车辆与结构相互作用随机动力分析
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摘要
一九六四年日本新干线的成功运营标志着现代高速铁路时代的来临。随后,德国的ICE高速列车和法国的TGV高速列车也相继投入使用。经过四十多年的发展,高速铁路已经成为世界上最经济、最具有效能的地面运输系统。随着车辆速度的提高,车辆与轨道、地基和桥梁等结构的动力相互作用问题也日益突出。一方面,车辆运行时会对周围环境产生冲击作用,引起周围结构的振动;另一方面,这些结构的振动又会对车辆运行的平稳性和安全性产生反作用。大量的实测数据和数值模拟表明,车辆与结构的相互作用具有很强的随机性。然而传统计算方法的低效率限制了人们采用随机振动理论对车辆与结构相互作用问题进行研究。迄今为止,关于车辆与结构随机振动分析的研究成果还不是很多。本博士学位论文在随机振动理论框架下将先进的计算力学方法,例如虚拟激励方法、精细积分方法、与哈密顿对偶体系密切相关的辛数学方法等,引入到车辆与结构的振动分析中,并且根据车辆与结构相互作用的特点对上述方法进行了扩展,提出了精确高效的分析方法,研究了车辆与结构耦合振动的统计特性。全文的主要研究内容如下:
1.车辆与轨道结构垂向耦合随机振动分析
对车辆与轨道结构耦合系统进行随机振动分析面临的主要问题是计算量大。若直接套用常规的线性随机振动公式来求解耦合系统的随机响应,不但因系统自由度众多而计算量庞大,而且计算过程也相当繁琐。本文将轨道视为周期性链式结构,将虚拟激励法和辛数学方法相结合来求解耦合系统的随机响应。亦即:首先使用虚拟激励法将随机轨道不平顺转化为一系列虚拟的简谐轨道不平顺,然后采用辛数学方法求解周期性轨道结构的频率响应特性,最后通过求解耦合系统的虚拟响应得到振动响应的功率谱和标准差。由于分析带有周期性边界条件的一个典型子结构就获得对整条轨道进行分析相同的结果,所以本方法的计算效率比通常基于有限元的方法高得多,对一典型算例表明可高出百倍以上。在数值算例中还讨论了车辆轨道耦合模型和车辆模型各自的适用范围,以及车辆速度、轨道结构阻尼等因素对耦合系统随机响应的影响。
2.交通荷载引起的非平稳随机波在分层土壤介质中的传播
车辆在运行时必然引起轨道结构的随机振动,并且这种振动会通过附近的地层向外传播,对邻近区域造成干扰。由于交通荷载具有随机性和移动性,所以就轨道结构附近地面任意的固定点而言,其振动响应是非平稳随机过程。这类非平稳随机波传播问题的计算存在较多困难,研究成果较少。本文建立了轨道—地层耦合分析模型。假定轨道结构由钢轨、枕木和道床等部分构成。地层结构为各向同性的分层介质。将随机波传播问题引入到哈密顿对偶体系进行求解。首先将虚拟激励法扩展到处理移动随机荷载作用下线性系统动力响应问题。然后证明在对偶体系下波传播的运动方程在频率—波数域内是线性哈密顿方程的两端边值问题。再然后应用精细积分法给出该问题在计算机意义下的精确解。最后根据地层结构各向同性的性质,给出了进一步加速计算的方法。在数值算例中研究了地面振动响应的统计特性,以及交通荷载之间的相关性对随机响应的影响。
3.结构动力响应的高效计算
当前处理移动荷载问题时常采用的Newmark法等都必须假定在每一时间步长内,荷载的大小和作用位置都是固定不变的。因此必须采用很小的时间步长来确保计算精度。但这是以增加计算量为代价的。为了克服上述限制,本文对精细积分法进行了推广,将它应用于处理移动荷载问题。在每一个积分步内,根据有限元方法的形函数或者平行力系平衡原理,提出了称为协调分解过程和简单分解过程的精细积分格式。这两种分解过程可以模拟荷载幅度的变化和作用位置的变化。因此非常显著地减小了因选取较大积分步长而引起的计算误差。当积分步长满足一定的条件时,协调分解过程甚至可以给出结构动力响应在计算机意义下的精确解。相比之下简单分解过程的精度虽略低些,但应用更为简便。根据不同的工程需求,对二者的应用可作灵活选择。大量的数值比较表明,本文所提出的二种精细积分格式在处理移动荷载问题时都比Newmark法优越得多。
4.移动随机荷载作用下桥梁动力响应分析
对车辆桥梁这种复杂的耦合系统进行随机分析,历来被认为是一件非常困难的工作。作为这项研究工作的基础,研究随机移动荷载作用下桥梁结构振动响应的统计特性,近年来引起了广泛的关注。目前对此虽然已经有较多研究成果发表,但多数以简单桥梁模型为研究对象,采用效率不高的数值方法或者解析方法求解随机响应。分析过程中普遍采用模态叠加法。但为了提高计算效率,计算结果中经常忽略掉模态响应之间的相关项。对多荷载激励问题更是鲜有讨论。本文采用有限元方法建立桥梁模型,借助高效的虚拟激励法和扩展精细积分法提出了一种切实可行的随机分析方法,克服了计算效率的瓶颈。在数值算例中,以单跨和多跨连续桥梁为例,研究了桥梁结构基本的随机振动特性,讨论了桥梁模态之间的相关性、荷载之间的相关性以及随机荷载统计特性对随机响应的影响。
5.车辆与桥梁相互作用随机振动分析
车辆通过桥梁时,由于轨道不平顺的存在,车辆与桥梁相互作用发生复杂的耦合随机振动。车桥系统的质量、刚度和阻尼分布都是随时间连续变化的。对于这样一个时变系统,目前尚无有效的随机振动方法。本文本着由易到难的原则将研究工作分为两个步骤:首先针对车桥系统的特点,推导了线性时变系统的虚拟激励法,用二维模型研究了车桥系统垂向振动响应的统计特性。然后将上述扩展的虚拟激励法引入到三维车桥系统的随机分析中,研究了三种轨道不平顺同时作用下车桥系统横向随机响应的统计特性。为了提高计算效率,本文还通过分析计算过程的特点,采用耦联法来求解车桥系统的运动方程。其中采用了杨永斌教授提出的车桥耦合单元(VBI)。在数值算例中,通过与Monte Carlo法的计算结果进行比较,表明本文方法不但计算效率高,而且精确可靠。在实际工程分析中,工程师们最为关注的是车桥系统动力响应的最大值。本文基于Gauss型随机变量的“3σ法则”提出了一种估计方法。数值模拟结果表明这样估计是准确可靠的。
