开尔文模型粘弹性Timoshenko梁的动力分析与应用

基于Kelvin模型张量形式本构关系导出粘弹性Timoshenko梁自由振动微分方程组,给出两端简支粘弹性梁的固有频率解析解。对粘弹性梁的振动特性进行了分析和比较。以此计算材料开尔文模型粘弹性阻尼系数,结果表明,该方法准确可靠。     

移动荷载作用下弹性半空间Timoshenko梁的临界速度

根据梁的波速和半空间波速的相对关系,将Timoshenko梁-半空问系统分成四种不同情况。在梁与半空间相互作用的等效刚度和Timoshenko梁-半空间的弥散方程的基础上。利用弥散曲线，研究了移动荷载的临界速度。这四种情况分别为：软梁-硬半空间系统。次软梁-硬半空间系统，次硬梁-软半空间系统，硬梁-软半空间系统。研究表明，Timoshenko梁在移动荷载作用下的临界速度取决于梁的波速和半空间波速的相对关系；半空间的Rayleigh波波速始终是一个临界速度，当荷载速度达到Rayleigh波波速时．系统响应会趋于无穷大；对软梁-硬半空间系统，梁的剪切波速和压缩波速也是临界速度；对次软梁-硬半空间系统，梁的剪切波速是临界速度，并且还存在一个最小临界速度；对（次）硬梁-软半空间系统．粱的波速不再是临界速度。但也存在一个最小临界速度。     

轴向运动黏弹性Timoshenko梁横向非线性强受迫振动

研究轴向运动黏弹性Timoshenko梁横向非线性强受迫振动的稳态响应。由广义Hamilton变分原理推导出轴向运动黏弹性Timoshenko梁横向振动的控制方程及相应的边界条件。模型中考虑剪切模量、转动惯量对梁的影响。黏弹性本构关系中运用Kelvin模型并引入物质时间导数。对控制方程施用直接多尺度法,建立强受迫共振的可解性条件,得到稳态响应振幅与激励频率关系曲线。应用Routh-Hurwitz判据判断稳态响应振幅的稳定性。利用数值结果给出不同参数下,如非线性系数、激励振幅与黏弹性阻尼等对稳态幅频响应及稳定性影响。     

Arrays of dynamic circular loads moving on an infinite plate

Steady‐state dynamic responses of a Kirchhoff plate resting on a viscoelastic Winkler foundation subject to arrays of moving constant and harmonic loads with uniform circular distribution is studied in this paper using Fourier transform and generalized Duhamel's integral. A computational schema is constructed based on a two‐fold fast Fourier transform in a moving co‐ordinate system. Numerical case studies using a single moving load as well as four moving loads mimicking four wheel‐loads of a moving vehicle are conducted. Parameters of the plate are typical values of structural and material properties of highway pavements. Displacement responses of the plate to these five loading speeds seem to be very close to each other, though the vertical velocity and acceleration responses show considerable differences. The effect of load frequency on amplitude of response of plate is insignificant for parameters used in this study. Copyright © 2006 John Wiley & Sons, Ltd.     

弹性支持的无限铁木辛柯梁对移动振动质量的响应

针对具有弹性基础的无限梁在移动的振动质量激励下的响应问题，采用考虑剪切变形和转动惯量的铁木辛柯梁理论建立梁的微分方程，并利用双重傅立叶变换求解，得到梁的运动方程。最后，通过一个数值计算的实例，分析了振动质量以及其移动的速度对梁的响应的影响。结果表明，振动质量本身对梁的响应的影响不可忽视。     

移动荷载作用下轨道基础刚度突变对轨道振动的影响

通过建立移动荷载作用下具有基础弹性刚度突变连续梁的振动微分方程，得到了梁的变形解析表达式。利用该解析解和叠加原理，以车辆通过铁路轨道为例，研究了轨道基础弹性突变对轨道振动的影响。分析了单轮对和一台TGV高速动车通过五种轨道刚度比时的轨道动力响应。计算表明，基础弹性突变对梁的振动有较大影响，其影响随着移动荷载速度的增加而增加。     

Dynamic stiffness vibration analysis using a high‐order beam model

This work presents the derivation of the exact dynamic stiffness matrix for a high‐order beam element. The terms are found directly from the solutions of the differential equations that describe the deformations of the cross‐section according to the high‐order theory, which include cubic variation of the axial displacements over the cross‐section of the beam. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments. Using the dynamic stiffness matrix exact vibration frequencies for beams with various combinations of boundary conditions are tabulated and compared with results from the Bernoulli–Euler and Timoshenko beam models. Copyright © 2003 John Wiley & Sons, Ltd.     

