的地表振动竖向位移，与基于Biot模型计算方法的相应结果对比，论证了本文发展的饱和层状场地动力格林函数方法计算地下水埋藏较浅的场地上城市轨道交通环境振动的可行性，同时也验证了文献[5]基于Biot模型发展的计算方法是可靠的。门福录模型参数的物理意义明确，实际应用更为方便。
最后，借助项目组发展的频率-波数域计算轨道交通激励下列车-轨道-地基三维耦合系统地表反应的方法，探讨、论证了充分发挥台阵多次观测数据的优势反演激励源函数的可行性。通过虚拟反演，选取5个波长对应的PSD幅值表征激励源函数，与列车的速度及道砟路堤层无阻尼刚度一起共同构成待反演参数向量及其取值范围。用四个点振动的加速度级构造目标函数，采用早上班高峰期和接近中午平峰期的两次列车通过的观测记录数据，增加约束，反演了北京城轨13号线某一段的轮轨不平（圆）顺谱。与另一观测点实测数据对比，分析、验证了反演激励源函数的可靠性。完善了根据紧邻城市轨道交通线地表振动的台阵观测数据反演轮-轨不平（圆）顺谱的新方法。
总之，本文在城市轨道交通引起的环境振动及传播规律研究的六个方面得出了创新成果，包括时域计算方法的局限性、才观测记录中去除本底振动、从F-K谱中提取振动源的移动速度、轨道交通环境振动衰减局部放大的机理、基于门福录模型的饱和成层场地频率-波数域动力格林函数、根据多次观测数据联合反演轨道不平（圆）顺谱参数等，验证、完善并了项目组发展计算轨道交通激励下列车-轨道-地基三维耦合系统地表反应的频率-波数域动力格林函数方法，和根据紧邻城市轨道交通线地表振动的台阵观测数据反演作为系统激励源的轮-轨不平（圆）顺谱参数的创新思路。
As a summary of the last part research work of the Key Project by NationalNatural Science Foundation "Environment vibration caused by urban rail traffic andits propagation feature"(50538030), the dissertation demonstrates a mentionedconclusion which is needed to make deep research, improves the data processing toremove the background vibration from the observed environment vibration causedby urban rail traffic and discusses the application of frequency-wavenumberspectrum, demonstrates and perfects the development the dynamic Green Functionmethod in frequency-wavenumber domain to calculate3dimensional response oftrain-track-ground system and its capability to inverse the wheel-track spectrum, onthe basis of the breakthrough achieved by the whole team of the project.
Firstly, the feasibility of time-space domain method to simulate theenvironment vibration caused by urban rail traffic in frequency band10Hz to80Hz,in general, and60Hz to80Hz in near field, is demonstrated. With unit harmonicexcitation with frequencies2Hz,10Hz,30Hz and60Hz on surface of a threedimension model with single layer overlaid on half space, the vertical displacementamplitudes at the points with distance5meters-20meters from the exciting point arecalculated by wave finite element with transmission boundary conditions. For thedemonstration, the calculations are carried out repeatedly with four discrete gridsizes from1m,2m,5m to10m. From the comparison of the results with those by thedynamic Green Function method in frequency-wave number domain, the error incalculation in time-space domain is showed obviously. For a given excitingfrequency, the smaller the grid, the smaller error by the finite element calculation;for a given grid size, the lower the frequency, the smaller error the finite elementcalculation; the reliability of the three dimension finite element simulation of wavewith frequency in band60Hz-80Hz with grid size from stability study mainly byone-dimension analysis, is not ensured even the maximum size rule is now widelyadopted in two or three dimensional analysis. A preliminary quantitative rule forthree dimension analysis is that the maximum grind size should be less than1/25ofthe minimum interesting wavelength. Mean while, it is also pointed out that the calculation stability is also depends on the discrete step in time domain, themaximum step should be0.0001second if the discrete grid is taken as1m, forsimulation of common urban rail traffic caused environment vibration. That meansthe calculated amount must be very large, the time consumed must be quite long inthe case, so that it may take one week on a personal computer for simple casecalculation.