In 1964, the successful operation of Shinkansen in Japan signifies the coming of the era of modern high speed railway transportation. Later, France and Germany develop TGV and ICE high speed train, respectively. After more than 40 years of development, such high speed transportation has become the most economical and efficient land transportation system. With the increasing of vehicle velocity, the dynamic interaction between vehicles and neighbouring structures such as the rails, groundbasis or nearby buildings, is received more and more attention. This is because, on one hand , the running of vehicles will induce vibration of surrounding structures. On the other hand, the dynamic reaction of such structures will affect the stable and safe operation of the vehicles considerable. A great deal of testing data and simulation-based numerical results show that dynamic interactions between vehicles and neighbouring structures are essentially random. Unfortunately, the complexity and low efficiency of the conventional random vibration approaches has considerably restricted the development of the relevant research work. So far, not many papers in this field can be found in the literature. In the present ph.d thesis, based on the theoretical framework of random vibration, some advanced computational mechanics methodology, such as the pseudo excitation method for random vibration, the precise integration methods in the time domain or the space domain, the symplectic geometric scheme, which is closely related to Hamiltonian duality system, and others, are introduced into the present research on dynamic analyses of various coupled vehicle-structure systems. Some innovative schemes based on the above methodology have been further developed, which can be summarized as follows:
1. Vertical random vibration analysis of vehicle-track coupling systems
The main difficulty in the random vibration analysis of vehicle-track systems is the prohibitive computational effort required when using the conventional approaches. In addition, the process is also quite complicated. In the present paper, the tack is regarded as an infinitely long periodic structure. The pseudo excitation method (PEM) and the symplectic geometric method are combined to perform the random vibration analysis. That means, the random unevenness of the rail-surfaces are firstly transformed into the summation of a series of harmonic unevenness in terms of the PEM, then the symplectic approach is used in the solution of the frequency-response characteristic of the periodic railway structure, finally the random responses of the couples vehicle-railway system are computed by means of the PEM accurately and efficiently. It is noted that analyzing a typical substructure with periodical boundary conditions will produce the frequency-response characteristic of the entire railway, therefore the efficiency of this approach is much higher that usual FEM-based approached, even reaches more than 100 times for a typical numerical example. Numerical comparisons also show the suitability of the coupled model and the traditional rigid track model, as well as the influences of track damping and vehicle velocity on the system random vibration.
