车桥系统耦合振动和地震响应的随机分析
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摘要
随着高速铁路的迅速发展以及列车速度的不断提高,列车与支承系统的动力相互作用成为十分重要的问题,尤其是列车与桥梁系统的耦合振动分析。为了满足高速铁路线路平顺性和稳定性的要求,可能要建造连续几公里甚至几十公里的高架桥,例如京沪高速铁路中桥梁总长就占到全线长度的80.5%。大量的实测数据和数值模拟表明,由于轨道不平顺的存在,列车与桥梁之间的相互作用具有很强的随机性。而传统计算方法的低效率和复杂性限制了人们采用随机振动理论对车桥耦合系统的振动特性进行研究。迄今为止,关于车桥耦合系统随机振动分析的研究成果还比较有限。
另外,随着桥梁在高速铁路中的大量应用,地震发生时列车在桥上的几率大大增加。因此地震作用下车桥耦合系统的动力响应以及列车运行安全性分析就成为一项十分重要的研究课题。地震动的强随机性势必使车桥系统表现出很强的随机振动,但由于车桥系统本身的时变性以及采用传统随机振动理论进行分析的复杂性,真正意义上的地震作用下车桥耦合系统的非平稳随机振动分析论著还很难找到。
鉴于此,本博士学位论文在随机振动理论框架下将虚拟激励法、精细积分法等力学领域近年来出现的创新性研究成果引入到车桥耦合系统的随机振动分析中,并根据车辆与桥梁相互作用的特点对这些方法进行了发展,提出了各种独具特色的高效精确计算方法。利用这些强有力的分析工具对车桥耦合系统的随机振动特性以及随机地震作用下的车桥响应进行了精确细致的分析,详细地研究了列车运行速度、轨道不平顺质量、地震场地条件、地震视波速等因素对系统随机振动的影响。全文的主要研究内容如下：
1.虚拟激励法在车桥时变系统中的发展
虚拟激励法原来仅适用于高层建筑、大跨度结构等时不变系统,以及受单维平稳随机激励的时变系统。鉴于车桥时变系统随机振动分析的需要,本文通过严格的理论推导,系统地证明了虚拟激励法同样适用于受多维或者非平稳随机激励的时变系统,并分别构造了单维／多维、单点／多点完全相干、平稳／非平稳、均匀调制／非均匀调制等多种随机激励的虚拟激励形式。这些工作为虚拟激励法应用于高速铁路领域奠定了良好的理论基础,同时也有力地推动了随机振动理论在工程领域的广泛应用。
2.移动质量过桥问题的精细积分高效求解
当前求解移动质量过桥动力响应时常采用的Newmark法等传统逐步积分法都必须假定在每个积分步内移动质量的位置和惯性力的作用位置与大小都是固定不变的。因此必须采用很小的时间步长才能确保计算精度,但却严重降低了计算效率。为了克服以上限制,本文推广精细积分法来对移动质量通过时桥梁的动力响应进行高效精确求解。在每一个积分步内,首先对前后积分时刻单元节点的加速度进行时间线性插值,生成随时间连续变化的惯性力；然后根据平行力系平衡原理或者有限元的形函数分别构造基于简单分解、协调分解或混合分解方式的精细积分格式；最后将系统运动方程中的时变项移到方程右端,并采用精细积分格式进行迭代求解。这种方法可以更为真实地模拟移动质量惯性力在时间域和空间域上的连续变化,因此采用大得多的积分步长就能得到具有很高精度的计算结果。大量的数值比较表明,本文所提出的精细积分迭代求解方法在处理移动质量过桥问题时比Newmark法在计算效率和精度上均有重大改进。
3.车桥时变系统的非平稳随机振动研究
由于轨道不平顺的存在,当列车通过桥梁时,车桥系统会发生复杂的耦合随机振动。并且系统运动方程的质量、刚度和阻尼矩阵都是随时间连续变化的。对于这样一个时变系统,目前大多采用计算精度较差的时间历程法进行分析。本文将时变系统的虚拟激励法和扩展的精细积分迭代格式相结合,建立了一种车桥时变系统非平稳随机振动分析的高效精确算法——虚拟激励-精细积分法(PEM-PIM)。按照由易到难的原则,将研究工作分为三个步骤：首先,将时变系统的虚拟激励法分别和基于简单、协调、混合分解的精细积分迭代格式相结合,通过独轮车-桥梁系统的非平稳随机振动分析,比较了它们的计算效率和精度。然后,针对车桥垂向振动模型,考虑各车轮间轨道不平顺激励的相位差,采用PEM-PIM简单分解迭代格式研究了车桥系统垂向随机响应的统计特性；最后,将PEM-PIM算法引入到三维车桥系统的随机振动分析中,研究了三种轨道不平顺作用下三维车桥系统的非平稳随机振动特性。通过与Monte Carlo法的计算结果相比较,表明本文建立的PEM-PIM方法精确可靠。在此基础上,详细分析了列车行车速度、轨道不平顺等级和类型等因素对车桥系统随机振动的影响,得到了一些颇具参考价值的结论。
4.地震作用下车桥系统的随机振动分析
桥梁在高速铁路中的大量使用使得地震发生时列车在桥上的几率显著增大,研究地震作用下车桥系统的随机振动特性具有十分重要的现实意义。对于这样一个受双重随机激励的复杂时变系统,目前时间历程分析法是唯一的求解途径。本文假设水平地震动为单点均匀调制随机激励,轨道不平顺为多维多点平稳随机激励,采用PEM-PIM算法成功地实现了双重随机激励作用下车桥时变系统的非平稳随机振动分析,方便地得到了系统响应的时变功率谱和标准差等统计量。同样采用Monte Carlo法对其正确性进行了验证,并重点分析了地震场地条件、列车速度等对系统随机响应的影响。由于在实际工程分析中工程师们最为关心的是车桥系统地震响应的最大值,本文基于首次超越理论提出
With the rapid development of high-speed railway and the continuous increase of train speed, the dynamic interaction between trains and the supporting system has become a problem of great importance, particularly for the coupled vibration analysis of train-bridge systems. To meet the requirements on the smoothness of track and hence the safety and stability of running trains, elevated bridges can stretch for dozens of kilometers, e.g. bridges account for about 80.5% of the length of the Beijing-Shanghai express railway. A great deal of testing data and simulation-based numerical results show that dynamic interactions between trains and bridges are essentially random due to track irregularities. Unfortunately, the complexity and low efficiency of the conventional random vibration methods has considerably restricted the development of the relevant research work. So for, not many papers in this field can be found in the literature.