Removing background vibration noise in observed records of trafficenvironment vibration is an important step to increase the Signal to Noise Ratio anddemonstrate the characteristic of traffic environment vibration. The shortcomings ofexisting methods to remove the background vibration noise in observed records oftraffic environment vibration is analyzed, an idea to take into account thecross-relativity between background vibration and the observed vibration is putforward, the auto-cross spectrum method is developed as a new method to removebackground vibration and increase the ratio of signal to noise. A case fordemonstration is built as that the vibration recorded in time period long before thetrain passing is taken as background vibration, that recorded while the train ispassing as environment vibration, and that superposed by the two as observedvibration. An environment vibration is obtained by the background vibrationremoved from the observed vibration from the auto-cross spectrum method. Themethod is then verified by comparing the result vibration with the environmentvibration in this case, as the result shows that the background noise can removedmore effectively, the power spectrum, time history, vibration level VAL andweighted level VLz of the obtained environment vibration are all closer to those ofthe environment vibration.
The superiority of array observation is that the auto-correlation andcross-correlation characteristic of each point can reflect the spatial structure ofvibration field of urban rail traffic. In order to discuss the effect of source of urbanrail traffic on spatial structure of the vibration field, total power spectrum andpower spectra of some short time periods are calculated by maximum likelihoodmethod from environment vibrations recorded at6points on a rectangular arraywhile a train is passing it, and frequency-wave number (F-K in brief) spectra ofvibration fields in the short time periods are then calculated, the variation of the features of those power spectra in one by one time period is analyzed. Theorientation of the train to the array center (corresponding to the position of the trainon the rail) in each time period is determined from the corresponding F-K spectrum.The result shows the capability to infer the passing process of the train from the F-Kspectra of vibration field on array, and to compute the train speed. It demonstratesthat the moving speed of exciting source of traffic environment vibration couldacquired directly from the above F-K spectra, if the short time periods are taken asshort enough. Ground vertical displacements on model of single soil layer overlaidon half space with excitation on surface are calculated by means of the dynamicGreen Function method in frequency-wave number domain.
The capability of the method to deal with the local amplification in attenuationof ground vibration that phenomenon has been observed in some measurements ofenvironment vibration caused by rail traffic, is demonstrated. The results show thatlocal amplification is controlled by an important burried velocity interface, thelocation and intensity of the amplification depends on the ratio of elastic modului ofthe media in the both sides of the interface (i. e. impedance ratio), the thickness ofthe overburden layer, exciting frequency and damping ratio of the overburden layer,from analysis of influence on the amplification of factors as modulus ratio of halfspace to overburden layer media with seven values1,5,6,7,8,9and10,overburden layer thickness with four values3,5,7and9meters, exciting frequencywith seven values16Hz,48Hz,56Hz,64Hz,72Hz,80Hz and108Hz, overburdenlayer damping ratio with three values0.025,0.05and0.1. In general, the largerimpedance at the interface, the more obvious amplifying; keeping the otherconditions the same, the closer of directly arriving time of surface wave to thearriving time of refraction wave from the interface governed by the overburdenlayer thickness, the more obvious amplifying; the closer of the peaks of the wavesgoverned also by the exciting frequency, the more obvious amplifying; the smallerdamping ratio of the overburden layer, the more obvious amplifying. A mechanismof the local amplification in environment vibration attenuation is explained from theresults with a brief analysis of wave propagation paths. The reliability of calculationof rail traffic environment vibration by the dynamic Green Function method infrequency-wave number domain is verified by the way.
In order to fit the requirement to analyze the environment vibration at site withshallow groundwater, the dynamic Green Function in frequency-wave numberdomain on saturated layered ground is derived in rectangular Cartesian coordinatesystem based on the model suggested by Men FuLu for wave propagation insaturated porous medium, and by Haskell-Thomson transfer matrix method. Groundvertical displacements of four site models are calculated from a degeneration byreplacing the values of saturated soil parameters as ρ2=0.0001kg/m~3, f=1,Ew=0.0001Pa, b=0.0001Pa·s/m~2, so that the site models are very close to those insingle phase medium. The formulas and the program of the dissertation are verifiedby comparing the results with the published results of the corresponding singlephase medium models. Ground vertical displacements of model with singlesaturated layer overlaid on half space by unit harmonic excitation with eightfrequencies10Hz,20Hz,30Hz,40Hz,50Hz,60Hz,70Hz and80Hz are calculated.The feasibility of the developed dynamic Green Function method infrequency-wave number domain to simulate the vibration field by urban rail trafficat site with shallow ground water is demonstrated by comparing the results withthose published by Wang (2011) based on Biot model for the same models. Meantime, the reliability of the method developed by Wang is also demonstrated. Thephysical meaning of parameters in Men Fulu's model is unambiguous, thus themethod of this paper is more convenient in practice application.
