Running and Standing Waves of Timoshenko Beam

Standing waves of a Timoshenko beamof finite length and their connectionwith running waves for an infinite beam are considered. It is shown that the principle of a “closed cycle” of a running wave is completely identical to the usual procedure of direct satisfaction from the side of the general solution for an infinite Timoshenko beam, to the boundary conditions of a beam of finite length. The question of the existence of a second frequency spectrum under arbitrary boundary conditions of a beam is discussed. A “relaxed approach” to the concept of the second frequency spectrum is proposed. The results of the theoretical analysis are confirmed by numerical calculations for the Timoshenko beam with elastic supports and elastic sealing of its end sections.     

无约束修正Timoshenko梁的冲击问题

介绍了修正后的Timoshenko梁运动方程,并比较了修正Timoshenko梁与经 典Timoshenko梁的运动方程. 推导了考虑剪切变形引起的转动惯量的修正Timoshenko 梁的正交条件,推导了集中质量对无约束修正Timoshenko梁的正碰撞对梁所引起的瞬态冲 击响应公式,并用算例进行了分析,且与集中质量对经典的无约束Timoshenko梁的正碰撞 对梁所引起的冲击响应进行了比较,另外还用算例分析了梁的刚度的变化和冲击质量比对其 冲击响应产生的影响.     

Timoshenko beam theory: A perspective based on the wave-mechanics approach

This paper reviews Timoshenko beam theory from the point of view of wave mechanics. Vibration of beam structures can be studied in terms of either normal modes or propagating waves. The latter wave approach has two distinct features: first, it gives rise to clear physical understanding of beam vibration; second, it leads to exact methods for vibration analysis of beam structures, especially in the mid-frequency range. In this paper, the work on wave solutions of an infinite Timoshenko beam is first discussed. The work on the splitting effect of spinning on wave solutions is also reviewed. The wave is treated as constitutive components of standing waves (i.e. normal modes), and a discussion on how the wave components formulate various standing waves is presented. Finally, several numerical examples are presented to illustrate the pros and cons of using different wave approaches to tackle vibration analysis of finite-length Timoshenko beams.     

任意边界条件下Timoshenko梁及其修正理论的自振特性分析

提出一种求解任意边界条件下经典Timoshenko梁以及修正Timoshenko梁自振频率和振型的新方法。利用改进的傅立叶级数消除传统傅立叶级数的边界不收敛问题,然后通过Rayleigh-Ritz法导出Timoshenko梁的拉格朗日泛函,根据Hamilton原理将原问题转化为求解矩阵广义特征值问题。通过与解析解对比,本文采用的方法具有较好的收敛性以及较高的计算精度;通过数值计算发现,经典Timoshenko梁的自振频率略高于修正的Timoshenko梁,随着振型阶数的提高,经典Timoshenko梁的计算结果逐渐偏离文献解和有限元结果,而修正的Timoshenko梁能够保持较好的一致性;对于不同边界条件下修正Timoshenko梁的计算结果均能与有限元的计算结果吻合得很好。最后运用MATLAB编程软件将程序设计为App,对于不同情形的梁只需要修改参数即可,可为实际工程提供高效便捷的计算方案和可靠理论依据。     

确定性载荷作用下Timoshenko薄壁梁的弯扭耦合动力响应

建立了一种普遍的解析理论用于求解确定性载荷作用下Timoshenko薄壁梁的弯扭耦合动力响应。首先通过直接求解单对称均匀Timoshenko薄壁梁单元弯扭耦合振动的运动偏微分方程，给出了计算其自由振动的精确方法，并导出了Timoshenko弯扭耦合薄壁梁自由振动主模态的正交条件。然后利用简正模态法研究了确定性载荷作用下单对称Timoshenko薄壁梁的弯扭耦合动力响应，该弯扭耦合梁所受到的荷载可以是集中载荷或沿着梁长度分布的分布载荷。最后假定确定性载荷是谐波变化的，得到了各种激励下封闭形式的解，并对动力弯曲位移和扭转位移的数值结果进行了讨论。     

从一个怪例看细长梁理论的基本假定

分别采用欧拉和铁木辛柯梁理论分析了均匀分布力偶作用下的两端固支等截面匀质细长 梁, 并通过ABAQUS有限元分析了一个实例, 验证了铁木辛柯梁理论分析的结果. 对比证明在 这种载荷及边界条件下即使细长梁, 也必须考虑剪切效应的影响.     

