车辆—轨道耦合系统高效随机振动分析及优化
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摘要
随着能源环境问题的日益突出和不断增加的交通运输需求,世界铁路建设呈现出了客运高速化、货运重载化的发展趋势,这也日益加剧了车辆与轨道、桥梁、地基、弓网以及周边建筑等结构的动力相互作用,直接关系到列车与线路的安全性、平顺性、疲劳、噪声等一系列问题。轨道不平顺是引起铁路机车车辆振动的公认的主要激励源,它具有明显的随机性。对轨道不平顺作用下车辆—轨道耦合系统进行随机振动分析,尤其是对其功率谱的计算,在铁路运输的动力计算中占据极其重要的地位。
然而,由于轨道结构庞大的计算自由度和传统频域和时域随机振动分析方法的低效,对上述问题进行求解需要花费高额的计算成本,迄今为止对这一领域的研究还大都只能局限于采用较简单的分析模型,这显然无法满足现代铁路列车实际工程的需要,因此亟需发展新的计算理论和数值方法。
本博士学位论文立足于现代化铁路建设的具体需求,以发展有效的复杂耦合系统随机振动响应行为预测理论与数值方法为目标,实现了有限元车辆一轨道耦合系统的高效随机振动分析及其平顺性优化,具体研究内容如下：
1.随机载荷作用下无穷周期链式结构振动分析扩展的辛数学方法
本文扩展了辛数学方法的适用范围,使其可以用于计算任意载荷作用下无穷周期链式结构的动力响应。该方法只以受力子结构为研究对象,首先利用无穷周期链式结构传递系数的性质消去子结构中不独立的自由度,得到了系数矩阵中含有波数的凝聚运动方程。然后基于这类结构的色散关系和辛数学方法的基本假定提出了离散和闭合两种子结构运动方程的求解形式：前者是将波数在区间[0,2π)上均匀离散,得到一系列的通带传播系数,则结构的响应可以通过对通带频率响应进行累加而得到；后者则是将凝聚运动方程的系数矩阵和响应进行傅立叶展开,通过波数与时间的变量分离而得到。最后将该方法与虚拟激励法相结合,实现了任意平稳／非平稳随机载荷作用下无穷周期链式结构的随机振动分析。
2.周期系数时变系统随机振动分析的周期拟稳态方法
本文基于虚拟激励的具体形式、变量分离手段和Schur分解技巧,提出了一种用于周期系数时变系统随机振动分析的周期拟稳态方法。首先采用虚拟激励法将平稳／非平稳的随机激励转化为确定性的虚拟激励,建立了虚拟激励作用下时变系统的周期系数运动方程,并将其引入状态空间；然后将载荷向量中与频率相关的项进行分离,通过对系统进行一个周期内的逐步积分运算,求解出耦合系统状态空间下的周期状态转移矩阵和周期载荷系数向量；最后利用方程解的周期性和Schur分解技巧将传统计算方法中的逐步积分过程转化为了系数矩阵为上三角阵的线性方程组求解问题,从而将虚拟激励作用下系统的确定性时间历程分析转化为了简单的拟稳态响应分析,计算效率较之基于逐步积分方法的非平稳随机振动虚拟激励法又提高了1-2个数量级。
3.多体动力学车辆—轨道耦合系统高效随机振动分析
本文采用虚拟激励法、辛数学方法、对称性凝聚方法和周期拟稳态方法对高低、方向和水平三类轨道不平顺同时作用下二维垂向／三维多体动力学车辆—轨道耦合系统的随机动力响应进行了求解。具体做法是：首先采用虚拟激励法将上述三类轨道不平顺转化为一系列相应的虚拟简谐轨道不平顺,将时变耦合系统受多源多点完全相干平稳激励的随机振动分析问题转化为三类广义单点虚拟轨道不平顺分别单独作用下耦合系统确定性响应的求解问题。在此基础上,根据车辆定点和车辆移动两种多体动力学车辆—轨道耦合系统分析模型各自的特点,结合虚拟激励的具体形式,将辛数学方法、扩展的辛数学方法和对称性凝聚方法分别应用于建立耦合系统两种分析模型的二维垂向和三维运动方程,极大程度地降低了耦合系统的计算自由度。最后采用周期拟稳态方法将车辆移动模型的运动方程整理为拟稳态求解形式,通过求解线性方程组方便地得到了车辆定点和车辆移动两种模型的虚拟响应,进而得到了响应的功率谱和标准差。通过分析结果对车辆定点和车辆移动两种模型进行了对比,对车辆速度、轨道不平顺等级等参数进行了讨论,并对车辆—轨道耦合系统随机动力响应的传递机理进行了探讨。
4.精细有限元车辆—轨道耦合系统高效随机振动分析
