基于分层法的石墨烯增强功能梯度Timoshenko梁动力特性有限元分析

基于分层法对石墨烯增强功能梯度Timoshenko梁的动力特性问题进行有限元分析。在有限元建模过程中,首先将Timoshenko梁沿厚度方向分成若干层,然后利用改进的Halpin-Tsai细观力学模型计算各层的弹性模量,相应的泊松比和密度则根据混合率法则进行计算,最后对各层均采用4节点四边形板单元离散。通过数值算例分析分层数和单元尺寸比例的合理性,探究石墨烯片的分布形式、质量含量、几何形状和尺寸等因素对Timoshenko梁动力特性的影响。结果表明,添加少量的石墨烯能显著提高Timoshenko梁自由振动的频率,尤其在梁的上下部位分布较多正方形石墨烯片时效果最佳。随着石墨烯片长厚比的增加,Timoshenko梁的自由振动基频不断增大。当石墨烯片长厚比超过1 000之后,梁的基频变化不明显。

THERMAL POST-BUCKLING OF FUNCTIONALLY GRADED MATERIAL TIMOSHENKO BEAMS

Analysis of thermal post-buckling of FGM （Functionally Graded Material） Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam, geometrical nonlinear governing equations including seven basic unknown functions for functionally graded beams subjected to mechanical and thermal loads were formulated. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely nonuniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained. Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted. The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details. The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.

Bending of functionally graded nanobeams incorporating surface effects based on Timoshenko beam model

The bending responses of functionally graded （FG） nanobeams with simply supported edges are investigated based on Timoshenko beam theory in this article. The Gurtin-Murdoch surface elasticity theory is adopted to analyze the influences of surface stress on bending response of FG nanobeam. The material properties are assumed to vary along the thickness of FG nanobeam in power law. The bending governing equations are derived by using the minimum total potential energy principle and explicit formulas are derived for rotation angle and deflection of nanobeams with surface effects. Illustrative examples are implemented to give the bending deformation of FG nanobeam. The influences of the aspect ratio, gradient index, and surface stress on dimensionless deflection are discussed in detail.

Bending of Timoshenko beam with effect of crack gap based on equivalent spring model

Considering the effect of crack gap, the bending deformation of the Timoshenko beam with switching cracks is studied. To represent a crack with gap as a nonlinear unidirectional rotational spring, the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function. A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks. Three examples of bending of the Timoshenko beam are presented. The influence of the beam＇s slenderness ratio, the crack＇s depth, and the external load on the crack state and bending performances of the cracked beam is analyzed. It is revealed that a cusp exists on the deflection curve, and a jump on the rotation angle curve occurs at a crack location. The relation between the beam＇s deflection and load is bilinear, each part corresponding to an open or closed state of crack, respectively. When the crack is open, flexibility of the cracked beam decreases with the increase of the beam＇s slenderness ratio and the decrease of the crack depth. The results are useful in identifying non-destructive cracks on a beam.

Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories

The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.

EXPONENTIAL DECAY FOR A VISCOELASTICALLY DAMPED TIMOSHENKO BEAM

Of concern is a viscoelastic beam modelled using the Timoshenko theory. It is well-kimwn that the system is exponentially stable if the kernel in the memory term is sub- exponential. That is, if the product of the kernel with an exponential function is a summable function. In this article we address the questions： What if the kernel is tested against a different function （say Gamma） other than the exponential function？ Would there still be stability？ In the affirmative, what kind of decay rate we get？ It is proved that for a non- decreasing function ＂Gamma＂ whose ＂logarithmic derivative＂ is decreasing to zero we have a decay of order Gamma to some power and in the case it decreases to a different value than zero then the decay is exponential.

Nonlinear flexural waves and chaos behavior in finite-deflection Timoshenko beam

Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov＇s method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.

