Natural Frequencies of Bernoulli–Euler Beams Resting on Two Elastic Supports: Application to Railway Vehicles

SUMMARY

The purpose of the present study is to investigate the effect of the different parameters of the elastic supports upon which a deformable beam lies, upon the natural frequencies of the system.

The influence of support parameters is obtained by exploiting characteristic transcendental equations. The natural flexural vibrations of the beam are only affected by the stiffness of the supports over a very restricted range; outside this range the frequencies are those of a completely free beam or of a beam supported by two simple rigid supports (infinite stiffness).     

宽幅下承式空间多索面异型系杆拱桥设计关键技术

以外秦淮河大桥为工程背景,采用Midas Civil对该异型拱桥进行仿真模拟,首先探讨了不同主梁刚度、拱肋刚度两个参数条件下的拱桥结构受力性能影响规律;其次考虑异型拱桥结构最不利情况,进行了强度、刚度验算分析;另外计算了桥梁结构前五阶振型及自振频率,分析了其动力特性;最后针对拱肋可能会发生的失稳情况,进行了拱肋稳定性设计分析。结果表明：主梁与拱肋的变形、应力均满足设计要求;在10%~20%范围内改变主梁、拱肋结构刚度对结构受力影响较小,外秦淮河大桥作为城市交通景观桥梁,主梁与拱肋构造设计仍有一定的优化空间;桥梁前5阶的自振频率在0.49~1.25 Hz之间,动力性能较好,为抗震设计提供了参考;该异型拱桥结构具有较好的稳定性能。     

基于模型修正的公路简支板梁桥荷载横向分布系数计算方法

为了使得理论计算的桥梁荷载横向分布系数与试验结果更加吻合，并且降低静载试验成本，在铰接板法基础上，提出一种基于模型修正技术的公路简支板梁桥荷载横向分布系数简化分析方法。该方法建立了一种可以考虑板间接缝剪切变形的简化分析模型，并推导了相应的静力和动力方程。针对简化模型的结构特点，提出了以待测桥梁动力试验测得的自振频率和桥梁跨中振型构造目标函数，以竖向弹簧刚度k、扭转弹簧刚度Ψ以及剪切弹簧刚度k_q为待识别参数的模型修正方法。通过提出的模型修正方法，得到实际状态下桥梁的主要参数，以简化模型的荷载横向分布系数影响线为基础，可计算各板梁的横向分布系数；验证了不考虑板梁间接缝剪切变形时，基于简化模型的横向分布系数分析结果与铰接板梁法相同，从而证明了所提简化模型和分析方法的可靠性。最后以一座桥梁为对象，进行了动力测试，识别了简化模型的物理参数。模型修正之后的模态频率和实测值吻合良好，同时振型之间的MAC（模态保证准则）系数也接近于1，从而表明利用所提的模型修正方法可以有效识别简化模型的物理参数，使理论模型和实际桥梁吻合。     

高速铁路斜交刚构连续梁桥车桥耦合振动分析

分别采用板单元、四片式、两片式和单片式梁单元模型来计算高速铁路斜交刚构连续梁桥的空间自振频率，通过比较和分析，推荐采用四片式梁单元进行车桥耦合振动分析。计算结果表明斜交刚构连续梁桥具有足够的刚度和良好的整体性，可以满足高速列车行车时的安全性和舒适性要求。     