2. Traffic induced non-stationary random wave propagation in layered soil media.
Moving vehicles will induce track random vibration, which will propagate through thenearby ground, typically layered soil media. Such kind of non-stationary random wave propagation problems is in general quite difficult to solve. In this thesis, a track-ground coupling system is set up, by which the rail is modeled as a single infinite Euler beam connected to sleepers and hence to ballast. This ballast rests on the ground, which is assumed to consist of layered transversely isotropic soil. The random wave propagation problem is introduced into a Hamiltonian duality system. First, the non-stationary power spectral density (PSD) and the time-dependent standard deviation is derived conveniently by means of PEM. Then it is proved that in the frequency-wave number domain the coefficient matrix of wave propagation state equation is Hamiltonian, which can be solve accurately by using the precise integration method (PIM). At last in order to decrease the computational effort, an improved computational procedure is proposed, which takes advantage of the transverse isotropic property of the layered soil to reduce the threefold iteration process into a twofold one. In the numerical examples, the statistical characteristic of the ground response and the effect of correlation between loads are studied.
3. An accurate and efficient algorithm for dynamic analysis of FE structures subjected to moving loads
When dealing with moving loads, generally direct integration methods, such as the Newmark method, are used which require the position and magnitude of the load to be invariant in each integration step, so that a very small integration step size is needed to ensure sufficient precision, witch can increase computational effort considerable. In order to overcome this shortcoming, the precise integration method (PIM) is extended to deal with dynamic analysis of FE structures subjected to moving loads.
In each time step, based on FEM shape functions or the force equilibrium principle, two alternative decomposition procedures, i.e. consistent decomposition and simplified decomposition, are proposed. The resulting methods allow the position of the load and its magnitude to vary with time in each integration step and thus reduce the computational error considerable. When the time step stratified some conditions, the consistent decomposition procedure can give results up to computer precision. The simplified decomposition procedure is relatively less accurate, but very convenient for using. Numerous numerical comparisons show that either proposed method is greatly superior to the Newmark method.
4. Non-stationary random vibration of bridges subjected to moving loads
It is well know that the random vibration analysis of vehicle-bridge coupling system is a very difficult problem. As the fundamental of this problem, the dynamic analysis of bridge subjected to random moving loads is received much attention in recent years. Some research papers have been published. However, most of these papers used relatively simple bridge models and adopted analytical solution or inefficient numerical method to get stochastic response. The modal superposition is used in the analysis, but the cross-correlation between modal responses is usually neglected in order to decrease the computational effort. Moreover the investigation of the influence of the correlation between loads for multi-load problems is lacking. In the present paper, a FE bridge model is used, An efficient method based on PEM and PIM is proposed, which overcome the computation bottleneck. Numerical examples investigate the dynamic statistical characteristic of a simply supported beam and of a three non-uniform span beam, and discuss the effect of the cross-correlation between modal responses and that of between loads, and the type of random loads.