SMA纤维混杂层合梁的材料阻尼

研究一类由形状记忆合金（SMA）和普通纤维混杂而成的层合梁的阻尼特性,基于最大应变能理论提出SAM混杂层合梁的等效材料阻尼预测的数学模型,其中,单层材料的弹性性能和阻尼性能分别采用多胞模型及其阻尼细观力学分析模型确定,利用正交各向异性层合梁的铁木辛柯理论分析梁的变形,通过数值算例分析了SMA含量,纤维铺设角对梁的等效阻尼比的影响。     

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将理性有限元法引入到Timoshenko梁问题中,提出了一种理性Timoshenko梁单元,克服了 剪切锁死现象. 在推导控制方程时,与传统有限元方法采用Lagrange插值不同, 理性有限元法用Timoshenko梁弯曲问题的基本解逼近单元内部场. 运用该梁单元分析 Timoshenko梁时,无需缩减积分,就能避免剪切锁死,并且极大地提高了计算精度,说明 理性有限元法具有广泛的应用前景.     

Transient wave analysis of a cantilever Timoshenko beam subjected to impact loading by Laplace transform and normal mode methods

This study applies two analytical approaches, Laplace transform and normal mode methods, to investigate the dynamic transient response of a cantilever Timoshenko beam subjected to impact forces. Explicit solutions for the normal mode method and the Laplace transform method are presented. The Durbin method is used to perform the Laplace inverse transformation, and numerical results based on these two approaches are compared. The comparison indicates that the normal mode method is more efficient than the Laplace transform method in the transient response analysis of a cantilever Timoshenko beam, whereas the Laplace transform method is more appropriate than the normal mode method when analyzing the complicated multi-span Timoshenko beam. Furthermore, a three-dimensional finite element cantilever beam model is implemented. The results are compared with the transient responses for displacement, normal stress, shear stress, and the resonant frequencies of a Timoshenko beam and Bernoulli–Euler beam theories. The transient displacement response for a cantilever beam can be appropriately evaluated using the Timoshenko beam theory if the slender ratio is greater than 10 or using the Bernoulli–Euler beam theory if the slender ratio is greater than 100. Moreover, the resonant frequency of a cantilever beam can be accurately determined by the Timoshenko beam theory if the slender ratio is greater than 100 or by the Bernoulli–Euler beam theory if the slender ratio is greater than 400.     

Theoretical analysis of transient waves in a simply-supported Timoshenko beam by ray and normal mode methods

The dynamic transient responses of a simply-supported Timoshenko beam subjected to an impact force are investigated by two theoretical approaches – ray and normal mode methods. The mathematical methodology proposed in this study for the ray method enable us to construct the solution for the interior source problem and to extend to solve the complicated problem for the multi span of the Timoshenko beam. Numerical results based on these two approaches are compared. The comparison in this study indicates that the normal mode method is more computationally efficient than the ray method except for very short time after the impact. The long-time transient responses are easily calculated using the normal mode method. It is shown that the average long-time transient response converges to the corresponding static value. The Timoshenko beam theory is more accurate than the Bernoulli–Euler beam theory because it includes shear and rotary inertia. This study also provides the slender ratio for which the Bernoulli–Euler beam can be used for the transient-response analysis of the displacement. Moreover, the resonant frequencies obtained from finite element calculation based on the three-dimensional model are compared with the results calculated using the Timoshenko beam and Bernoulli–Euler beam theories. It is noted in this study that the resonant frequency can be accurately determined by the Timoshenko beam theory if the slender ratio is larger than 100, and by the Bernoulli–Euler beam theory if the slender ratio is larger than 400.     

Modeling and bifurcation analysis of the centre rigid-body mounted on an external Timoshenko beam

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A microstructure-dependent Timoshenko beam model based on a modified couple stress theory

A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli-Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli-Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.     

Vibration analysis of a stepped laminated composite Timoshenko beam

The goal of this study is to investigate the vibration characteristics of a stepped laminated composite Timoshenko beam. Based on the first order shear deformation theory, flexural rigidity and transverse shearing rigidity of a laminated beam are determined. In order to account for the effect of shear deformation and rotary inertia of the stepped beam, Timoshenko beam theory is then used to deduce the frequency function. Graphs of the natural frequencies and mode shapes of a T300/970 laminated stepped beam are given, in order to illustrate the influence of step location parameter exerts on the dynamic behavior of the beam.     