装配式PC箱梁刚度退化及识别的足尺试验

为了明确装配式预应力混凝土（PC）梁刚度在损伤状态下的退化规律，针对3片30 m预应力混凝土箱梁开展了足尺模型试验研究。首先，建立了损伤梁的结构静动刚度识别模型，基于跨中挠度提出了结构等代静刚度计算式，基于自振频率提出了结构等代动刚度的计算式；其次，对足尺箱梁进行了多工况的静力逐级加载和动力交替试验，分析了箱梁裂缝、挠度、频率等特征参数随荷载的演化规律，研究了损伤状态对箱梁特征参数的影响；再次，通过有限元模型修正给出了箱梁动刚度衰减系数，研究了结构静动刚度随损伤状态的衰减规律，分析了静动刚度差异的原因；最后，建立了结构静动刚度衰减系数和裂缝特征参数之间的经验回归式，并与现有研究成果进行了对比。结果表明：初始损伤状况对箱梁受荷后的裂缝统计分布特征参数有一定影响；箱梁开裂后在相同荷载下的结构等代静、动刚度衰减程度并不相同，其中最终的静刚度衰减系数为0.30，最终的动刚度衰减系数为0.80；足尺梁与室内缩尺模型梁的静刚度衰减系数并不相同。所提出的静、动刚度衰减公式较为简洁，可用于实际损伤箱梁的结构性能评估。     

基于扩展离散元法的钢筋混凝土梁裂缝分析方法

针对传统数值模拟方法难以准确模拟钢筋混凝土(RC)梁开裂的问题,提出一种改进的扩展离散元法(EDEM)。该方法引入拉伸-剪切破坏准则,建立六边形块体模型,块体接触处存在潜在裂缝,根据EDEM计算流程,将满足破坏准则的潜在裂缝转化真实裂缝,实现裂缝的模拟。采用该方法建立二维RC梁模型,开展RC梁四点弯曲试验,结合理论计算,对其开裂模式、跨中位移及固有频率变化规律进行对比。结果表明:随着荷载增大,RC梁底部产生裂缝,当裂缝穿过梁宽度方向3/4时,向加载点方向延伸;基于RC梁跨中位移变化和固有频率连续下降的特征,可确定裂缝扩展和纵向受拉钢筋屈服导致的结构刚度降低过程;EDEM数值模拟结果与理论、试验结果基本吻合,该方法可有效模拟RC梁开裂和扩展过程。     

Dynamic and Vibration Analysis of a Vehicle Rear Axle System

A detailed finite element model for the rear axle system of a sport utility vehicle is developed in this investigation. The axle system is treated as a multibody system that consists of nine bodies that include the input shaft, two output shafts, the carrier and tube system, four control arms and a track bar. The rotating input and output shafts are mounted on the carrier and tube system using six bearings. The four control arms and the track bar are connected to the carrier system and the frame of the vehicle using rubber bushings. In the model developed in this investigation, three dimensional beam elements are used to develop the finite element model for the input and output axle shafts, the control arms, and the track bar. A non-conventional finite element formulation is used to develop the equations of motion of the rotating input and output shafts in order to account for the effect of their angular velocities. These equations are expressed in terms of inertia shape integrals that depend on the assumed displacement field. The inertia shape integrals are first evaluated for each finite element. The inertia shape integrals of the rotating shafts are obtained by assembling the inertia shape integrals of its finite elements using a standard finite element assembly procedure. A conventional finite element formulation is used for the control arms and the track bar. The model developed in this investigation includes the effect of the bearing stiffness, the effect of the stiffness of the helical springs of the suspension system, and the effect of the stiffness of the tires. Using the Lagrangian dynamics and the finite element method, the equations of motion of the axle system are developed and expressed in terms of the nodal coordinates of the shafts, the control arms and the track bar as well as the degrees of freedom of the carrier. This finite dimensional model is used to determine the mode shapes and the natural frequencies of the axle system. The discrepancies between several of the natural frequencies predicted using the dynamic model developed in this investigation and natural frequencies determined experimentally are found to be less than 2%. A parametric study is performed in order to investigate the effect of the axle system parameters on the natural frequencies and mode shapes. Using the modal transformation, a set of differential equations of motion of the axle system is developed and used to examine the system dynamics under given loading conditions. The solutions of the resulting equations of motion are obtained using numerical methods.     