本文基于多体动力学车辆—轨道耦合系统随机振动的分析方法,在普通的个人电脑上首次实现了具有数十万自由度的精细有限元车辆—轨道耦合系统精确、高效的随机振动分析。车辆各构件采用有限元软件Ansys建模,利用其自由模态、通过一系、二系悬挂装置将它们连接在一起；轨道采用三维三层离散点支承梁模型。首先应用虚拟激励法将高低、方向、水平和轨距四类轨道不平顺转化为相应的简谐虚拟轨道不平顺,从根本上解决了对精细有限元模型进行随机振动分析时计算效率极其低下的问题；在此基础上,将有限元车辆运动方程与低自由度的轨道结构辛运动方程耦合在一起,求解得到了耦合系统的虚拟响应,进而得到了响应的功率谱和标准差。最后对车辆各构件弹性振动的发生机理进行了深入的研究,并绘制了车辆各构件的随机动力响应标准差云图。通过这些云图可以直观地查看轨道不平顺作用下车辆各构件不同部位的随机振动总体情况,以此确定车辆结构中振动较强的薄弱部位,为实际工程中车辆的设计和维护提供了有力的依据。
5.基于有限元车辆—轨道耦合系统的车辆平顺性优化
本文在精细有限元车辆—轨道耦合系统高效随机振动分析的基础上,采用具有百万量级自由度的精细车体有限元刚—柔混合车辆—轨道耦合系统分析模型实现了国产某高速列车头车悬架系统的平顺性优化设计。优化中选取车体地板上54个热点作为研究对象,采用IS02631国际标准对其舒适度进行了评估；提出了以—系和二系悬挂装置的刚度和阻尼共8个参数为设计变量、以各点舒适度指标的最大值为目标函数的最小—最大优化问题；采用K-S函数对目标函数进行拟合,使其具备良好的整体性、光滑性和可微性；综合运用虚拟激励法、辛数学方法和对称性凝聚方法对目标函数值进行了精确、高效的求解,并进一步发展了这些方法,得到了目标函数的1阶和2阶解析敏度,由此很好地解决了上述优化中的困难。采用该优化方法将某国产高速列车头车的平顺性指标最大值降低了58.34%,并对不同车速、不同轨道谱等级的优化结果进行了对比。
Due to the outstanding energy and environmental issues and growing transportation needs, the development of high-speed and heavy haul technologies are becoming the main directions on railway development. This considerably increases the interaction between the vehicle and the track, the bridge, the ground, the catenary and the surrounding buildings, which has a direct impact on a range of issues of the train and the railway line, e.g. safety, ride comfort, fatigue, noise, etc. The track irregularity is known as the most important source of random excitation of the railway vehicle, hence the random vibration analysis of the coupled vehicle-track systems subjected to the track irregularity, especially the calculation of response power spectral densities (PSDs) plays a very important role in the dynamic analysis for railway transportation.
However, the degrees of freedom (DOFs) of the track models are usually considerably large and the computational efficiencies of both the conventional frequency-domain and time-domain random vibration analysis methods are very low, which undoubtedly makes the random vibration analysis of the complicated coupled vehicle-track systems particularly challenging. As a result, only very simple models have previously been used and it is in urgent need to develop new theories and numerical methods.