移动荷载作用下周期支承Timoshenko梁动力响应

针对移动荷载作用下周期支承Timoshenko梁的具体结构,将其任一梁单元分解为一个支撑单元和两个非支撑单元。在轨道结构动力响应的周期解析解的基础上,采用传递矩阵法,应用Laplace变换等理论,推导周期支承Timoshenko梁的传递矩阵。将激振点的状态变量乘以一系列梁单元的传递矩阵,推得拾振点的传递函数,进而给出移动荷载作用下周期支承Timoshenko梁上任一点的动力响应解。

用Timoshenko梁修正理论研究功能梯度材料梁的动力响应

采用Timoshenko梁修正理论研究了功能梯度材料梁的动力响应问题,利用静力方程确定了功能梯度材料梁的中性轴位置,在此基础上应用Timoshenko梁修正理论建立了功能梯度材料梁的振动方程,求得其自振频率表达式及其在简谐荷载作用下强迫振动的解析解。分析了中性面位置、梯度指数等因素对功能梯度材料梁的动力响应的影响,并用有限元法验证Timoshenko梁修正理论。通过实例计算,得到了中性轴位置对功能梯度材料梁动力响应有较大影响。

解析型Timoshenko梁有限单元

为提高深梁结构内力及变形的计算精度和效率,以 Timoshenko 梁理论为基础,建立了深梁位移控制方程,进而构造了深梁挠度、截面弯曲转角和剪切角的解析位移形函数.采用势能原理建立了深梁的势能泛函,利用势能变分原理得到了解析型单元列式,进而给出了解析型单元总刚度矩阵,将其与理论解、插值多项式深梁单元进行对比分析.结果表明:构造的解析型单元只需划分为一个单元即可保证计算的深梁挠度和转角与理论解一致,采用插值多项式单元确定的挠度和转角与理论解的相对误差最大可达到 19.785%.同时,为验证剪切变形对深梁位移影响,将构造的单元与 Euler 梁单元的计算结果进行对比.对比表明:对于承受均布荷载作用的悬臂梁,基于 Euler 梁计算的位移与基于 Timoshenko 梁理论构造的解析型单元计算的位移偏差可达到 50%;对于承受端部集中弯矩作用的简支梁,基于 Euler 梁计算的位移与基于 Timoshenko 梁理论构造的解析型单元计算的位移偏差可达到 10.769%.本文构造的单元满足了高精度、高效率的要求;该解析型梁单元可适用于浅梁分析,且不存在剪切闭锁的问题.

基于Timoshenko梁模型的车辆-轨道耦合系统垂向随机振动分析

将钢轨视为无限长Timoshenko梁,由两层弹簧阻尼系统连续支撑,在频域建立车辆-轨道垂向耦合动力学模型。提出采用格林函数法求解钢轨运动偏微分方程,可在较宽频域内得到轨道动力响应避免模态截断频率限制,结合车辆方程求解点导纳及传递导纳,运用虚拟激励法将真实轨道谱激励作为系统输入,求解车辆-轨道系统随机振动响应,并将该弹性轨道与传统刚性轨道、简化弹簧轨道模型结果进行对比。研究结果表明,采用格林函数法求解无限长Timoshenko梁弹性轨道模型可快速实现全频域计算,得到轨道系统频率响应特性。利用虚拟激励法及叠加法,可得到轮轨多点接触工况下的车辆与轨道结构随机振动响应。采用刚性轨道结构模型会导致过高估计车辆结构在高频的振动,整个耦合系统振动响应均对速度较敏感。考虑轨道弹性影响的弹性轨道模型更符合实际,采用格林函数法求解轨道模型较为快速精确。