Dynamic and Vibration Analysis of a Vehicle Rear Axle System

A detailed finite element model for the rear axle system of a sport utility vehicle is developed in this investigation. The axle system is treated as a multibody system that consists of nine bodies that include the input shaft, two output shafts, the carrier and tube system, four control arms and a track bar. The rotating input and output shafts are mounted on the carrier and tube system using six bearings. The four control arms and the track bar are connected to the carrier system and the frame of the vehicle using rubber bushings. In the model developed in this investigation, three dimensional beam elements are used to develop the finite element model for the input and output axle shafts, the control arms, and the track bar. A non-conventional finite element formulation is used to develop the equations of motion of the rotating input and output shafts in order to account for the effect of their angular velocities. These equations are expressed in terms of inertia shape integrals that depend on the assumed displacement field. The inertia shape integrals are first evaluated for each finite element. The inertia shape integrals of the rotating shafts are obtained by assembling the inertia shape integrals of its finite elements using a standard finite element assembly procedure. A conventional finite element formulation is used for the control arms and the track bar. The model developed in this investigation includes the effect of the bearing stiffness, the effect of the stiffness of the helical springs of the suspension system, and the effect of the stiffness of the tires. Using the Lagrangian dynamics and the finite element method, the equations of motion of the axle system are developed and expressed in terms of the nodal coordinates of the shafts, the control arms and the track bar as well as the degrees of freedom of the carrier. This finite dimensional model is used to determine the mode shapes and the natural frequencies of the axle system. The discrepancies between several of the natural frequencies predicted using the dynamic model developed in this investigation and natural frequencies determined experimentally are found to be less than 2%. A parametric study is performed in order to investigate the effect of the axle system parameters on the natural frequencies and mode shapes. Using the modal transformation, a set of differential equations of motion of the axle system is developed and used to examine the system dynamics under given loading conditions. The solutions of the resulting equations of motion are obtained using numerical methods.     

无粘结预应力混凝土梁桥自振频率的RBF网络识别

为研究采用神经网络的方法识别无粘结预应力混凝土梁桥的自振频率,收集以往PC梁的动力试验数据,并在此基础上补充制作5根PC梁进行动力试验,采集相关数据。构建径向基(RBF)神经网络,采用泛化回归神经网络(GRNN)进行函数逼近,径向基函数的光滑因子取为0.15。筛选9个影响PC梁自振频率的关键参数作为神经网络的输入参数,用收集到的试验数据对神经网络进行训练,并预留出1根PC梁的试验数据对网络进行仿真。仿真结果表明,采用所研究的神经网络方法识别无粘结预应力混凝土梁桥的自振频率是可行的,这种网络具有很好的预测能力和泛化能力。     

基于灵敏度分析的动力总成悬置系统优化

针对某车动力总成悬置系统固有频率偏高和振动耦合严重的问题,建立了该车动力总成悬置系统6自由度模型,并对悬置元件的主刚度参数进行了灵敏度分析.以分析找出的少量敏感参数为设计变量,结合振动解耦和固有频率匹配理论,对该悬置系统参数进行了优化,并对优化前后的动力总成悬置系统进行了台架试验.结果表明,优化后的悬置系统隔振能力有了...     