5. Efficient dynamic analysis for non-stationary random vibration of vehicle-bridge systems
When vehicles cross bridges, due to track irregularity the interaction between vehicle and bridge will generate complicated random vibration. The mass, stiffness and damping of vehicle bridge system is time dependent. For such a system, there is no mature random vibration formulation to deal with it. According to the principle of "From easiness to difficulty", the present paper takes two steps in the research work.First, the pseudo excitation method (PEM) is extended to handle the random analysis of time-dependent systems, in which the PSD and deviation is derived. The statistical characteristics of vertical dynamic responses for vehicle bridge system are investigated. Then, the extended PEM is introduced into 3D dynamic analysis, in which three types track irregularity are considered. To reduce the computational effort a vehicle bridge interaction (VBI) element is adopted by considering the feature of stochastic analysis procedure. In the numerical examples, the proposed method is justified by comparing with Monte Carlo simulation results. Also, a method, based on the 3σrule for Gaussian stochastic processes, to estimate maximum responses is suggested. Examples include a train moving across both simply supported and three-span continuous bridges and some observed phenomena are discussed.
1.车辆与轨道结构垂向耦合随机振动分析
对车辆与轨道结构耦合系统进行随机振动分析面临的主要问题是计算量大。若直接套用常规的线性随机振动公式来求解耦合系统的随机响应,不但因系统自由度众多而计算量庞大,而且计算过程也相当繁琐。本文将轨道视为周期性链式结构,将虚拟激励法和辛数学方法相结合来求解耦合系统的随机响应。亦即:首先使用虚拟激励法将随机轨道不平顺转化为一系列虚拟的简谐轨道不平顺,然后采用辛数学方法求解周期性轨道结构的频率响应特性,最后通过求解耦合系统的虚拟响应得到振动响应的功率谱和标准差。由于分析带有周期性边界条件的一个典型子结构就获得对整条轨道进行分析相同的结果,所以本方法的计算效率比通常基于有限元的方法高得多,对一典型算例表明可高出百倍以上。在数值算例中还讨论了车辆轨道耦合模型和车辆模型各自的适用范围,以及车辆速度、轨道结构阻尼等因素对耦合系统随机响应的影响。
2.交通荷载引起的非平稳随机波在分层土壤介质中的传播
车辆在运行时必然引起轨道结构的随机振动,并且这种振动会通过附近的地层向外传播,对邻近区域造成干扰。由于交通荷载具有随机性和移动性,所以就轨道结构附近地面任意的固定点而言,其振动响应是非平稳随机过程。这类非平稳随机波传播问题的计算存在较多困难,研究成果较少。本文建立了轨道—地层耦合分析模型。假定轨道结构由钢轨、枕木和道床等部分构成。地层结构为各向同性的分层介质。将随机波传播问题引入到哈密顿对偶体系进行求解。首先将虚拟激励法扩展到处理移动随机荷载作用下线性系统动力响应问题。然后证明在对偶体系下波传播的运动方程在频率—波数域内是线性哈密顿方程的两端边值问题。再然后应用精细积分法给出该问题在计算机意义下的精确解。最后根据地层结构各向同性的性质,给出了进一步加速计算的方法。在数值算例中研究了地面振动响应的统计特性,以及交通荷载之间的相关性对随机响应的影响。
3.结构动力响应的高效计算
当前处理移动荷载问题时常采用的Newmark法等都必须假定在每一时间步长内,荷载的大小和作用位置都是固定不变的。因此必须采用很小的时间步长来确保计算精度。但这是以增加计算量为代价的。为了克服上述限制,本文对精细积分法进行了推广,将它应用于处理移动荷载问题。在每一个积分步内,根据有限元方法的形函数或者平行力系平衡原理,提出了称为协调分解过程和简单分解过程的精细积分格式。这两种分解过程可以模拟荷载幅度的变化和作用位置的变化。因此非常显著地减小了因选取较大积分步长而引起的计算误差。当积分步长满足一定的条件时,协调分解过程甚至可以给出结构动力响应在计算机意义下的精确解。相比之下简单分解过程的精度虽略低些,但应用更为简便。根据不同的工程需求,对二者的应用可作灵活选择。大量的数值比较表明,本文所提出的二种精细积分格式在处理移动荷载问题时都比Newmark法优越得多。
4.移动随机荷载作用下桥梁动力响应分析
对车辆桥梁这种复杂的耦合系统进行随机分析,历来被认为是一件非常困难的工作。作为这项研究工作的基础,研究随机移动荷载作用下桥梁结构振动响应的统计特性,近年来引起了广泛的关注。目前对此虽然已经有较多研究成果发表,但多数以简单桥梁模型为研究对象,采用效率不高的数值方法或者解析方法求解随机响应。分析过程中普遍采用模态叠加法。但为了提高计算效率,计算结果中经常忽略掉模态响应之间的相关项。对多荷载激励问题更是鲜有讨论。本文采用有限元方法建立桥梁模型,借助高效的虚拟激励法和扩展精细积分法提出了一种切实可行的随机分析方法,克服了计算效率的瓶颈。在数值算例中,以单跨和多跨连续桥梁为例,研究了桥梁结构基本的随机振动特性,讨论了桥梁模态之间的相关性、荷载之间的相关性以及随机荷载统计特性对随机响应的影响。
5.车辆与桥梁相互作用随机振动分析
车辆通过桥梁时,由于轨道不平顺的存在,车辆与桥梁相互作用发生复杂的耦合随机振动。车桥系统的质量、刚度和阻尼分布都是随时间连续变化的。对于这样一个时变系统,目前尚无有效的随机振动方法。本文本着由易到难的原则将研究工作分为两个步骤:首先针对车桥系统的特点,推导了线性时变系统的虚拟激励法,用二维模型研究了车桥系统垂向振动响应的统计特性。然后将上述扩展的虚拟激励法引入到三维车桥系统的随机分析中,研究了三种轨道不平顺同时作用下车桥系统横向随机响应的统计特性。为了提高计算效率,本文还通过分析计算过程的特点,采用耦联法来求解车桥系统的运动方程。其中采用了杨永斌教授提出的车桥耦合单元(VBI)。在数值算例中,通过与Monte Carlo法的计算结果进行比较,表明本文方法不但计算效率高,而且精确可靠。在实际工程分析中,工程师们最为关注的是车桥系统动力响应的最大值。本文基于Gauss型随机变量的“3σ法则”提出了一种估计方法。数值模拟结果表明这样估计是准确可靠的。