Finally, the feasibility of inversing excitation source function with theadvantage in array data observed from several trains is tested and demonstrated bymeans of the method developed by project team to calculate vibration on threedimensional train-rail-ground coupled system in the frequency-wave numberdomain. PSD amplitudes corresponding five wavelengths are selected tocharacterize exciting source function, speed of the train and the un-damped stiffnessof ballast and embankment are selected to build inversion parameter vector with theabove five together, and the value ranges of the seven are estimated from a virtualinversion. The objective function is constructed by the vibration acceleration levelsof four points, and the observation data of the two trains during rush hour inmorning and close to the flat peak period at noon are used, and constraint hasincreased. The wheel/rail uneven (round) spectrum of Beijing No.13rail line is inversed. The reliability of the inversion excitation source function is analyzed andverified though comparison of calculation result used another point measured data.The new method to inverse wheel and rail uneven (round) spectrum of the urban railtraffic according to the surface of the vibration on array observation data close tothe rail is improved.
In conclusion, six innovation achievements on environment vibration causedby urban rail traffic and its propagation are worked out, such as limitations of timedomain calculation, the improvement to remove the background vibration noise, andacquiring of moving speed of the vibration source from F-K spectra, mechanism oflocal amplification in environment vibration attenuation, dynamic Green Functionin frequency-wave number domain in saturated layered ground based on Men FuLumodel and joint inversion of wheel-rail uneven (round) spectrum parameters fromarray data observed from several trains. The dynamic Green Function method tocalculate the response of three dimensional system of train-track-ground infrequency-wave number domain and the innovational idea to inverse wheel-railuneven (round) spectrum as the exciting source from data observed on array close tourban rail traffic according to the surface of the vibration on array observation datathe rail are demonstrated and completed.
城市轨道交通引起环境振动的卓越频带在10Hz-80Hz，近距离处主要在60Hz-80Hz。在文献[4]提出广泛应用的有限元方法计算这一高频振动在土层中传播有局限性的基础上，本文通过具体的算例论证时域-空间域方法模拟高频波传播的适用性。设定单覆盖层半空间模型表面作用2Hz、10Hz、30Hz和60Hz四种频率的单位简谐激励，采用波动有限元加透射边界计算1米、2米、5米和10米四种尺寸网格模型在距离振源5米-20米处的竖向振动位移幅值。通过与频率-波数域动力格林函数法的结果比较，指出了在许多条件下时间-空间域方法的计算误差比较明显；对于给定频率的激励，网格越小，有限元计算结果误差越小；对于给定的网格尺寸，频率越低，有限元计算结果误差越小；目前广泛采用的、基于一维离散网格模型分析稳定性研究提出的最大离散网格尺寸限定准则不能够确保三维计算结果在60Hz-80Hz频段的可靠性。归纳得到初步的定量结论为最大的离散网格不能超过感兴趣的最小波长的1/25。同时指出了计算的稳定性与离散的时间步长有关，对于一般的城市轨道交通环境振动分析，网格尺寸取1米时，时间步长需要取到0.0001秒，致使计算量很大，计算时间非常长，普通的个人计算机往往需要一周才能计算一个简单例子。