考虑防火涂料的PC箱梁时空温度场及刚度退化试验研究

为了明确涂有防火涂料的预应力混凝土（PC）箱梁火灾温度场及火灾后的刚度退化性能,针对3片有、无膨胀型防火涂料的PC简支箱梁开展了火灾模型试验研究。基于预应力混凝土梁受火性状推导了刚度衰变率与挠度衰变率关系方程,通过分析混凝土箱梁截面时空温度场分布状态,基于分区域理论提出了一种PC箱梁受火损伤后初始截面等效刚度简化计算方法,并基于自振频率推导了受火梁动刚度计算公式;对模型梁进行了火灾后的静力逐级加载和动力性能试验,分析了受火梁挠度、频率等静、动力参数的变化规律,研究了受火损伤状态对箱梁静、动力特征参数的影响;基于理论计算和试验数据对比分析了受火梁静、动力刚度在不同损伤状态下的退化规律。研究结果表明：膨胀型防火涂料的使用显著降低了预应力混凝土箱梁的高温敏感性,混凝土温升速率明显下降,大幅降低了混凝土结构温度,有效减少了混凝土结构开裂和爆裂现象;受火试验梁的截面静刚度随荷载增加而逐渐减小,且初始损伤越大,进入弹塑性阶段发展越快,越早达到结构极限承载力;初始损伤状态对结构截面静刚度的影响大于对结构动刚度的影响,且截面静刚度衰减速率比动刚度衰减速率更快。提出的受火梁静、动刚度及固有频率的简化计算方法可为火损箱梁的结构静、动力性能初评估提供理论依据。     

大跨度提篮式人行拱桥的振动病害分析与改造

针对柔性桥面系动力性能不佳,在行人步伐荷载作用下振动过大,甚至可能发生人桥共振的问题,对一实例桥梁进行自振特性分析,提出提篮式拱桥的前几阶竖向自振频率主要由拱肋性能决定,竖向高阶频率主要由桥面系性能决定,提高桥面系刚度不能有效错开行人步频和结构竖向自振频率,但能有效地减弱动力响应,提高桥梁的使用性能和安全性。根据此类桥梁的这一动力特性,可以有效地诊断出其振动病害并寻求合理的改造加固措施,通常有增加桥面系刚度或增设耗能减振系统两种方法,后者更有效,也可以将这两种方法结合使用。动力响应分析法能有效地进行人行桥振动病害分析和加固前后动力性能对比分析。     

商用车车架弯曲刚度研究

商用车由于使用用途、承载等设计参数的不同,再考虑到成本、重量等因素,车架的弯曲刚度要与设计目的匹配在一定的合理范围内,文章针对这一"合理范围"进行研究讨论。从梁的弯曲理论,到实际车架的弯曲刚度理论推导及有限元方法边界条件的验证,最后讨论了"轴距-弯曲刚度"及"整备质量-弯曲刚度"统计图的应用。结果表明,目前应用的车架有限元弯曲刚度计算的边界条件是合理的,通过对不同车型的车架弯曲刚度进行有限元分析结果的数据统计形成"轴距-弯曲刚度"及"整备质量-弯曲刚度"图,能够形成整车设计指标数据,为车架前期设计提供数据支持。     

The dynamic response of rail support

This research reviews principles behind the dynamic response of rail supports, and introduces a method of analysis to find the maximum response in a realistic setting. Assuming a time-dependent, moving mass with massive wheels is essential, because the ratio of the moving mass to the rail mass is significant. However, the dynamic response of the track is not affected by dynamic properties of the train other than its unsprung mass, because the natural frequencies of the train suspension and track are significantly different. A numerical method is developed to model the dynamic response based on these principles, and applied to the Korean urban transit. The dynamic response includes multiple peaks with a large amplitude range, creating noise while the wheel passes the support. The dynamic impact factor (DIF) for the rail support depends mainly on the stiffness and damping of the rail support. The DIF for the rail moment is below the code value, whether this value is based on numerical analysis or on-site measurements. However, our numerical analysis results in a DIF for support settlement that is greater than the code value, if the damping is less than 3%.     