In order to meet the specific needs of practical engineering, some valuable exploration and research for random vibration analysis and optimization of coupled vehicle-track systems are performed in this thesis. The main work can be summarized as follows:
1. Generalized symplectic random vibration analysis for infinite long chain-type structures
In this thesis, the symplectic mathematical method is generalized for the first time to investigate the transient response of an infinitely long chain-type structure subjected to arbitrary loads. This method simply needs to establish the equation of motion of the loaded substructure. The dependent DOFs are firstly condensed into the independent ones according to the properties of the wave propagation constants. Consequently, the condensed equation of motion, whose coefficient matrices are functions of the wave number, is derived. Two solving forms, i.e. the discrete form and closed form, are then performed based on the dispersion relations of such structures and the basic assumptions of symplectic method to obtain the transient responses. The former form is established by desecrating the wave number evenly in the interval [0,2n), so that the corresponding propagation constants are derived. This enables the response of the infinitely periodic structure to be obtained by accumulating the pass-band frequency responses. The latter form is established by applying Fourier expansions to these coefficient matrices, so that the response vectors, the time and wave number variables are easily separated accordingly. Finally, the resulting equations are combined with PEM for stationary or non-stationary random response analysis.
2. Pseudo-steady state approach for random vibration analysis of periodic time-dependent systems
A pseudo-steady state approach for random vibration analysis of periodic time-dependent systems is developed based on PEM, separation of variables and the Schur decomposition scheme. The periodically time-varying equation of motion of the coupled system subjected to pseudo-excitation is firstly established by transforming the stationary/non-stationary random excitation into deterministic pseudo-excitation using PEM, which is then rewritten as a first order linear differential equation group with periodic coefficients in state-space. Since the frequency-dependent terms are separated from the load vector to avoid repeated computations for different frequencies associated with the pseudo-excitations, the periodic state transition matrix and periodic load vector of the state-space equation of motion are then derived using a step-by-step integration scheme over only one period. Finally, based on the periodicity of the solution, the Schur decomposition scheme is performed to further transform the former pseudo response problem for this time-dependent system into a set of linear equations whose coefficient matrix is upper triangular. Thus the conventional periodically time-history analysis is transformed in to a pseudo-steady state response analysis and an improvement of the computational efficiency by above1-2orders of magnitude can be achieved as compared with the non-stationary PEM approach based on numerical integration method.
3. Efficient random vibration analysis of coupled multi-body vehicle-track systems
The random responses of the2-dimentional (2D) vertical and3-dimentional (3D) multi-body models of coupled vehicle-track systems subjected to three types of rail irregularities, i.e. the longitudinal level irregularity, alignment irregularity and the cross level irregularity are achieved using the PEM, the symplectic mathematical method, the symmetric condensation approach and the pseudo-steady state approach. PEM is firstly applied to convert the complicated random vibration analysis subjected to multi-sourse multi-point fully-coherent non-stationary random excitations into simple deterministic response analysis subjected to generalized single-point harmonic excitations by transforming the above three types of rail irregularities into corresponding harmonic pseudo-excitations. The symplectic method, the generalized symplectic method and the symmetric condensation approach are then applied to establish the equations of motion of2D/3D F-V/M-V models based on the concrete forms of pseudo-excitations and the characteristics of F-V/M-V models, which considerably reduces the computational DOFs of the coupled system. Finally, the pseudo-steady state approach is applied to transform the equations of motion of the M-V models into pseudo-steady state form so that the pseudo-responses of both the F-V and M-V models can be calculated by solving linear equations, after which the response PSDs and the standard deviations can be derived conveniently. Based on the calculation results, the F-V model and the M-V model are compared; the influences of vehicle velocity and class of track on system responses are discussed; the transmission mechanism of random vibration in the coupled system is also investigated.