去除环境振动观测记录中的本底振动是提高信噪比，揭示轨道交通环境振动特征的一个重要环节。本文分析现有的从轨道交通环境振动观测记录中去除本底振动方法的不足，进一步考虑本底振动与观测振动互相关性，提出了一个去除本底振动的新方法——自互功率谱法，推导了全套计算公式。选定一组列车达到前足够久的观测记录和列车通过时的观测记录为本底振动和轨道交通“环境振动”，再从两者叠加得到的“观测记录”中去除本底振动，与原来选定的“环境振动”比较，通过实际算例论证了方法的可靠性。结果表明，本文提出的自互功率谱法能够更有效地去除观测数据中的本底干扰，获得更贴近实际的轨道交通环境振动的功率谱、时程、振动级VAL以及加权振级VLz。尤其是在本底振动占优势的低频段，比现有方法有明显的改善。
台阵观测的优势在于各观测点振动的自相关及互相关特性在一定程度上体现了城市轨道交通振动场的空间结构。为了探讨振动源对这一空间结构的影响，对一台阵6个点的环境振动观测数据进行总体功率谱分析及分时段功率谱分析，采用最大似然法计算了列车通过台阵一侧不同时段振动场的F-K谱，逐一时段分析了F-K谱的特征变化。据不同时段F-K谱的特征，分析指出了相应时刻列车相对台阵的不同方位（与列车在轨道上的位置一一对应），展现了借助台阵振动场的F-K谱判断列车运行过程的能力，并推算出列车运行的速度。结果表明，只要时段划分得足够小，就可以直接从上述F-K谱中提取轨道交通激励源的移动速度。
接着，本文采用频率-波数域动力格林函数计算地表激励下单一覆盖层-半空间表面的竖向位移，论证了表达轨道交通环境振动现场观测中多次发现的地面振动衰减局部放大现象的能力。通过七个半空间与覆盖层的模量比1、5、6、7、8、9和10，四个覆盖层厚度3、5、7和9米，7个激励频率16Hz、48Hz、56Hz、64Hz、72Hz、80Hz及108Hz，三个覆盖层阻尼比，0.025、0.05和0.1等诸多因素对局部放大影响的分析，指出了局部放大受埋藏的主要速度界面控制，局部放大区的位置和放大的强度与该界面两侧的模量比（波阻抗比）、覆盖层厚度、激励的频率及覆盖土层的阻尼比等因素有关。一般来说，速度界面的波阻抗越大，局部放大越强烈；给定其他条件，使直达的表面波与半空间表面折射波越接近同时到达的覆盖层厚度使局部放大越为明显，使两者波峰越靠近的激励频率会引起更为明显的局部放大；覆盖层阻尼比越小，局部放大越强烈。结合简单的波传播路径分析，给出了城市轨道交通环境振动衰减局部放大机理的一个解释，亦从一个侧面验证了本文采用的频率-波数域的动力格林函数方法计算轨道交通环境振动的可靠性。
为了适应地下水埋藏较浅场地环境振动分析的需要，基于饱和多孔介质动力分析的门福录模型，采用Haskell-Thomson传递矩阵法，在直角坐标系下推导了饱和层状场地频率-波数域动力格林函数。通过饱和土层参数取ρ2=0.0001kg/m~3，f=1，Ew=0.0001Pa，b=0.0001Pa·s/m~2，使其十分接近单相介质来退化，计算4种场地模型的地表竖向位移，与相应单相介质模型计算的结果比较，论证了本文推导的计算公式和编程的可靠性。计算了8个频率10Hz、20Hz、30Hz、40Hz、50Hz、60Hz、70Hz和80Hz单位简谐点源激励下单一饱和土覆盖层-半空间模型的地表振动竖向位移，与基于Biot模型计算方法的相应结果对比，论证了本文发展的饱和层状场地动力格林函数方法计算地下水埋藏较浅的场地上城市轨道交通环境振动的可行性，同时也验证了文献[5]基于Biot模型发展的计算方法是可靠的。门福录模型参数的物理意义明确，实际应用更为方便。
最后，借助项目组发展的频率-波数域计算轨道交通激励下列车-轨道-地基三维耦合系统地表反应的方法，探讨、论证了充分发挥台阵多次观测数据的优势反演激励源函数的可行性。通过虚拟反演，选取5个波长对应的PSD幅值表征激励源函数，与列车的速度及道砟路堤层无阻尼刚度一起共同构成待反演参数向量及其取值范围。用四个点振动的加速度级构造目标函数，采用早上班高峰期和接近中午平峰期的两次列车通过的观测记录数据，增加约束，反演了北京城轨13号线某一段的轮轨不平（圆）顺谱。与另一观测点实测数据对比，分析、验证了反演激励源函数的可靠性。完善了根据紧邻城市轨道交通线地表振动的台阵观测数据反演轮-轨不平（圆）顺谱的新方法。
总之，本文在城市轨道交通引起的环境振动及传播规律研究的六个方面得出了创新成果，包括时域计算方法的局限性、才观测记录中去除本底振动、从F-K谱中提取振动源的移动速度、轨道交通环境振动衰减局部放大的机理、基于门福录模型的饱和成层场地频率-波数域动力格林函数、根据多次观测数据联合反演轨道不平（圆）顺谱参数等，验证、完善并了项目组发展计算轨道交通激励下列车-轨道-地基三维耦合系统地表反应的频率-波数域动力格林函数方法，和根据紧邻城市轨道交通线地表振动的台阵观测数据反演作为系统激励源的轮-轨不平（圆）顺谱参数的创新思路。
As a summary of the last part research work of the Key Project by NationalNatural Science Foundation "Environment vibration caused by urban rail traffic andits propagation feature"(50538030), the dissertation demonstrates a mentionedconclusion which is needed to make deep research, improves the data processing toremove the background vibration from the observed environment vibration causedby urban rail traffic and discusses the application of frequency-wavenumberspectrum, demonstrates and perfects the development the dynamic Green Functionmethod in frequency-wavenumber domain to calculate3dimensional response oftrain-track-ground system and its capability to inverse the wheel-track spectrum, onthe basis of the breakthrough achieved by the whole team of the project.