Moreover, the frequent use of elevated bridges in high-speed railways considerably increases the probability of earthquakes taking place when trains are crossing bridges. Therefore, the dynamic responses of train-bridge systems under the action of earthquakes and their effects on the running safety of trains have become an important research subject. The strong randomness of earthquakes certainly will make train-bridge systems vibrate randomly. However, due to the time variation of train-bridge systems and the complexity of the conventional random vibration algorithm, the non-stationary random vibration analysis of train-bridge coupled systems subjected to earthquakes is too hard to be performed.
In the present thesis, based on the theoretical framework of random vibration, some advanced computational mechanics methodology, such as the pseudo excitation method (PEM) for random vibration, the precise integration methods (PIM) in the time or space domain, and others, are introduced into the present research on random vibration analysis of train-bridge coupled systems. Some innovative schemes based on the above methodology have been further developed according to the characteristics of the dynamic interaction between trains and bridges. By using these powerful analysis tools, the random vibration and seismic responses of train-bridge coupled systems are investigated accurately and efficiently, and the influences of some important factors, such as train speeds, track irregularities, earthquake soil sites, and seismic apparent velocities, are discussed in detail. The main research contents are as follows:
1. The extension of PEM to time-dependent train-bridge systems
PEM was previously used only to time-independent systems, such as tall buildings or large span structures, and some time-dependent systems subjected to single-dimension stationary random excitations, In the present paper, to carry out the random vibration analysis for time-dependent train-bridge systems, PEM is strictly proven to be also applicable to the time-dependent systems that are subjected to multi-dimension or non-stationary random excitations. And the pseudo-excitations corresponding to the single-dimension or multi-dimension, single-point or multi-point fully coherent, stationary or non-stationary, uniformly modulated or non-uniformly modulated random excitations, are constituted respectively through theoretical derivation. This research work not only establishes theoretical foundation for applying PEM into the high-speed railway field, but also powerfully promotes the wide application of random vibration theory in project fields.