4. FEM-based random vibration analysis of coupled vehicle-track systems
In this thesis, an accurate and efficient random vibration analysis of a well-meshed FE coupled vehicle-track system with hundreds of thousands of DOFs is performed on a ordinary personal computer for the first time based on the multi-body model approach. The vehicle components are modeled by Ansys software and then connected by the1st and2nd suspension system using their free modals. The track is modeled as a3D discrete-supported structure with three layers. PEM is firstly used to transform the four types of rail irregularities, i.e. the longitudinal level irregularity, alignment irregularity, the cross level irregularity and the gauge irregularity, into corresponding harmonic pseudo-excitations, which fundamentally solves the difficulties in the random vibration analysis of well-meshed FE model. Based on this, the equation of motion of the FE vehicle model and the low-DOFs symplectic equation of motion of the3D track model are coupled and then solved to derive the pseudo-response, after which the response PSD and the standard deviation of the coupled system can be derived conveniently. Finally, the elastic vibration mechanism of the vehicle is studied. Some standard deviation nephograms of vehicle components are also drawn, through which the overall random vibration behavior and vulnerable parts of these components can be viewed intuitively. This provides a powerful foundation for vehicle design and maintenance in practical engineering.
5. FEM-based riding comfort optimization using a coupled vehicle-track system
Based on the above-mentioned FEM-based random vibration analysis of coupled vehicle-track systems, a riding comfort optimization approach on the suspension system of a domestic high-speed railway train is developed using a coupled FE body rigid-flexible hybrid vehicle-track model with millions of DOFs is achieved. The min-max optimization approach is utilized to improve the train riding comfort with related8parameters of the suspension structure adopted as design variables, in which54design points on the vehicle floor are chosen as estimation locations and the international standard ISO-2631is used to evaluate the riding comfort. The K-S function is applied to fit the objective function to make it smooth, differentiable and have superior integrity, which is then solved accurately and efficiently using the PEM, the symplectic method and the symmetric condensation approach. The1st and2nd order analytical sensitivities are derived by further developing these methods and so the difficulties in the optimization are fundamentally solved. The max weighted RMS acceleration among the54design points reduced by58.34%after optimization, the influences of vehicle velocities and class of track irregularity on optimization results are also discussed.
然而,由于轨道结构庞大的计算自由度和传统频域和时域随机振动分析方法的低效,对上述问题进行求解需要花费高额的计算成本,迄今为止对这一领域的研究还大都只能局限于采用较简单的分析模型,这显然无法满足现代铁路列车实际工程的需要,因此亟需发展新的计算理论和数值方法。
本博士学位论文立足于现代化铁路建设的具体需求,以发展有效的复杂耦合系统随机振动响应行为预测理论与数值方法为目标,实现了有限元车辆一轨道耦合系统的高效随机振动分析及其平顺性优化,具体研究内容如下：
1.随机载荷作用下无穷周期链式结构振动分析扩展的辛数学方法