Firstly, the feasibility of time-space domain method to simulate theenvironment vibration caused by urban rail traffic in frequency band10Hz to80Hz,in general, and60Hz to80Hz in near field, is demonstrated. With unit harmonicexcitation with frequencies2Hz,10Hz,30Hz and60Hz on surface of a threedimension model with single layer overlaid on half space, the vertical displacementamplitudes at the points with distance5meters-20meters from the exciting point arecalculated by wave finite element with transmission boundary conditions. For thedemonstration, the calculations are carried out repeatedly with four discrete gridsizes from1m,2m,5m to10m. From the comparison of the results with those by thedynamic Green Function method in frequency-wave number domain, the error incalculation in time-space domain is showed obviously. For a given excitingfrequency, the smaller the grid, the smaller error by the finite element calculation;for a given grid size, the lower the frequency, the smaller error the finite elementcalculation; the reliability of the three dimension finite element simulation of wavewith frequency in band60Hz-80Hz with grid size from stability study mainly byone-dimension analysis, is not ensured even the maximum size rule is now widelyadopted in two or three dimensional analysis. A preliminary quantitative rule forthree dimension analysis is that the maximum grind size should be less than1/25ofthe minimum interesting wavelength. Mean while, it is also pointed out that the calculation stability is also depends on the discrete step in time domain, themaximum step should be0.0001second if the discrete grid is taken as1m, forsimulation of common urban rail traffic caused environment vibration. That meansthe calculated amount must be very large, the time consumed must be quite long inthe case, so that it may take one week on a personal computer for simple casecalculation.
Removing background vibration noise in observed records of trafficenvironment vibration is an important step to increase the Signal to Noise Ratio anddemonstrate the characteristic of traffic environment vibration. The shortcomings ofexisting methods to remove the background vibration noise in observed records oftraffic environment vibration is analyzed, an idea to take into account thecross-relativity between background vibration and the observed vibration is putforward, the auto-cross spectrum method is developed as a new method to removebackground vibration and increase the ratio of signal to noise. A case fordemonstration is built as that the vibration recorded in time period long before thetrain passing is taken as background vibration, that recorded while the train ispassing as environment vibration, and that superposed by the two as observedvibration. An environment vibration is obtained by the background vibrationremoved from the observed vibration from the auto-cross spectrum method. Themethod is then verified by comparing the result vibration with the environmentvibration in this case, as the result shows that the background noise can removedmore effectively, the power spectrum, time history, vibration level VAL andweighted level VLz of the obtained environment vibration are all closer to those ofthe environment vibration.
The superiority of array observation is that the auto-correlation andcross-correlation characteristic of each point can reflect the spatial structure ofvibration field of urban rail traffic. In order to discuss the effect of source of urbanrail traffic on spatial structure of the vibration field, total power spectrum andpower spectra of some short time periods are calculated by maximum likelihoodmethod from environment vibrations recorded at6points on a rectangular arraywhile a train is passing it, and frequency-wave number (F-K in brief) spectra ofvibration fields in the short time periods are then calculated, the variation of the features of those power spectra in one by one time period is analyzed. Theorientation of the train to the array center (corresponding to the position of the trainon the rail) in each time period is determined from the corresponding F-K spectrum.The result shows the capability to infer the passing process of the train from the F-Kspectra of vibration field on array, and to compute the train speed. It demonstratesthat the moving speed of exciting source of traffic environment vibration couldacquired directly from the above F-K spectra, if the short time periods are taken asshort enough. Ground vertical displacements on model of single soil layer overlaidon half space with excitation on surface are calculated by means of the dynamicGreen Function method in frequency-wave number domain.