响应耦合子结构法的轴类工件频响函数预测

轴类工件切屑过程中,容易产生颤振影响加工稳定性,快速计算工件频响函数,便能为防止颤振提供依据。由于车削加工工件频繁更换,针对以往测量工件频响函数往往采用锤击法需要重复测量等问题,提出一种改进的子结构划分方法。该方法将主轴-卡盘-工件系统划分为主轴-卡盘-夹持部分、工件悬伸部分,使用子结构响应耦合法(RCSA)、有限差分法、Timoshenko梁模型求取轴类工件任意截面频响函数,并利用该方法探究了尾座对工件频响函数的影响。此划分方法避免了结合面的参数辨识,能够快速方便预测工件任意截面频响函数,与仿真所得频响函数进行对比,预测结果与仿真结果对比误差在5.38%以内,验证了该预测模型的有效性。

功能梯度材料Timoshenko梁的热过屈曲分析

研究了功能梯度材料Timoshenko梁在横向非均匀升温下的热过屈曲.在精确考虑轴线伸长和一阶横向剪切变形的基础上,建立了功能梯度Timoshenko梁在热_机械载荷作用下的几何非线性控制方程,将问题归结为含有7个基本未知函数的非线性常微分方程边值问题.其中,假设功能梯度梁的材料性质为沿厚度方向按照幂函数连续变化的形式.然后采用打靶法数值求解所得强非线性边值问题,获得了横向非均匀升温场内两端固定Timoshenko梁的静态非线性热屈曲和热过屈曲数值解.绘出了梁的变形随温度载荷及材料梯度参数变化的特性曲线,分析和讨论了温度载荷及材料的梯度性质参数对梁变形的影响.结果表明,由于材料在横向的非均匀性,均匀升温时的梁中存在拉_弯耦合变形.

功能梯度材料Timoshenko梁的非线性大变形分析

采用打靶法研究了两端不可移简支功能梯度Timoshenko梁在横向非均匀升温下的大挠度弯曲问题.在精确考虑轴线伸长和基于一阶横向剪切变形理论的基础上建立了功能梯度Timoshenko梁受热-机载荷作用时的几何非线性控制方程,其中功能梯度梁的材料性质采用了沿厚度方向按照幂函数连续变化的形式.用打靶法数值求解所得强非线性边值问题,获得了横向非均匀升温时Timoshenko梁的静态非线性大变形数值解.绘出了梁的变形随温度载荷及材料梯度参数变化的特性关系曲线,并分析和讨论了温度载荷及材料的梯度性质参数对梁变形的影响.结果表明,由于材料的非均匀性,功能梯度梁中存在拉-弯耦合变形.

Timoshenko梁模型下完整桩瞬态横向振动半解析解

文章采用Timoshenko梁模型下桩-土系统横向振动的动力学模型和数学模型,通过Laplace变换,得到桩顶横向振动速度在频域内的解析解,进行快速Fourier逆变换,得到桩顶速度时域内的表达式;根据试验桩参数绘制出速度导纳曲线及桩顶速度曲线,分析Bernoulli-Euler梁模型和Timoshenko梁模型的速度导纳曲线的特点,对比2种模型桩的理论速度曲线与试验桩实测速度曲线的差异;分析不同参数对桩顶横向动力响应的影响。结果表明,Timoshenko梁模型的结果比Bernoulli-Euler梁模型更接近于工程实际中低应变横向测桩的结果,所得半解析解结果能在频域内进行分析,具有很好的适应性和准确性。

解析型弹性地基Timoshenko梁单元

采用双参数弹性地基模型和Timoshenko深梁模型,建立了弹性地基一般梁挠度控制方程,求解得到了挠度方程解析通解,构建了双参数弹性地基深梁的挠度、截面弯曲转角及剪切角的解析位移形函数。建立了梁模型、梁基模型等两种势能泛函,利用最小势能原理,构造了两个双参数弹性地基深梁单元,给出了单元列式。分析表明:梁模型单元在均布荷载作用下误差为0.221%,非均布荷载作用下误差为0;梁基模型单元在均布荷载作用下误差为0,在两端集中力作用下误差为6.597%,在跨中集中力作用下误差为102.716%;同时,该文提出的双参数Timoshenko梁模型单元不存在剪切闭锁的问题。

