Non-uniform Euler-Bernoulli beams under a single moving oscillator: An approximate analytical solution in time domain

Dynamic responses of simply supported non-uniform beams traversed by a moving oscillator are analysed in this paper. An approximate analytical method based on Rayleigh-Ritz (R-R) formulation is developed. The fundamental approximate mode obtained from R-R method is used in the present formulation to determine the responses of the beam and the oscillator. Effects of surface irregularities on the displacement and acceleration responses of the beam and the vehicle are also analysed. The results are compared with those obtained using Finite element method (FEM). A numerical example is provided to illustrate the validity of the present method which shows that the proposed method is simple, computationally more efficient compared to FEM and gives fairly good results. Though the single-mode approach used in the present paper is a classical one and numerous studies on the responses of uniform beams under moving loads have been reported in the past, its application to non-uniform beams (for which there does not exist any closed form expression for mode shapes) under a moving load, especially a moving oscillator, is presented for the first time.     

基于广义正交函数和正则化的移动荷载识别法

为了识别桥上移动荷载,把车/桥系统抽象为承受一组移动荷载的简支梁,用有限元方法建立桥梁振动方程,根据测试的桥梁响应,由广义正交函数根据模态叠加原理确定模态响应及其导数,用正则化技巧得到稳定的识别结果。数值模拟和试验结果表明,该方法用于识别桥上移动荷载是有效的、可行的     

Critical speeds and the response of a tensioned beam on an elastic foundation to repetitive moving loads

A finite-length tensioned beam on a damped elastic foundation is acted upon by an infinite series of equally spaced and steadily moving concentrated transverse loads. The deflection response of the beam is obtained by an expansion in terms of the normal modes of vibration. Numerical results are determined for various values of the load-spacing, beam tension, foundation stiffness and damping, and for a range of load-speeds. It is found that the critical velocities for repetitive loading exist at significantly lower speeds than would be expected based upon the well-known critical speed for a single moving load. An interpretation in terms of forced vibration response is given.     

Dynamic analysis of composite beam subjected to harmonic moving load based on the third-order shear deformation theory

The response of an infinite Timoshenko beam subjected to a harmonic moving load based on the third-order shear deformation theory (TSDT) is studied. The beam is made of laminated composite, and located on a Pasternak viscoelastic foundation. By using the principle of total minimum potential energy, the governing partial differential equations of motion are obtained. The solution is directed to compute the deflection and bending moment distribution along the length of the beam. Also, the effects of two types of composite materials, stiffness and shear layer viscosity coefficients of foundation, velocity and frequency of the moving load over the beam response are studied. In order to demonstrate the accuracy of the present method, the results TSDT are compared with the previously obtained results based on first-order shear deformation theory, with which good agreements are observed.     

Dynamic response of a finite length euler-bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force

In this paper the dynamic response of a simply-supported, finite length Euler-Bernoulli beam with uniform cross-section resting on a linear and nonlinear viscoelastic foundation acted upon by a moving concentrated force is studied. The Galerkin method is utilized in order to solve the governing equations of motion. Results are compared with the finite element solution for the linear foundation model in order to validate the accuracy of the solution technique. A good agreement between the two solution techniques is observed. The effect of the nonlinearity of foundation stiffness on beam displacement is analyzed for different damping ratios and different speeds of the moving load. The results for the time response of the midpoint of the beam are presented graphically.     

Forced responses of cracked cantilever beams subjected to a concentrated moving load

An analytical method is developed to present the dynamic response of a cracked cantilever beam subject to a concentrated moving load. The cracked beam system is modeled as a two-span beam and each span of the continuous beam is assumed to obey Euler–Bernoulli beam theory. The crack is modeled as a rotational spring with sectional flexibility. Considering the compatibility requirements on the crack, the relationships between these two spans can be obtained. By using the analytical transfer matrix method, eigensolutions of this cracked system are obtained explicitly. The forced responses can be obtained by the modal expansion theory using the determined eigenfunctions. Some numerical results are shown to present the crack effects (crack extent, location of the crack) and are studied for different speeds of the moving load.     

由梁的响应识别移动荷载

介绍了由梁的响应识别梁上移动荷移的方法，讨论了测试误差、荷载数、荷间隔、荷载移动速度等因素对识别结果的影响。比较了单点多时，同时多点的位移响应或应变响应用以识别移动荷载的效果。     

基于ANN和动应变的梁桥移动荷载识别及试验

为了实现桥梁上车载参数的快速识别,基于欧拉梁动力解析解分析桥梁挠度和应变对移动荷载的敏感性,选择敏感性更强的应变作为输入参数,研究将人工神经网络(artificial neural networks,简称ANN)用于识别梁桥移动车载的理论和方法。对简支梁桥在移动车载作用下的动应变响应进行理论分析及数值模拟,选取不同工况下的模拟数据对网络进行训练,分析激活函数组合和训练方法对网络精度的影响及噪声水平对动荷载工况下正确识别率的影响。通过车-桥模型动力试验验证该方法的合理性和可用性。结果表明,不同激活函数组合对识别结果影响较小,而不同的训练算法对识别结果影响较大,在应用神经网络识别桥梁移动荷载时,可以通过桥梁的动应变,对车辆的位置、速度和动荷载进行识别。     