本文扩展了辛数学方法的适用范围,使其可以用于计算任意载荷作用下无穷周期链式结构的动力响应。该方法只以受力子结构为研究对象,首先利用无穷周期链式结构传递系数的性质消去子结构中不独立的自由度,得到了系数矩阵中含有波数的凝聚运动方程。然后基于这类结构的色散关系和辛数学方法的基本假定提出了离散和闭合两种子结构运动方程的求解形式：前者是将波数在区间[0,2π)上均匀离散,得到一系列的通带传播系数,则结构的响应可以通过对通带频率响应进行累加而得到；后者则是将凝聚运动方程的系数矩阵和响应进行傅立叶展开,通过波数与时间的变量分离而得到。最后将该方法与虚拟激励法相结合,实现了任意平稳／非平稳随机载荷作用下无穷周期链式结构的随机振动分析。
2.周期系数时变系统随机振动分析的周期拟稳态方法
本文基于虚拟激励的具体形式、变量分离手段和Schur分解技巧,提出了一种用于周期系数时变系统随机振动分析的周期拟稳态方法。首先采用虚拟激励法将平稳／非平稳的随机激励转化为确定性的虚拟激励,建立了虚拟激励作用下时变系统的周期系数运动方程,并将其引入状态空间；然后将载荷向量中与频率相关的项进行分离,通过对系统进行一个周期内的逐步积分运算,求解出耦合系统状态空间下的周期状态转移矩阵和周期载荷系数向量；最后利用方程解的周期性和Schur分解技巧将传统计算方法中的逐步积分过程转化为了系数矩阵为上三角阵的线性方程组求解问题,从而将虚拟激励作用下系统的确定性时间历程分析转化为了简单的拟稳态响应分析,计算效率较之基于逐步积分方法的非平稳随机振动虚拟激励法又提高了1-2个数量级。
3.多体动力学车辆—轨道耦合系统高效随机振动分析
本文采用虚拟激励法、辛数学方法、对称性凝聚方法和周期拟稳态方法对高低、方向和水平三类轨道不平顺同时作用下二维垂向／三维多体动力学车辆—轨道耦合系统的随机动力响应进行了求解。具体做法是：首先采用虚拟激励法将上述三类轨道不平顺转化为一系列相应的虚拟简谐轨道不平顺,将时变耦合系统受多源多点完全相干平稳激励的随机振动分析问题转化为三类广义单点虚拟轨道不平顺分别单独作用下耦合系统确定性响应的求解问题。在此基础上,根据车辆定点和车辆移动两种多体动力学车辆—轨道耦合系统分析模型各自的特点,结合虚拟激励的具体形式,将辛数学方法、扩展的辛数学方法和对称性凝聚方法分别应用于建立耦合系统两种分析模型的二维垂向和三维运动方程,极大程度地降低了耦合系统的计算自由度。最后采用周期拟稳态方法将车辆移动模型的运动方程整理为拟稳态求解形式,通过求解线性方程组方便地得到了车辆定点和车辆移动两种模型的虚拟响应,进而得到了响应的功率谱和标准差。通过分析结果对车辆定点和车辆移动两种模型进行了对比,对车辆速度、轨道不平顺等级等参数进行了讨论,并对车辆—轨道耦合系统随机动力响应的传递机理进行了探讨。
4.精细有限元车辆—轨道耦合系统高效随机振动分析
本文基于多体动力学车辆—轨道耦合系统随机振动的分析方法,在普通的个人电脑上首次实现了具有数十万自由度的精细有限元车辆—轨道耦合系统精确、高效的随机振动分析。车辆各构件采用有限元软件Ansys建模,利用其自由模态、通过一系、二系悬挂装置将它们连接在一起；轨道采用三维三层离散点支承梁模型。首先应用虚拟激励法将高低、方向、水平和轨距四类轨道不平顺转化为相应的简谐虚拟轨道不平顺,从根本上解决了对精细有限元模型进行随机振动分析时计算效率极其低下的问题；在此基础上,将有限元车辆运动方程与低自由度的轨道结构辛运动方程耦合在一起,求解得到了耦合系统的虚拟响应,进而得到了响应的功率谱和标准差。最后对车辆各构件弹性振动的发生机理进行了深入的研究,并绘制了车辆各构件的随机动力响应标准差云图。通过这些云图可以直观地查看轨道不平顺作用下车辆各构件不同部位的随机振动总体情况,以此确定车辆结构中振动较强的薄弱部位,为实际工程中车辆的设计和维护提供了有力的依据。
5.基于有限元车辆—轨道耦合系统的车辆平顺性优化
本文在精细有限元车辆—轨道耦合系统高效随机振动分析的基础上,采用具有百万量级自由度的精细车体有限元刚—柔混合车辆—轨道耦合系统分析模型实现了国产某高速列车头车悬架系统的平顺性优化设计。优化中选取车体地板上54个热点作为研究对象,采用IS02631国际标准对其舒适度进行了评估；提出了以—系和二系悬挂装置的刚度和阻尼共8个参数为设计变量、以各点舒适度指标的最大值为目标函数的最小—最大优化问题；采用K-S函数对目标函数进行拟合,使其具备良好的整体性、光滑性和可微性；综合运用虚拟激励法、辛数学方法和对称性凝聚方法对目标函数值进行了精确、高效的求解,并进一步发展了这些方法,得到了目标函数的1阶和2阶解析敏度,由此很好地解决了上述优化中的困难。采用该优化方法将某国产高速列车头车的平顺性指标最大值降低了58.34%,并对不同车速、不同轨道谱等级的优化结果进行了对比。
Due to the outstanding energy and environmental issues and growing transportation needs, the development of high-speed and heavy haul technologies are becoming the main directions on railway development. This considerably increases the interaction between the vehicle and the track, the bridge, the ground, the catenary and the surrounding buildings, which has a direct impact on a range of issues of the train and the railway line, e.g. safety, ride comfort, fatigue, noise, etc. The track irregularity is known as the most important source of random excitation of the railway vehicle, hence the random vibration analysis of the coupled vehicle-track systems subjected to the track irregularity, especially the calculation of response power spectral densities (PSDs) plays a very important role in the dynamic analysis for railway transportation.
However, the degrees of freedom (DOFs) of the track models are usually considerably large and the computational efficiencies of both the conventional frequency-domain and time-domain random vibration analysis methods are very low, which undoubtedly makes the random vibration analysis of the complicated coupled vehicle-track systems particularly challenging. As a result, only very simple models have previously been used and it is in urgent need to develop new theories and numerical methods.