2. Efficient precise integration calculation for dynamic responses of FE bridges subjected to a moving mass
When solving dynamic responses of bridges due to a moving mass, generally direct integration methods, such as the Newmark method, requires the position and magnitude of the inertia force of moving mass to be invariant in each integration step, so that a very small integration step size is needed to ensure sufficient precision, which will increase computational effort considerable. In order to overcome this shortcoming, PIM is extended to deal with dynamic analysis of FE bridge model subjected to a moving mass. In each time step, an iterative linear interpolation scheme based on the accelerations at the element nodes at the beginning and end of the time interval, firstly, is adopted to generate the interaction forces that vary continuously with time. And then according to the force equilibrium principle or FEM shape functions, three precise integration forms, based on the simplified, consistent and hybrid decompositions respectively, are proposed and used iteratively to calculate the bridge responses. These methods can well simulate the continuous varying of the inertia forces of moving mass both in the time and space domains, and thus can give high precision results by using large integration steps. Numerous numerical comparisons show that each proposed method is greatly superior to the Newmark method.
3. Non-stationary random vibration analysis of time-dependent train-bridge systems
When train cross bridges, due to track irregularity, the interaction between train and bridge will generate complicated random vibration. The mass, stiffness and damping of train-bridge system is time-dependent. For such a system, the most common analysis method is the time history method of low precision. In the present paper, by combining PEM of time-dependent system and extended PIM iterative forms, a new efficient and accurate algorithm for non-stationary random vibration analysis of time-dependent train-bridge systems is proposed, which is named as PEM-PIM. According to the principle of "From easiness to difficulty", the main research takes three steps. Firstly, PEM of time-dependent system is combined respectively with three PIM iterative forms, and then used in the one-wheel vehicle and bridge coupled system to analyse and compare their computational efficiency and precision. Secondly, for the vertical vehicle-bridge system taken phase-lags between successive wheels into account, the PEM-PIM scheme, based on simplified decomposition, is adopted to investigate the statistical characteristics of vertical random responses. Thirdly, this PEM-PIM scheme is introduced into the 3D random vibration analysis of train-bridge systems, in which three types track irregularities are considered. In the numerical examples, the proposed method is justified by comparing with Monte Carlo simulation results. Also, the influences of train speeds and track irregularities on system random vibrations are discussed.
4. Earthquake induced random vibration analysis for train-bridge systems
With more and more bridges used in high-speed railways, the probability of earthquakes taking place when trains are crossing bridges increases considerably, which establishes the importance of investigating the safety of train-bridge systems under the action of earthquakes. At present, for such a problem, the time history method is the only method to deal with it. In this thesis, the lateral horizontal earthquake is assumed to be a single-dimension uniformly modulated non-stationary random process, while the excitations due to track irregularity are assumed to be multi-dimension multi-phase stationary random ones. The non-stationary random responses of train-bridge systems due to such two random excitations are calculated by applying the PEM-PIM scheme, and their time-dependent PSD and standard deviations are obtained conveniently. Besides, a new formula to estimate maximum responses, based on the first-passage failure criterion, is suggested. The numerical results indicate its reliability.
5. The random seismic responses of train-bridge coupled systems with wave passage effect considered
It is well known that for long-span bridges subjected to earthquakes, it is very important to account for the phase-lags between ground joints, i.e. the so-called wave passage effect. The seismic responses with wave passage effect considered may be quite different from the results obtained based on the assumption of uniform ground motion. In similar fashion, it is necessary to consider the wave passage effect in the seismic response analysis of train-bridge coupled systems. However, no research on such random vibration analysis has been found up to now. The present paper, for the first time, establishes the time-dependent motion equation of the train-bridge system subjected to multi-point earthquake excitations based on the concept of pseudo-static displacement. With the horizontal and vertical earthquake excitations assumed to be 2D uniformly modulated, multi-point fully coherent random excitations, the accurate non-stationary random vibration analysis for such a system is performed successfully by using PEM of time-dependent system. The numerical results show the effectiveness and accuracy of the proposed method, and the influence of seismic apparent wave velocity is discussed.