附有滑动质量的旋转柔性梁动力学特性研究

提出用弹簧-质量系统抑制旋转柔性梁的振动,建立Euler-Bernoulli梁的动力学模型.对方程进行无量纲化,并在质量为慢时变运动、旋转角速度和角加速度为梁变形同阶小量时,对非线性方程进行近似简化,分析弹簧-质量系统对梁振动特性的影响机理.利用多尺度方法对非线性方程近似求解,在主共振、内共振条件下,研究梁和质量运动幅值随质量位置和调协参数的变化趋势.     

Numerical study on moving train parameter identification system through a simply supported bridge

This paper presents a new experimental system for moving train parameter identification on bridge. The numerical method is adopted to simulate the experimental system and to investigate the effect of the system on identification of the moving train parameters. Displacement responses are obtained by solving the equation of motion of a simply supported bridge model. Basic parameters of the train (i.e., moving speed, wheel axle numbers, and axle spacing) are recognized with a self-designed parameter recognition system. Axle load values on bridge are identified with a load identification system based on simulated annealing genetic algorithm and by minimizing the errors between measured and reconstructed displacements from the moving train loads at each time step. Measured data are simulated by adding artificial noise to the calculated data. A series of numerical simulations is performed to analyze the effect of the system under various train situations. The results show that the experimental system is feasible and valid for identification of moving train parameters, thereby providing effective preparation for practice.     

Tensionless contact of a finite beam resting on Reissner foundation

A more generalized model of a beam resting on a tensionless Reissner foundation is presented. Compared with the Winkler foundation model, the Reissner foundation model is a much improved one. In the Winkler foundation model, there is no shear stress inside the foundation layer and the foundation is assumed to consist of closely spaced, independent springs. The presence of shear stress inside Reissner foundation makes the springs no longer independent and the foundation to deform as a whole. Mathematically, the governing equation of a beam on Reissner foundation is sixth order differential equation compared with fourth order of Winkler one. Because of this order change of the governing equation, new boundary conditions are needed and related discussion is presented. The presence of the shear stress inside the tensionless Reissner foundation together with the unknown feature of contact area/length makes the problem much more difficult than that of Winkler foundation. In the model presented here, the effects of beam dimension, gap distance, loading asymmetry and foundation shear stress on the contact length are all incorporated and studied. As the beam length increases, the results of a finite beam with zero gap distance converge asymptotically to those obtained by the previous model for an infinitely long beam.     

A further study about the behaviour of foundation piles and beams in a Winkler–Pasternak soil

The natural vibrations and critical loads of foundation beams embedded in a soil simulated with two elastic parameters through the Winkler–Pasternak (WP) model are analysed. General end supports of the beam are considered by introducing elastic constraints to transversal displacements and rotations. The solution is tackled by means of a direct variational methodology previously developed by the authors who named it as whole element method. The solution is stated by means of extended trigonometric series. This method gives rise to theoretically exact natural frequencies and critical loads. A particular behaviour arises from the analysis of the lateral soil influence. It is found that the boundary conditions of the beam are influenced by the soil at the left and right sides of the beam. The possible alternatives are that the soil be cut or dragged by the non-fixed ends of the beam. In the standard WP model, the lateral soil influence is not considered. Natural frequencies and critical load numerical values are reported for beams and piles elastically supported and for various soil parameters. The results are found with arbitrary precision depending on the number of terms taken in the series. Some unexpected modes and eigenvalues are found when the different alternatives are studied. It should be noted that this special behaviour is present only when the Pasternak contribution is taken into account.     

Dynamic analysis of an axially moving beam subject to inner pressure using finite element method

A dynamic model of an axially moving flexible beam subject to an inner pressure is present. The coupling principle between a flexible beam and inner pressure is analyzed first, and the potential energy of the inner pressure due to the beam bending is derived using the principle of virtual work. A 1D hollow beam element contain inner pressure is established. The finite element method and Lagrange’s equation are used to derive the motion equations of the axially moving system. The dynamic responses are analyzed by Newmark-β time integration method. Based on the computed dynamic responses, the effects of inner pressure on beam dynamics are discussed. Some interesting phenomenon is observed.     