In order to meet the specific needs of practical engineering, some valuable exploration and research for random vibration analysis and optimization of coupled vehicle-track systems are performed in this thesis. The main work can be summarized as follows:
1. Generalized symplectic random vibration analysis for infinite long chain-type structures
In this thesis, the symplectic mathematical method is generalized for the first time to investigate the transient response of an infinitely long chain-type structure subjected to arbitrary loads. This method simply needs to establish the equation of motion of the loaded substructure. The dependent DOFs are firstly condensed into the independent ones according to the properties of the wave propagation constants. Consequently, the condensed equation of motion, whose coefficient matrices are functions of the wave number, is derived. Two solving forms, i.e. the discrete form and closed form, are then performed based on the dispersion relations of such structures and the basic assumptions of symplectic method to obtain the transient responses. The former form is established by desecrating the wave number evenly in the interval [0,2n), so that the corresponding propagation constants are derived. This enables the response of the infinitely periodic structure to be obtained by accumulating the pass-band frequency responses. The latter form is established by applying Fourier expansions to these coefficient matrices, so that the response vectors, the time and wave number variables are easily separated accordingly. Finally, the resulting equations are combined with PEM for stationary or non-stationary random response analysis.
2. Pseudo-steady state approach for random vibration analysis of periodic time-dependent systems
A pseudo-steady state approach for random vibration analysis of periodic time-dependent systems is developed based on PEM, separation of variables and the Schur decomposition scheme. The periodically time-varying equation of motion of the coupled system subjected to pseudo-excitation is firstly established by transforming the stationary/non-stationary random excitation into deterministic pseudo-excitation using PEM, which is then rewritten as a first order linear differential equation group with periodic coefficients in state-space. Since the frequency-dependent terms are separated from the load vector to avoid repeated computations for different frequencies associated with the pseudo-excitations, the periodic state transition matrix and periodic load vector of the state-space equation of motion are then derived using a step-by-step integration scheme over only one period. Finally, based on the periodicity of the solution, the Schur decomposition scheme is performed to further transform the former pseudo response problem for this time-dependent system into a set of linear equations whose coefficient matrix is upper triangular. Thus the conventional periodically time-history analysis is transformed in to a pseudo-steady state response analysis and an improvement of the computational efficiency by above1-2orders of magnitude can be achieved as compared with the non-stationary PEM approach based on numerical integration method.