The capability of the method to deal with the local amplification in attenuationof ground vibration that phenomenon has been observed in some measurements ofenvironment vibration caused by rail traffic, is demonstrated. The results show thatlocal amplification is controlled by an important burried velocity interface, thelocation and intensity of the amplification depends on the ratio of elastic modului ofthe media in the both sides of the interface (i. e. impedance ratio), the thickness ofthe overburden layer, exciting frequency and damping ratio of the overburden layer,from analysis of influence on the amplification of factors as modulus ratio of halfspace to overburden layer media with seven values1,5,6,7,8,9and10,overburden layer thickness with four values3,5,7and9meters, exciting frequencywith seven values16Hz,48Hz,56Hz,64Hz,72Hz,80Hz and108Hz, overburdenlayer damping ratio with three values0.025,0.05and0.1. In general, the largerimpedance at the interface, the more obvious amplifying; keeping the otherconditions the same, the closer of directly arriving time of surface wave to thearriving time of refraction wave from the interface governed by the overburdenlayer thickness, the more obvious amplifying; the closer of the peaks of the wavesgoverned also by the exciting frequency, the more obvious amplifying; the smallerdamping ratio of the overburden layer, the more obvious amplifying. A mechanismof the local amplification in environment vibration attenuation is explained from theresults with a brief analysis of wave propagation paths. The reliability of calculationof rail traffic environment vibration by the dynamic Green Function method infrequency-wave number domain is verified by the way.
In order to fit the requirement to analyze the environment vibration at site withshallow groundwater, the dynamic Green Function in frequency-wave numberdomain on saturated layered ground is derived in rectangular Cartesian coordinatesystem based on the model suggested by Men FuLu for wave propagation insaturated porous medium, and by Haskell-Thomson transfer matrix method. Groundvertical displacements of four site models are calculated from a degeneration byreplacing the values of saturated soil parameters as ρ2=0.0001kg/m~3, f=1,Ew=0.0001Pa, b=0.0001Pa·s/m~2, so that the site models are very close to those insingle phase medium. The formulas and the program of the dissertation are verifiedby comparing the results with the published results of the corresponding singlephase medium models. Ground vertical displacements of model with singlesaturated layer overlaid on half space by unit harmonic excitation with eightfrequencies10Hz,20Hz,30Hz,40Hz,50Hz,60Hz,70Hz and80Hz are calculated.The feasibility of the developed dynamic Green Function method infrequency-wave number domain to simulate the vibration field by urban rail trafficat site with shallow ground water is demonstrated by comparing the results withthose published by Wang (2011) based on Biot model for the same models. Meantime, the reliability of the method developed by Wang is also demonstrated. Thephysical meaning of parameters in Men Fulu's model is unambiguous, thus themethod of this paper is more convenient in practice application.
Finally, the feasibility of inversing excitation source function with theadvantage in array data observed from several trains is tested and demonstrated bymeans of the method developed by project team to calculate vibration on threedimensional train-rail-ground coupled system in the frequency-wave numberdomain. PSD amplitudes corresponding five wavelengths are selected tocharacterize exciting source function, speed of the train and the un-damped stiffnessof ballast and embankment are selected to build inversion parameter vector with theabove five together, and the value ranges of the seven are estimated from a virtualinversion. The objective function is constructed by the vibration acceleration levelsof four points, and the observation data of the two trains during rush hour inmorning and close to the flat peak period at noon are used, and constraint hasincreased. The wheel/rail uneven (round) spectrum of Beijing No.13rail line is inversed. The reliability of the inversion excitation source function is analyzed andverified though comparison of calculation result used another point measured data.The new method to inverse wheel and rail uneven (round) spectrum of the urban railtraffic according to the surface of the vibration on array observation data close tothe rail is improved.
In conclusion, six innovation achievements on environment vibration causedby urban rail traffic and its propagation are worked out, such as limitations of timedomain calculation, the improvement to remove the background vibration noise, andacquiring of moving speed of the vibration source from F-K spectra, mechanism oflocal amplification in environment vibration attenuation, dynamic Green Functionin frequency-wave number domain in saturated layered ground based on Men FuLumodel and joint inversion of wheel-rail uneven (round) spectrum parameters fromarray data observed from several trains. The dynamic Green Function method tocalculate the response of three dimensional system of train-track-ground infrequency-wave number domain and the innovational idea to inverse wheel-railuneven (round) spectrum as the exciting source from data observed on array close tourban rail traffic according to the surface of the vibration on array observation datathe rail are demonstrated and completed.
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