Dynamic analysis of an axially moving sandwich beam with viscoelastic core

A development of the beam model of the axially moving sandwich continua with elastic faces and the core characterized by viscoelastic properties is presented in this paper. Two-parameter Kelvin–Voigt rheological model is used to describe material properties of the core. The Galerkin method is used to solve the governing partial differential equation. Dynamic analysis of the composite with two aluminum facings and a polyurethane core is carried out. The effect of the transport speed, the core thickness and the internal damping of the core material on the dynamic behavior of the system is investigated in undercrtitical and supercritical range of transport speed.     

Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load

This paper studies the dynamic response of functionally graded beams with an open edge crack resting on an elastic foundation subjected to a transverse load moving at a constant speed. It is assumed that the material properties follow an exponential variation through the thickness direction. Theoretical formulations are based on Timoshenko beam theory to account for the transverse shear deformation. The cracked beam is modeled as an assembly of two sub-beams connected through a linear rotational spring. The governing equations of motion are derived by using Hamilton’s principle and transformed into a set of dynamic equations through Galerkin’s procedure. The natural frequencies and dynamic response with different end supports are obtained. Numerical results are presented to investigate the influences of crack location, crack depth, material property gradient, slenderness ratio, foundation stiffness parameters, velocity of the moving load and boundary conditions on both free vibration and dynamic response of cracked functionally graded beams.     

A new method of shear stiffness prediction of periodic Timoshenko beams

This article interprets the new implementation of an asymptotic homogenization method for effective bending stiffness of heterogeneous beam structures with periodic microstructure along its axial direction in an intuitionistic way. With this interpretation, the authors then develop a new method of evaluating effective shear stiffness for their Timoshenko beam model. This method can be easily implemented numerically in commercial software. Different kinds of elements and modeling techniques available in commercial software can be applied to model the unit cell. Several examples are given to demonstrate the effectiveness of this new method.     

Non‐linear spatial Timoshenko beam element with curvature interpolation

The paper presents a spatial Timoshenko beam element with a total Lagrangian formulation. The element is based on curvature interpolation that is independent of the rigid‐body motion of the beam element and simplifies the formulation. The section response is derived from plane section kinematics. A two‐node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence. The formulation is extended to N‐node elements with polynomial curvature interpolation. Models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step. Generalized second‐order stress resultants are identified and the section response takes into account non‐linear material behaviour. Green–Lagrange strains are expressed in terms of section curvature and shear distortion, whose first and second variations are functions of node displacements and rotations. A symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials. The comparison of analytical results with numerical simulations in the literature demonstrates the consistency, accuracy and superior numerical performance of the proposed element. Copyright © 2001 John Wiley & Sons, Ltd.     

Fractional visco‐elastic Timoshenko beam deflection via single equation

This paper deals with the response determination of a visco‐elastic Timoshenko beam under static loading condition and taking into account fractional calculus. In particular, the fractional derivative terms arise from representing constitutive behavior of the visco‐elastic material. Further, taking advantages of the Mellin transform method recently developed for the solution of fractional differential equation, the problem of fractional Timoshenko beam model is assessed in time domain without invoking the Laplace‐transforms as usual. Further, solution provided by the Mellin transform procedure will be compared with classical Central Difference scheme one, based on the Grunwald–Letnikov approximation of the fractional derivative. Moreover, Timoshenko beam response is generally evaluated by solving a couple of differential equations. In this paper, expressing the equation of the elastic curve just through a single relation, a more general procedure, which allows the determination of the beam response for any load condition and type of constraints, is developed. Copyright © 2015 John Wiley & Sons, Ltd.     

经典边界条件黏弹性Pasternak地基上Bernoulli-Euler梁横向自振特性分析

自振特性在结构的动力分析中具有重要的意义。将回传射线矩阵法(MRRM)推广到地基梁自振特性的研究中,通过节点力平衡和位移协调方程及对偶局部坐标系下单元相位关系,建立两端简支、两端自由、两端固支、简支-自由、简支-固支及固支-自由这六种边界条件下黏弹性Pasternak地基上的Bernoulli-Euler梁的回传射线矩阵,进而得到其频率方程。根据单一局部坐标系下的边界条件,推导出模态函数解析表达式,进一步根据正交归一化条件求解模态函数表达式中的未知参数。通过具体算例验证了回传射线矩阵法求解的正确性,并对不同边界条件下的自振频率、衰减系数及模态函数进行了分析。为黏弹性地基梁的振动特性研究提供理论基础。     

Dynamic stiffness for piecewise non‐uniform Timoshenko column by power series—part II: Follower force

A follower force is an applied force whose direction changes according to the deformed shape during the course of deformation. The dynamic stiffness matrix of a non‐uniform Timoshenko column under follower force is formed by the power‐series method. The dynamic stiffness matrix is unsymmetrical due to the non‐conservative nature of the follower force. The frequency‐dependent mass matrix is still symmetrical and positive definite according to the extended Leung theorem. An arc length continuation method is introduced to find the influence of a concentrated follower force, distributed follower force, end mass and stiffness, slenderness, and taper ratio on the natural frequency and stability. It is found that the power‐series method can handle a very wide class of dynamic stiffness problem. Copyright © 2001 John Wiley & Sons, Ltd.     

