Vibration of beams with two overlapping delaminations in prebuckled states

An analytical solution for the free vibrations of beams with two overlapping delaminations in prebuckled states is presented. The delaminated beam is analyzed as seven interconnected Euler–Bernoulli beams. The continuity and boundary conditions are satisfied between adjoining beams. Both the ‘free mode’ and the ‘constrained mode’ assumption in delamination buckling and vibration are used. A parametric study is performed to investigate the influence of the axial compressive load on the natural frequency and the mode shape of the delaminated beam. A monotonic relation between the natural frequency and the compressive load is observed. Comparisons with the analytical results reported for delamination buckling and vibration verify the validity of the present solution.     

Energy expressions and free vibration analysis of a rotating double tapered Timoshenko beam featuring bending–torsion coupling

In this study, free vibration analysis of a rotating, double tapered Timoshenko beam featuring coupling between flapwise bending and torsional vibrations is performed. At the beginning of the study, kinetic and potential energy expressions of a rotating Timoshenko beam having single cross-sectional symmetry are derived in a detailed way by using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion. The parameters for the hub radius, rotational speed, rotary inertia, shear deformation, slenderness ratio, bending–torsion coupling and taper ratio are incorporated into the equations of motion. In the solution part, an efficient mathematical technique, called the differential transform method (DTM), is used to solve the governing differential equations of motion. Using the computer package, Mathematica, the mode shapes are plotted, the effects of the incorporated parameters on the natural frequencies are investigated. The calculated results are tabulated in several tables and plotted in several graphics.     

变截面功能梯度旋转轴的自由振动特性

基于微分求积有限元(DQFEM)和铁木辛柯梁理论研究轴向变截面功能梯度旋转轴自由振动特性.首先利用上述理论和方法建立理论分析模型;然后将基于该模型所得的计算结果与已有文献和有限元软件ANSYS进行对比验证,结果表明该模型具有准确性、稳定性和通用性;最后采用上述模型分别研究变截面锥度系数和梯度系数对转轴固有频率和临界转速...     

旋转几何非线性复合材料薄壁梁的自由振动分析

研究具有几何非线性的旋转复合材料薄壁梁的自由振动。梁的变形引入了Von Kármán几何非线性, 基于Hamilton原理和变分渐进法 (Variational-Asymptotical Method -VMA),导出旋转复合材料薄壁梁的非线性振动偏微分方程组。采用Galerkin法将振动方程离散化为常微分方程组。借助于谐波平衡法 (Harmonic Balance Method -HBM) 建立自由振动的振幅-非线性固有频率关系方程。将上述方程化为非线性特征值问题,采用迭代算法进行求解。将所建立的旋转复合材料薄壁梁非线性自由振动分析模型和计算方法,应用于周向均匀刚度配置(Circumferentially Uniform Stiffness –CUS) 构型复合材料薄壁梁,通过数值计算揭示了纤维铺层角、旋转速度对非线性振动固有频率-振幅关系的影响。     

Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam

In this article, we present the nonlocal, nonlinear finite element formulations for the case of nonuniform rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress effects are also taken into consideration. Nonlocal stress resultants are obtained by employing Eringen’s nonlocal differential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the effect of nonlocal parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the effects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied.     

复合材料变截面旋转薄壁悬臂梁自由振动分析

基于拉格朗日方程推导出复合材料封闭变截面旋转薄壁梁的自由振动方程。与基于哈密顿原理的动力学建模方法相比,该文所采用的方法更为简洁。此外,在薄壁梁的结构模型中还考虑除横向剪切外的扭转、拉伸和弯曲引起的翘曲,具有考虑翘曲因素多的特点。给出了两种刚度配置下的变矩形截面旋转悬臂直梁的自由振动方程简化形式及其相应的迦辽金法求解的固有频率。基于大型通用有限元软件ANSYS,计算了薄壁变截面旋转悬臂梁的固有频率,并且与迦辽金法的求解结果进行了对比。分析了复合材料的弹性耦合、铺层角度、截面变化和旋转速度对薄壁梁的自由振动的影响。     

弹性支撑旋转Timoshenko梁动力学特性

为研究弹性支撑旋转梁动力学特性随转速及弹性支撑参数变化规律,考虑剪切效应、转动惯量和陀螺效应,采用Hamilton原理推导旋转Timoshenko梁动力学方程,应用Chebyshev谱方法获得系统涡动频率与模态振型数值解。结果表明,在高速转动状态下陀螺效应、支撑结构刚度对Timoshenko梁动力学特性有显著影响;各阶固有频率随着转速增加而分成正向涡动频率与反向涡动频率,高阶频率变化幅度更大;涡动频率随支撑结构直线刚度增加而呈阶梯状变化,当直线刚度增加到一定值后系统涡动频率将保持稳定;随着支撑结构转动刚度增加,涡动频率出现一个最小值与最大值,前者低于自由边界条件下频率值,后者高于固定边界条件下频率值。相关结果可用于各类旋转梁机构的设计与优化。     