In 1964, the successful operation of Shinkansen in Japan signifies the coming of the era of modern high speed railway transportation. Later, France and Germany develop TGV and ICE high speed train, respectively. After more than 40 years of development, such high speed transportation has become the most economical and efficient land transportation system. With the increasing of vehicle velocity, the dynamic interaction between vehicles and neighbouring structures such as the rails, groundbasis or nearby buildings, is received more and more attention. This is because, on one hand , the running of vehicles will induce vibration of surrounding structures. On the other hand, the dynamic reaction of such structures will affect the stable and safe operation of the vehicles considerable. A great deal of testing data and simulation-based numerical results show that dynamic interactions between vehicles and neighbouring structures are essentially random. Unfortunately, the complexity and low efficiency of the conventional random vibration approaches has considerably restricted the development of the relevant research work. So far, not many papers in this field can be found in the literature. In the present ph.d thesis, based on the theoretical framework of random vibration, some advanced computational mechanics methodology, such as the pseudo excitation method for random vibration, the precise integration methods in the time domain or the space domain, the symplectic geometric scheme, which is closely related to Hamiltonian duality system, and others, are introduced into the present research on dynamic analyses of various coupled vehicle-structure systems. Some innovative schemes based on the above methodology have been further developed, which can be summarized as follows:
1. Vertical random vibration analysis of vehicle-track coupling systems
The main difficulty in the random vibration analysis of vehicle-track systems is the prohibitive computational effort required when using the conventional approaches. In addition, the process is also quite complicated. In the present paper, the tack is regarded as an infinitely long periodic structure. The pseudo excitation method (PEM) and the symplectic geometric method are combined to perform the random vibration analysis. That means, the random unevenness of the rail-surfaces are firstly transformed into the summation of a series of harmonic unevenness in terms of the PEM, then the symplectic approach is used in the solution of the frequency-response characteristic of the periodic railway structure, finally the random responses of the couples vehicle-railway system are computed by means of the PEM accurately and efficiently. It is noted that analyzing a typical substructure with periodical boundary conditions will produce the frequency-response characteristic of the entire railway, therefore the efficiency of this approach is much higher that usual FEM-based approached, even reaches more than 100 times for a typical numerical example. Numerical comparisons also show the suitability of the coupled model and the traditional rigid track model, as well as the influences of track damping and vehicle velocity on the system random vibration.