A 2.5D finite/infinite element approach for modelling visco‐elastic bodies subjected to moving loads

The objective of this study is to propose a 2.5D finite/infinite element procedure for dealing with the ground vibrations induced by moving loads. Besides the two in‐plane degrees of freedom (DOFs) per node conventionally used for plane strain elements, an extra DOF is introduced to account for the out‐of‐plane wave transmission. The profile of the half‐space is divided into a near field and a semi‐infinite far field. The near field containing loads and irregular structures is simulated by the finite elements, while the far field covering the soils extending to infinity by the infinite elements with due account taken of the radiation effects for moving loads. Enhanced by the automated mesh expansion procedure proposed previously by the writers, the far field impedances for all the lower frequencies are generated repetitively from the mesh created for the highest frequency considered. Finally, the accuracy of the proposed method is verified through comparison with a number of analytical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

An accurate solution for the responses of circular curved beams subjected to a moving load

In this paper, an accurate and effective solution for a circular curved beam subjected to a moving load is proposed, which incorporates the dynamic stiffness matrix into the Laplace transform technique. In the Laplace domain, the dynamic stiffness matrix and equivalent nodal force vector for a moving load are explicitly formulated based on the general closed‐form solution of the differential equations for a circular curved beam subjected to a moving load. A comparison with the modal superposition solution for the case of a simply supported curved beam confirms the high accuracy and applicability of the proposed solution. The internal reactions at any desired location can easily be obtained with high accuracy using the proposed solution, while a large number of elements are usually required for using the finite element method. Furthermore, the jump behaviour of the shear force due to passage of the load is clearly described by the present solution without the Gibb's phenomenon, which cannot be achieved by the modal superposition solution. Finally, the present solution is employed to study the dynamic behaviour of circular curved beams subjected to a moving load considering the effects of the loading characteristics, including the moving speed and excitation frequency, and the effects of the characteristics of curved beams such as the radius of curvature, number of spans, opening angles and damping. The impact factors for displacement and internal reactions are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Dynamic stiffness for piecewise non‐uniform Timoshenko column by power series—part I: Conservative axial force

The dynamic stiffness method uses the solutions of the governing equations as shape functions in a harmonic vibration analysis. One element can predict many modes exactly in the classical sense. The disadvantages lie in the transcendental nature and in the need to solve a non‐linear eigenproblem for the natural modes, which can be solved by the Wittrick–William algorithm and the Leung theorem. Another practical problem is to solve the governing equations exactly for the shape functions, non‐uniform members in particular. It is proposed to use power series for the purpose. Dynamic stiffness matrices for non‐uniform Timoshenko column are taken as examples. The shape functions can be found easily by symbolic programming. Step beam structures can be treated without difficulty. The new contributions of the paper include a general formulation, an extended Leung's theorem and its application to parametric study. Copyright © 2001 John Wiley & Sons, Ltd.     

轴向运动黏弹性梁横向非线性受迫振动

运用微分求积法数值研究不同边界条件下轴向运动黏弹性梁受到简谐外激励的横向受迫     

黏弹性Pasternak地基梁振动的复模态分析

运用复模态分析研究了有限长黏弹性Pasternak地基梁的振动特性,将梁的振动方程写成状态方程,利用复模态的正交性解耦为常微分方程组,得出复频率和复模态及任意初始条件下外激励的响应。通过两个具体算例对比,分析了简支边界条件下的Pasternak地基梁的固有频率和模态函数的特征,并通过文中给出的复模态函数,计算了两种典型外激励作用下的动力响应。     

Free vibration behavior of a thermally post-buckled FG Timoshenko beam under large deflection using a tangent stiffness-based method

For thermally postbuckled configurations, the free vibration behavior of functionally graded (FG) Timoshenko beams are investigated. The postbuckling configurations are obtained through a geometrically nonlinear static problem. The free vibration problem around the postbuckled configuration is formulated using its tangent stiffness. The energy based governing equations are solved following the Ritz method. The elements of the tangent stiffness matrix are obtained using the Ritz coefficients. The results are shown to exhibit the effects of FG material, material profile parameter, and length-thickness ratio. The comparative results are presented for both the cases of the physical neutral surface and the geometrical neutral surface.