The study of structural system dynamic problems by recursive estimation method

This paper presents the recursion-relation algorithm to determine the input forces of beam structures and the individual node displacement. The system state equations of the beam structures are based on the average acceleration method, and a dynamic beam matrix equation is constructed by using the finite element method. Implementation of this method involves handling various systems' parameters such as the measurement noise, the modeling error, and the external disturbance. The Kalman filter is used to predict the applied forces and adjusted at each time step according to the measured dynamic response data of structures. The estimator employs a recursive least-squares algorithm to compute the magnitudes of the impulsive loads. The accuracy of the present method is demonstrated by the numerical simulations for loads applied along the axis of a beam on different time frames.     

Vibration analysis of a simply supported beam under moving mass based on moving finite element method

In this paper, a moving finite element (MFE) method is proposed to perform the dynamic analysis of a simply supported beam for a moving mass (MM). The MFE method treats the moving mass as a moving part of the entire system, so that the transverse inertial effects caused by the moving mass may easily be taken into account. The solution to the beam’s dynamic behaviors including its displacement is obtained via a Newmark-β method; the effects of the velocity and weight of the MM on the beam’s dynamic behaviors are further discussed. The numerical examples show that the inertial effects of the MM significantly affect the transverse responses of the simply supported beam.     

Repair of a cracked Timoshenko beam subjected to a moving mass using piezoelectric patches

This paper presents an analytical method for the application of piezoelectric patches for the repair of cracked beams subjected to a moving mass. The beam equations of motion are obtained based on the Timoshenko beam theory by including the dynamic effect of a moving mass traveling along a vibrating path. The criterion used for the repair is altering the first natural frequency of the cracked beam towards that of the healthy beam using a piezoelectric patch. Conceptually, an external voltage is applied to actuate a piezoelectric patch bonded on the beam. This affects the closure of the crack so that the singularity induced by the crack tip will be decreased. The equations of motion are discretized by using the assumed modes method. The cracked beam is modeled as number of segments connected by two massless springs at the crack locations (one, extensional and the other, rotational). The relationships between any two spans can be obtained by considering the compatibility requirements on the crack section and on the ends of the piezoelectric patch. Using the analytical transfer matrix method, eigensolutions of the system can be calculated explicitly. Finally, numerical simulations are performed with respect to different conditions such as the moving load velocity. It is seen that when the piezoelectric patch is used, the maximum deflection of the cracked beam approaches maximum deflection of the healthy beam.     

Vibration analysis of a simply supported beam under moving mass based on moving finite element method

In this paper, a moving finite element (MFE) method is proposed to perform the dynamic analysis of a simply supported beam for a moving mass (MM). The MFE method treats the moving mass as a moving part of the entire system, so that the transverse inertial effects caused by the moving mass may easily be taken into account. The solution to the beam’s dynamic behaviors including its displacement is obtained via a Newmark-β method; the effects of the velocity and weight of the MM on the beam’s dynamic behaviors are further discussed. The numerical examples show that the inertial effects of the MM significantly affect the transverse responses of the simply supported beam.     

移动质量激励下梁动态响应分析与试验研究

以移动质量激励下的梁为研究对象,建立了移动质量-简支梁和移动质量-悬臂梁耦合振动模型.考虑了移动质量的重力、Corilous加速度和惯性加速度对梁横向振动的影响,利用得到的级数解编制程序分析了移动质量参数变化对梁动态响应的影响.建立了移动质量激励下的梁动态响应试验平台,通过试验得到了各种规格质量块以不同运动速度激励作用下的梁动态响应曲线.试验结果表明了移动质量激励下梁动态响应模型的正确性.     

Minimum weight design of beams against failure under uncertain loading by convex analysis

Under operational conditions, some loads acting on a beam are known (deterministic loads), but there usually exist other loads the magnitude and distribution of which are unpredictable (uncertain loads). If the uncertainty in the loading is not taken into account in the design, the likelihood of failure increases. In the present study beams are designed for minimum weight subject to maximum stress and buckling load criteria and under deterministic and uncertain transverse loads. The uncertain load, which is subject to a constraint on its L ₂ norm, is determined to maximize the normal stress using a convex analysis. The location of the maximum stress is determined under the combination of deterministic and worst-case uncertain loads. The minimum weight design is obtained by determining the minimum cross-sectional area subject to stress and buckling load constraints. Results are given for a number of problem parameters including the axial load, elastic foundation modulus and uncertainty levels.     

移动质量简支梁耦合时变系统建模与实验设计

建立了移动质量简支梁耦合时变系统的动力学模型,通过数值仿真分析了移动质量速度及加速度对耦合时变系统模态参数的影响,得到移动质量诱导产生的附加阻尼。设计并搭建移动质量简支梁实验系统,通过参考实验得到实验系统的初始阻尼,并分别采用频域和时域模态参数辨识方法对质量块不同移动速度下的实验系统进行辨识。结果表明,所建立动力学模型能够对移动质量问题进行准确描述,实验系统可为时变结构动力学分析的理论研究提供实验支持,特别是对时变结构模态参数辨识方法进行实验验证。