大客车空气弹簧动态特性的试验分析

空气弹簧具有变刚度特性，其振动频率要比钢板弹簧低得多。对客车用空气簧动态特性进行了试验分析。结果表明，空气弹簧在多种载荷工况下特性曲线具有相似性，说明一种空气弹簧适合于多种载荷的汽车；空气弹簧的动刚度主要取决于激励频率，在低频范围内幅值比较小，高频范围内幅值比较大。     

空间梁单元显式切线刚度矩阵推导

在开展桥梁、房屋、网壳、网架等杆系结构几何非线性分析或极限承载力计算时,需要空间梁单元的切线刚度矩阵。基于非线性问题的一般平衡方程和空间梁单元的非线性几何方程,推导应力应变一般线弹性关系下的空间梁单元显式切线刚度矩阵,该刚度矩阵中包含了由初应力和初应变产生的初应力刚度矩阵,为空间有限元程序的编制奠定了基础。     

发动机振动传递系统建模及刚度参数影响分析

推导了某类轿车发动机悬置振动系统模型，分析了发动机对车身振动噪声的传递特性，说明该类轿车前梁和副车架是引起噪声放大的主要环节。根据移频减振降噪原理，借助实验优化设计，分析了振动传递系统的六个刚度对抑制振动噪声传递的主次关系。分析说明：降低发动机对轿车的振动噪声最有效的刚度修改措施是调整车前梁与车身联接点的刚度，比直接修改发动机三个悬置点刚度更有效。     

Investigation on the sensitive and insensitive zones of the rail support stiffness for the dynamic response of a vehicle system under low excitation frequencies

The variation of the rail support stiffness is an inherent issue of railway tracks. There is still no consensus on the influence of the rail support stiffness variation on the dynamic response of the vehicle–track system. One view indicates that changes of the support stiffness do not have considerable influence on the vehicle dynamic response. The main influence factor is the rail deflection. However, the opposite view presents that the influence of the support stiffness on the system dynamic response is obvious. Reasons that lead to the dispute of previous studies are the neglect of the influence of the excitation frequencies and a lack of understanding of stiffness sensitive zones. In this study a vehicle–track coupling model with equivalent overall support stiffness is employed to investigate the response of the vehicle to changes of the track stiffness and excitation frequencies. Results show that for each of frequencies (1–40?Hz) the dynamic response of the vehicle is only sensitive to a certain range of the support stiffness. A stiffness sensitive zone for each excitation frequency can be observed. In order to further study the influence of the subgrade on the vehicle system dynamic response a vehicle–track-subgrade model is utilised. The subgrade stiffness belonging to the stiffness sensitive zone has a significant influence on vehicle vibrations. For overall support stiffness of the rail higher than 20?kN/mm, the stiffness sensitive zones of low excitation frequencies can be avoided.     

A stochastic quarter-car model for dynamic analysis of vehicles with uncertain parameters

A quarter-car model is used to investigate the vibration response of cars with uncertainty under random road input excitations in this paper. The sprung mass, unsprung mass, suspension damping, suspension stiffness, and tyre stiffness are considered as random variables. The road irregularity is considered a Gaussian random process and modelled by means of a simple exponential power spectral density. The power spectral density, mean value, standard deviation, and variation coefficient of the vehicle's natural frequencies and mode shapes are obtained by using the Monte Carlo simulation method. The computational expressions for the numerical characteristics of the mean square value of the vehicle's random response in the frequency domain are developed by means of the random variable's functional moment method. The influences of the randomness of the vehicle's parameters on the vehicle's dynamic response are investigated in detail using a practical example, and some useful conclusions are obtained.     

模数式桥梁伸缩缝水平向动力学响应分析

运用有限元分析软件对模数式桥梁伸缩缝进行水平向动力学研究,建立了其水平向有限元动力学模型,研究了车轮对中梁的水平冲击以及车速、中梁弹性支承刚度及预压量、滑动摩擦系数和剪切弹簧刚度的变化对中梁水平位移的响应特性。研究表明,当车速高于100 km/h,中梁弹性支承刚度小于70 000 N/mm时,应考虑车轮对中梁的水平冲击,当车速低于120 km/h,中梁弹性支承刚度及预压量分别大于80 000 N/mm和0.3 mm,滑动摩擦系数大于0.03,剪切弹簧刚度大于400 N/mm时,此时中梁水平位移小于0.5 mm,且车轮对中梁的水平冲击也可不考虑。