非对称混杂边界轴向运动Timoshenko梁横向振动分析

研究两端带有扭转弹簧且弹簧系数均可任意变化的非对称混杂边界下的轴向运动Timoshenko梁的横向振动.利用非对称混杂边界条件推导对应任意弹簧系数的系统超越方程以及特征函数.运用数值方法计算系统的固有频率及其相应的模态函数,并研究确定梁的刚度、轴向速度以及边界处扭转弹簧的刚度的影响.通过数值算例,比较Timoshenko梁、瑞利梁、剪切梁和欧拉梁的固有频率随轴向速度的变化,分析转动惯量和剪切变形的影响.

轴向随动分布载荷作用下FGMs Timoshenko梁的后屈曲特性

研究了Timoshenko功能梯度材料梁在随动分布载荷作用下的后屈曲问题。在考虑轴线伸长和一阶横向剪切变形基础上,建立了在轴向分布随动载荷作用下一端简支一端固定Timoshenko功能梯度梁的过屈曲控制方程。其中假设功能梯度材料性质只沿厚度方向变化,并以成分含量的幂指数函数形式变化。采用打靶法求解了所得线性常微分两点边值问题,获得了随动载荷作用下Timoshenko功能梯度梁的过屈曲平衡路径和平衡构形。对比了Timoshenko梁和Euler梁的后屈曲行为,并分析了材料的体积分数指数和长细比对梁屈曲行为的影响。结果表明:考虑剪切变形的Timoshenko梁的后屈曲行为与Euler梁的后屈曲行为明显不同;体积分数指数一定时,随着长细比的增加,梁的临界载荷减小;长细比一定时,随着体积分数指数的增加,梁的临界载荷也减小。

粘贴压电层功能梯度材料Timoshenko梁的热过屈曲分析

研究了上下表面粘贴压电层的功能梯度材料Timoshenko梁在升温及电场作用下的过屈曲行为。在精确考虑轴线伸长和一阶横向剪切变形的基础上,建立了压电功能梯度Timoshenko层合梁在热-电-机械载荷作用下的几何非线性控制方程。其中,假设功能梯度的材料性质沿厚度方向按照幂函数连续变化,压电层为各向同性均匀材料。采用打靶法数值求解所得强非线性边值问题,获得了在均匀电场和横向非均匀升温场内两端固定Timoshenko梁的静态非线性屈曲和过屈曲数值解。并给出了梁的变形随热、电载荷及材料梯度参数变化的特性曲线。结果表明,通过施加电压在压电层产生拉应力可以有效地提高梁的热屈曲临界载荷,延缓热过屈曲发生。由于材料在横向的非均匀性,即使在均匀升温和均匀电场作用下,也会产生拉-弯耦合效应。但是对于两端固定的压电-功能梯度材料梁,在横向非均匀升温下过屈曲变形仍然是分叉形的。

基于裂纹扭转弹簧模型的Timoshenko裂纹梁动力特性分析

忽略裂纹对梁剪切变形的影响,基于开裂纹的等效扭转弹簧模型,建立了Timoshenko裂纹梁动力弯曲的控制方程,得到了一种新的裂纹梁动力弯曲控制方程的求解方法,得出了具有任意条裂纹Timoshenko梁自振模态的统一显式表达式。数值分析了简支、悬臂、两端固支Timoshenko裂纹梁的自振频率和振动模态,考察了裂纹数目和裂纹深度等因素对裂纹梁动力特性的影响。结果表明:随着裂纹数目和深度的增加,裂纹梁的自振频率减小,且当裂纹较深时,裂纹深度对自振频率的影响显著;裂纹梁的挠度模态曲线和转角模态曲线在裂纹处分别呈现尖点和跳跃现象,且尖点效应和转角跳跃随裂纹深度的增加而愈加明显。另外,当裂纹处的弯矩为零时,裂纹对梁的自振频率和振动模态没有影响。这些结果可对梁的安全性评估及裂纹损伤检测提供理论指导。