3. Efficient random vibration analysis of coupled multi-body vehicle-track systems
The random responses of the2-dimentional (2D) vertical and3-dimentional (3D) multi-body models of coupled vehicle-track systems subjected to three types of rail irregularities, i.e. the longitudinal level irregularity, alignment irregularity and the cross level irregularity are achieved using the PEM, the symplectic mathematical method, the symmetric condensation approach and the pseudo-steady state approach. PEM is firstly applied to convert the complicated random vibration analysis subjected to multi-sourse multi-point fully-coherent non-stationary random excitations into simple deterministic response analysis subjected to generalized single-point harmonic excitations by transforming the above three types of rail irregularities into corresponding harmonic pseudo-excitations. The symplectic method, the generalized symplectic method and the symmetric condensation approach are then applied to establish the equations of motion of2D/3D F-V/M-V models based on the concrete forms of pseudo-excitations and the characteristics of F-V/M-V models, which considerably reduces the computational DOFs of the coupled system. Finally, the pseudo-steady state approach is applied to transform the equations of motion of the M-V models into pseudo-steady state form so that the pseudo-responses of both the F-V and M-V models can be calculated by solving linear equations, after which the response PSDs and the standard deviations can be derived conveniently. Based on the calculation results, the F-V model and the M-V model are compared; the influences of vehicle velocity and class of track on system responses are discussed; the transmission mechanism of random vibration in the coupled system is also investigated.
4. FEM-based random vibration analysis of coupled vehicle-track systems
In this thesis, an accurate and efficient random vibration analysis of a well-meshed FE coupled vehicle-track system with hundreds of thousands of DOFs is performed on a ordinary personal computer for the first time based on the multi-body model approach. The vehicle components are modeled by Ansys software and then connected by the1st and2nd suspension system using their free modals. The track is modeled as a3D discrete-supported structure with three layers. PEM is firstly used to transform the four types of rail irregularities, i.e. the longitudinal level irregularity, alignment irregularity, the cross level irregularity and the gauge irregularity, into corresponding harmonic pseudo-excitations, which fundamentally solves the difficulties in the random vibration analysis of well-meshed FE model. Based on this, the equation of motion of the FE vehicle model and the low-DOFs symplectic equation of motion of the3D track model are coupled and then solved to derive the pseudo-response, after which the response PSD and the standard deviation of the coupled system can be derived conveniently. Finally, the elastic vibration mechanism of the vehicle is studied. Some standard deviation nephograms of vehicle components are also drawn, through which the overall random vibration behavior and vulnerable parts of these components can be viewed intuitively. This provides a powerful foundation for vehicle design and maintenance in practical engineering.
5. FEM-based riding comfort optimization using a coupled vehicle-track system
Based on the above-mentioned FEM-based random vibration analysis of coupled vehicle-track systems, a riding comfort optimization approach on the suspension system of a domestic high-speed railway train is developed using a coupled FE body rigid-flexible hybrid vehicle-track model with millions of DOFs is achieved. The min-max optimization approach is utilized to improve the train riding comfort with related8parameters of the suspension structure adopted as design variables, in which54design points on the vehicle floor are chosen as estimation locations and the international standard ISO-2631is used to evaluate the riding comfort. The K-S function is applied to fit the objective function to make it smooth, differentiable and have superior integrity, which is then solved accurately and efficiently using the PEM, the symplectic method and the symmetric condensation approach. The1st and2nd order analytical sensitivities are derived by further developing these methods and so the difficulties in the optimization are fundamentally solved. The max weighted RMS acceleration among the54design points reduced by58.34%after optimization, the influences of vehicle velocities and class of track irregularity on optimization results are also discussed.
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