另外,随着桥梁在高速铁路中的大量应用,地震发生时列车在桥上的几率大大增加。因此地震作用下车桥耦合系统的动力响应以及列车运行安全性分析就成为一项十分重要的研究课题。地震动的强随机性势必使车桥系统表现出很强的随机振动,但由于车桥系统本身的时变性以及采用传统随机振动理论进行分析的复杂性,真正意义上的地震作用下车桥耦合系统的非平稳随机振动分析论著还很难找到。
鉴于此,本博士学位论文在随机振动理论框架下将虚拟激励法、精细积分法等力学领域近年来出现的创新性研究成果引入到车桥耦合系统的随机振动分析中,并根据车辆与桥梁相互作用的特点对这些方法进行了发展,提出了各种独具特色的高效精确计算方法。利用这些强有力的分析工具对车桥耦合系统的随机振动特性以及随机地震作用下的车桥响应进行了精确细致的分析,详细地研究了列车运行速度、轨道不平顺质量、地震场地条件、地震视波速等因素对系统随机振动的影响。全文的主要研究内容如下：
1.虚拟激励法在车桥时变系统中的发展
虚拟激励法原来仅适用于高层建筑、大跨度结构等时不变系统,以及受单维平稳随机激励的时变系统。鉴于车桥时变系统随机振动分析的需要,本文通过严格的理论推导,系统地证明了虚拟激励法同样适用于受多维或者非平稳随机激励的时变系统,并分别构造了单维／多维、单点／多点完全相干、平稳／非平稳、均匀调制／非均匀调制等多种随机激励的虚拟激励形式。这些工作为虚拟激励法应用于高速铁路领域奠定了良好的理论基础,同时也有力地推动了随机振动理论在工程领域的广泛应用。
2.移动质量过桥问题的精细积分高效求解
当前求解移动质量过桥动力响应时常采用的Newmark法等传统逐步积分法都必须假定在每个积分步内移动质量的位置和惯性力的作用位置与大小都是固定不变的。因此必须采用很小的时间步长才能确保计算精度,但却严重降低了计算效率。为了克服以上限制,本文推广精细积分法来对移动质量通过时桥梁的动力响应进行高效精确求解。在每一个积分步内,首先对前后积分时刻单元节点的加速度进行时间线性插值,生成随时间连续变化的惯性力；然后根据平行力系平衡原理或者有限元的形函数分别构造基于简单分解、协调分解或混合分解方式的精细积分格式；最后将系统运动方程中的时变项移到方程右端,并采用精细积分格式进行迭代求解。这种方法可以更为真实地模拟移动质量惯性力在时间域和空间域上的连续变化,因此采用大得多的积分步长就能得到具有很高精度的计算结果。大量的数值比较表明,本文所提出的精细积分迭代求解方法在处理移动质量过桥问题时比Newmark法在计算效率和精度上均有重大改进。
3.车桥时变系统的非平稳随机振动研究
由于轨道不平顺的存在,当列车通过桥梁时,车桥系统会发生复杂的耦合随机振动。并且系统运动方程的质量、刚度和阻尼矩阵都是随时间连续变化的。对于这样一个时变系统,目前大多采用计算精度较差的时间历程法进行分析。本文将时变系统的虚拟激励法和扩展的精细积分迭代格式相结合,建立了一种车桥时变系统非平稳随机振动分析的高效精确算法——虚拟激励-精细积分法(PEM-PIM)。按照由易到难的原则,将研究工作分为三个步骤：首先,将时变系统的虚拟激励法分别和基于简单、协调、混合分解的精细积分迭代格式相结合,通过独轮车-桥梁系统的非平稳随机振动分析,比较了它们的计算效率和精度。然后,针对车桥垂向振动模型,考虑各车轮间轨道不平顺激励的相位差,采用PEM-PIM简单分解迭代格式研究了车桥系统垂向随机响应的统计特性；最后,将PEM-PIM算法引入到三维车桥系统的随机振动分析中,研究了三种轨道不平顺作用下三维车桥系统的非平稳随机振动特性。通过与Monte Carlo法的计算结果相比较,表明本文建立的PEM-PIM方法精确可靠。在此基础上,详细分析了列车行车速度、轨道不平顺等级和类型等因素对车桥系统随机振动的影响,得到了一些颇具参考价值的结论。
4.地震作用下车桥系统的随机振动分析
桥梁在高速铁路中的大量使用使得地震发生时列车在桥上的几率显著增大,研究地震作用下车桥系统的随机振动特性具有十分重要的现实意义。对于这样一个受双重随机激励的复杂时变系统,目前时间历程分析法是唯一的求解途径。本文假设水平地震动为单点均匀调制随机激励,轨道不平顺为多维多点平稳随机激励,采用PEM-PIM算法成功地实现了双重随机激励作用下车桥时变系统的非平稳随机振动分析,方便地得到了系统响应的时变功率谱和标准差等统计量。同样采用Monte Carlo法对其正确性进行了验证,并重点分析了地震场地条件、列车速度等对系统随机响应的影响。由于在实际工程分析中工程师们最为关心的是车桥系统地震响应的最大值,本文基于首次超越理论提出
With the rapid development of high-speed railway and the continuous increase of train speed, the dynamic interaction between trains and the supporting system has become a problem of great importance, particularly for the coupled vibration analysis of train-bridge systems. To meet the requirements on the smoothness of track and hence the safety and stability of running trains, elevated bridges can stretch for dozens of kilometers, e.g. bridges account for about 80.5% of the length of the Beijing-Shanghai express railway. A great deal of testing data and simulation-based numerical results show that dynamic interactions between trains and bridges are essentially random due to track irregularities. Unfortunately, the complexity and low efficiency of the conventional random vibration methods has considerably restricted the development of the relevant research work. So for, not many papers in this field can be found in the literature.