A homogenization-based theory for anisotropic beams with accurate through-section stress and strain prediction

This paper presents a homogenization-based theory for three-dimensional anisotropic beams. The proposed beam theory uses a hierarchy of solutions to carefully-chosen beam problems that are referred to as the fundamental states. The stress and strain distribution in the beam is expressed as a linear combination of the fundamental state solutions and stress and strain residuals that capture the parts of the solution not accounted for by the fundamental states. This decomposition plays an important role in the homogenization process and provides a consistent method to reconstruct the stress and strain distribution in the beam in a post-processing calculation. A finite-element method is presented to calculate the fundamental state solutions. Results are presented demonstrating that the stress and strain reconstruction achieves accuracy comparable with full three-dimensional finite element computations, away from the ends of the beam. The computational cost of the proposed approach is three orders of magnitude less than the computational cost of full three-dimensional calculations for the cases presented here. For isotropic beams with symmetric cross-sections, the proposed theory takes the form of classical Timoshenko beam theory with Cowper’s shear correction factor and additional load-dependent corrections. The proposed approach provides an extension of Timoshenko’s beam theory that handles sections with anisotropic construction.     

基于等效黏性弹簧的黏弹性Timoshenko裂纹梁弯曲解析解

考虑裂纹的缝隙和黏性效应,将梁中横向裂纹等效为黏弹性扭转弹簧,利用广义Delta函数,给出了Laplace变换域内裂纹梁的等效抗弯刚度,得到了具有任意开闭裂纹数目且满足标准线性固体黏弹性本构的Timoshenko梁在时间域内的弯曲变形显式解析通解．在此基础上,通过两个数值算例,分析了时间、梁跨高比和裂纹深度等参数对黏弹性Timoshenko开裂纹梁弯曲变形的影响．结果表明：裂纹黏性对Timoshenko裂纹梁的弯曲具有显著的影响．相比于裂纹的弹性扭转弹簧模型,考虑裂纹黏性效应的黏弹性Timoshenko裂纹梁在裂纹处挠度尖点和转角跳跃现象十分明显．另外,由于横向剪切引起的附加变形,Timoshenko裂纹梁的稳态挠度与Euler-Bernoulli梁挠度的差值为常数,其大小与裂纹模型、梁跨高比或裂纹深度无关,这些结果对梁裂纹无损检测具有指导意义．     

A micro scale Timoshenko beam model based on strain gradient elasticity theory

A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.     

饱和多孔弹性Timoshenko梁的大挠度分析

基于微观不可压饱和多孔介质理论和弹性梁的大挠度变形假设,考虑梁剪切变形效应,在梁轴线不可伸长和孔隙流体仅沿轴向扩散的限定下,建立了饱和多孔弹性Timoshenko梁大挠度弯曲变形的非线性数学模型.在此基础上,利用Galerkin截断法,研究了两端可渗透简支饱和多孔Timoshenko梁在突加均布横向载荷作用下的拟静态弯曲,给出了饱和多孔 Timoshenko梁弯曲变形时固相挠度、弯矩和孔隙流体压力等效力偶等随时间的响应.比较了饱和多孔Timoshenko梁非线性大挠度和线性小挠度理论以及饱和多孔 Euler-Bernoulli梁非线性大挠度理论的结果,揭示了他们间的差异,指出当无量纲载荷参数q＞l0时,应采用饱和多孔Timoshenko梁或Euler-Bernoulli梁的大挠度数学模型进行分析,特别的,当梁长细比λ＜30时,应采用饱和多孔Timoshenko梁大挠度数学模型进行分析.     

基于径向基函数配点法的梁板弯曲问题分析

采用径向基函数配点法分析考虑剪切效应的梁板弯曲问题,该方法利用径向基函数作为近似函数,基于配点法离散方程,通过最小二乘法求解。径向基函数配点法在离散和计算过程中不需要任何形式的网格划分,是一种真正的无网格法;径向基函数可以用一元函数来描述多元函数,存在明显的储存和运算简单的特点;而基于配点法求解不需要积分,提高了计算效率。分析考虑剪切效应的薄梁板问题时,传统的有限元法或无网格法求解均会存在剪切锁闭问题,而径向基函数在全域内存在无限连续性,能够准确地满足Kirchhoff约束条件,因此径向基函数配点法能够消除剪切锁闭现象,而且不会出现应力波动。该方法的优势在于,其不仅易于离散、精度高,而且具有指数收敛率,计算效率高。数值算例验证了上述结论和该方法的稳定性。     

Analysis of micro/nanobridge test based on nonlocal elasticity

Based on the nonlocal Timoshenko beam theory, we develop a mechanics approach to analyze the micro/nanobridge test. This approach considers the shear deformation, the strain gradients, the substrate deformation, and the contact deformation between the indenter bar tip and a tested beam, resulting in an analytic solution of beam deflection versus applied load involving other parameters of material intrinsic length, film residual stress, and cylinder bar radius. The same approach was further developed to analyze the delamination test, giving explicit formulas for the energy release rate and the phase angle.     

Quasi-static and dynamical analysis for viscoelastic Timoshenko beam with fractional derivative constitutive relation

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