2. Traffic induced non-stationary random wave propagation in layered soil media.
Moving vehicles will induce track random vibration, which will propagate through thenearby ground, typically layered soil media. Such kind of non-stationary random wave propagation problems is in general quite difficult to solve. In this thesis, a track-ground coupling system is set up, by which the rail is modeled as a single infinite Euler beam connected to sleepers and hence to ballast. This ballast rests on the ground, which is assumed to consist of layered transversely isotropic soil. The random wave propagation problem is introduced into a Hamiltonian duality system. First, the non-stationary power spectral density (PSD) and the time-dependent standard deviation is derived conveniently by means of PEM. Then it is proved that in the frequency-wave number domain the coefficient matrix of wave propagation state equation is Hamiltonian, which can be solve accurately by using the precise integration method (PIM). At last in order to decrease the computational effort, an improved computational procedure is proposed, which takes advantage of the transverse isotropic property of the layered soil to reduce the threefold iteration process into a twofold one. In the numerical examples, the statistical characteristic of the ground response and the effect of correlation between loads are studied.
3. An accurate and efficient algorithm for dynamic analysis of FE structures subjected to moving loads
When dealing with moving loads, generally direct integration methods, such as the Newmark method, are used which require the position and magnitude of the load to be invariant in each integration step, so that a very small integration step size is needed to ensure sufficient precision, witch can increase computational effort considerable. In order to overcome this shortcoming, the precise integration method (PIM) is extended to deal with dynamic analysis of FE structures subjected to moving loads.
In each time step, based on FEM shape functions or the force equilibrium principle, two alternative decomposition procedures, i.e. consistent decomposition and simplified decomposition, are proposed. The resulting methods allow the position of the load and its magnitude to vary with time in each integration step and thus reduce the computational error considerable. When the time step stratified some conditions, the consistent decomposition procedure can give results up to computer precision. The simplified decomposition procedure is relatively less accurate, but very convenient for using. Numerous numerical comparisons show that either proposed method is greatly superior to the Newmark method.
4. Non-stationary random vibration of bridges subjected to moving loads
It is well know that the random vibration analysis of vehicle-bridge coupling system is a very difficult problem. As the fundamental of this problem, the dynamic analysis of bridge subjected to random moving loads is received much attention in recent years. Some research papers have been published. However, most of these papers used relatively simple bridge models and adopted analytical solution or inefficient numerical method to get stochastic response. The modal superposition is used in the analysis, but the cross-correlation between modal responses is usually neglected in order to decrease the computational effort. Moreover the investigation of the influence of the correlation between loads for multi-load problems is lacking. In the present paper, a FE bridge model is used, An efficient method based on PEM and PIM is proposed, which overcome the computation bottleneck. Numerical examples investigate the dynamic statistical characteristic of a simply supported beam and of a three non-uniform span beam, and discuss the effect of the cross-correlation between modal responses and that of between loads, and the type of random loads.
5. Efficient dynamic analysis for non-stationary random vibration of vehicle-bridge systems
When vehicles cross bridges, due to track irregularity the interaction between vehicle and bridge will generate complicated random vibration. The mass, stiffness and damping of vehicle bridge system is time dependent. For such a system, there is no mature random vibration formulation to deal with it. According to the principle of "From easiness to difficulty", the present paper takes two steps in the research work.First, the pseudo excitation method (PEM) is extended to handle the random analysis of time-dependent systems, in which the PSD and deviation is derived. The statistical characteristics of vertical dynamic responses for vehicle bridge system are investigated. Then, the extended PEM is introduced into 3D dynamic analysis, in which three types track irregularity are considered. To reduce the computational effort a vehicle bridge interaction (VBI) element is adopted by considering the feature of stochastic analysis procedure. In the numerical examples, the proposed method is justified by comparing with Monte Carlo simulation results. Also, a method, based on the 3σrule for Gaussian stochastic processes, to estimate maximum responses is suggested. Examples include a train moving across both simply supported and three-span continuous bridges and some observed phenomena are discussed.
引文
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