Moreover, the frequent use of elevated bridges in high-speed railways considerably increases the probability of earthquakes taking place when trains are crossing bridges. Therefore, the dynamic responses of train-bridge systems under the action of earthquakes and their effects on the running safety of trains have become an important research subject. The strong randomness of earthquakes certainly will make train-bridge systems vibrate randomly. However, due to the time variation of train-bridge systems and the complexity of the conventional random vibration algorithm, the non-stationary random vibration analysis of train-bridge coupled systems subjected to earthquakes is too hard to be performed.
In the present thesis, based on the theoretical framework of random vibration, some advanced computational mechanics methodology, such as the pseudo excitation method (PEM) for random vibration, the precise integration methods (PIM) in the time or space domain, and others, are introduced into the present research on random vibration analysis of train-bridge coupled systems. Some innovative schemes based on the above methodology have been further developed according to the characteristics of the dynamic interaction between trains and bridges. By using these powerful analysis tools, the random vibration and seismic responses of train-bridge coupled systems are investigated accurately and efficiently, and the influences of some important factors, such as train speeds, track irregularities, earthquake soil sites, and seismic apparent velocities, are discussed in detail. The main research contents are as follows:
1. The extension of PEM to time-dependent train-bridge systems
PEM was previously used only to time-independent systems, such as tall buildings or large span structures, and some time-dependent systems subjected to single-dimension stationary random excitations, In the present paper, to carry out the random vibration analysis for time-dependent train-bridge systems, PEM is strictly proven to be also applicable to the time-dependent systems that are subjected to multi-dimension or non-stationary random excitations. And the pseudo-excitations corresponding to the single-dimension or multi-dimension, single-point or multi-point fully coherent, stationary or non-stationary, uniformly modulated or non-uniformly modulated random excitations, are constituted respectively through theoretical derivation. This research work not only establishes theoretical foundation for applying PEM into the high-speed railway field, but also powerfully promotes the wide application of random vibration theory in project fields.
2. Efficient precise integration calculation for dynamic responses of FE bridges subjected to a moving mass
When solving dynamic responses of bridges due to a moving mass, generally direct integration methods, such as the Newmark method, requires the position and magnitude of the inertia force of moving mass to be invariant in each integration step, so that a very small integration step size is needed to ensure sufficient precision, which will increase computational effort considerable. In order to overcome this shortcoming, PIM is extended to deal with dynamic analysis of FE bridge model subjected to a moving mass. In each time step, an iterative linear interpolation scheme based on the accelerations at the element nodes at the beginning and end of the time interval, firstly, is adopted to generate the interaction forces that vary continuously with time. And then according to the force equilibrium principle or FEM shape functions, three precise integration forms, based on the simplified, consistent and hybrid decompositions respectively, are proposed and used iteratively to calculate the bridge responses. These methods can well simulate the continuous varying of the inertia forces of moving mass both in the time and space domains, and thus can give high precision results by using large integration steps. Numerous numerical comparisons show that each proposed method is greatly superior to the Newmark method.
3. Non-stationary random vibration analysis of time-dependent train-bridge systems
When train cross bridges, due to track irregularity, the interaction between train and bridge will generate complicated random vibration. The mass, stiffness and damping of train-bridge system is time-dependent. For such a system, the most common analysis method is the time history method of low precision. In the present paper, by combining PEM of time-dependent system and extended PIM iterative forms, a new efficient and accurate algorithm for non-stationary random vibration analysis of time-dependent train-bridge systems is proposed, which is named as PEM-PIM. According to the principle of "From easiness to difficulty", the main research takes three steps. Firstly, PEM of time-dependent system is combined respectively with three PIM iterative forms, and then used in the one-wheel vehicle and bridge coupled system to analyse and compare their computational efficiency and precision. Secondly, for the vertical vehicle-bridge system taken phase-lags between successive wheels into account, the PEM-PIM scheme, based on simplified decomposition, is adopted to investigate the statistical characteristics of vertical random responses. Thirdly, this PEM-PIM scheme is introduced into the 3D random vibration analysis of train-bridge systems, in which three types track irregularities are considered. In the numerical examples, the proposed method is justified by comparing with Monte Carlo simulation results. Also, the influences of train speeds and track irregularities on system random vibrations are discussed.
4. Earthquake induced random vibration analysis for train-bridge systems
With more and more bridges used in high-speed railways, the probability of earthquakes taking place when trains are crossing bridges increases considerably, which establishes the importance of investigating the safety of train-bridge systems under the action of earthquakes. At present, for such a problem, the time history method is the only method to deal with it. In this thesis, the lateral horizontal earthquake is assumed to be a single-dimension uniformly modulated non-stationary random process, while the excitations due to track irregularity are assumed to be multi-dimension multi-phase stationary random ones. The non-stationary random responses of train-bridge systems due to such two random excitations are calculated by applying the PEM-PIM scheme, and their time-dependent PSD and standard deviations are obtained conveniently. Besides, a new formula to estimate maximum responses, based on the first-passage failure criterion, is suggested. The numerical results indicate its reliability.
5. The random seismic responses of train-bridge coupled systems with wave passage effect considered
It is well known that for long-span bridges subjected to earthquakes, it is very important to account for the phase-lags between ground joints, i.e. the so-called wave passage effect. The seismic responses with wave passage effect considered may be quite different from the results obtained based on the assumption of uniform ground motion. In similar fashion, it is necessary to consider the wave passage effect in the seismic response analysis of train-bridge coupled systems. However, no research on such random vibration analysis has been found up to now. The present paper, for the first time, establishes the time-dependent motion equation of the train-bridge system subjected to multi-point earthquake excitations based on the concept of pseudo-static displacement. With the horizontal and vertical earthquake excitations assumed to be 2D uniformly modulated, multi-point fully coherent random excitations, the accurate non-stationary random vibration analysis for such a system is performed successfully by using PEM of time-dependent system. The numerical results show the effectiveness and accuracy of the proposed method, and the influence of seismic apparent wave velocity is discussed.
引文
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