几何参数对旋转薄壁圆柱壳振动特性的影响

摘 要: 本文基于Love壳体理论对旋转薄壁圆柱壳的自由振动特性进行分析,讨论几何参数对圆柱壳振动特性的影响。针对五种边界条件,分别分析厚度与半径之比、半径与长度之比对旋转薄壁圆柱壳固有频率特性的影响;探讨半径与长度之比对系统振型比的影响。分析结果表明：圆柱壳固有频率随着厚度与半径之比的增加而单调增加,增加幅度较小。半径与长度之比对圆柱壳固有频率与振型比的影响显著,且随其增大固有频率及振型比均不单调变化。由此可知,细长圆柱壳的振动特性与短粗圆柱壳的振动特性差异显著,研究结果为圆柱壳的动力学分析提供理论依据。     

用动态刚度法分析旋转变截面梁横向振动特性

通过引入动态刚度法分析旋转变截面梁的振动特性。首先基于欧拉-伯努利梁理论给出旋转变截面梁自由振动方程,然后通过动态刚度法推导该旋转梁的动态刚度矩阵,最后运用MATLAB中的fzero函数求解特征值方程得到旋转梁横向振动的固有频率和模态振型。数值计算结果证明了动态刚度法的精度和有效性,同时分析了轮毂半径、转速以及渐变系数对固有频率的影响。     

Free vibrations of a multi-span Timoshenko beam carrying multiple spring-mass systems

Structural elements supporting motors or engines are frequently seen in technological applications. The operation of a machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler single-span beams carrying a number of spring-mass system and Bernoulli-Euler multi-span beams carrying multiple spring-mass systems are plenty, but on Timoshenko multi-span beams carrying multiple spring-mass systems is fewer. This paper aims at determining the natural frequencies and mode shapes of a Timoshenko multi-span beam. The model allows to analyse the influence of the shear effect and spring-mass systems on the dynamic behaviour of the beams by using Timoshenko Beam Theory (TBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The natural frequencies of Timoshenko multi-span beam calculated by using secant method for non-trivial solution are compared with the natural frequencies of multi-span beam calculated by using Bernoulli-Euler Beam Theory (EBT) in literature; the mode shapes are presented in graphs.     

弹性螺旋桨-轴系建模及振动特性的解析方法研究

为考虑螺旋桨自身弹性对桨-轴系统振动特性的影响,建立了一套基于Timoshenko梁理论的解析方法。将螺旋桨、轴系均用Timoshenko梁建模,结合桨叶与轴系连接处的协调条件及其边界条件,给出系统横向、纵向自由振动的控制方程;在同有限元结果对比表明本方法具有良好精度基础上,分析了桨叶弹性对系统模态的影响及桨-轴系统的力传递特性。研究表明:桨-轴系统的模态振型中螺旋桨叶片和轴系的弹性变形同时发生且相互影响,叶片弯曲模态会加剧轴系振动;作用于桨叶的激励引起的桨-轴系统轴承处的纵向传递力被桨叶弯曲和轴系纵振两阶模态显著放大,而横向传递力主要由桨叶及轴系的弯曲模态控制。     

含多裂纹损伤圆弧曲梁自由振动扰动的有限元网格自适应分析

该文建立圆弧形曲梁裂纹的截面损伤缺陷比拟方案,实施微裂纹损伤诱发截面弱化,实现多裂纹深度、位置、数目的模拟。引入变截面Timoshenko梁的h型有限元网格自适应分析方法,求解含裂纹损伤圆弧曲梁自由振动问题,得到优化的网格和满足预设误差限的高精度自振频率和振型解答,研究多裂纹损伤对圆弧曲梁振型的扰动行为。数值算例表明,该算法中网格非均匀加密可适应裂纹损伤引起的振型变化,应用于各类曲梁夹角和裂纹损伤分布工况下的自由振动研究,定量分析了多裂纹损伤深度、数目、分布对圆弧曲梁自振频率和振型的扰动影响,检验了该文算法的精确性和实用性。     

旋转复合材料薄壁轴的不平衡非线性弯曲振动

研究几何非线性复合材料薄壁轴在偏心激励作用下的非线性振动特性。在轴的应变位移关系中引入Von Kármán几何非线性,基于Hamilton原理和变分渐进法(VAM)导出复合材料传动轴的拉-弯-扭耦合非线性振动偏微分方程组。为了着重研究轴的横向弯曲非线性振动特性,在上述模型中忽略轴向变形和扭转变形,得到轴的横向弯曲非线性振动偏微分方程,其中考虑了黏滞外阻和内阻的影响。采用Galerkin法,将偏微分方程转离散化为常微分方程,在此基础上利用四阶Runge-Kutta法对常微分方程组进行数值模拟,获得位移时间响应图、相平面图和功率谱图,研究了外阻、内组、偏心距和转速对非线性振动响应的影响,发现旋转复合材料薄壁轴存在混沌运动。     

Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams

This paper investigates the nonlinear free vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Kármán geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, vibration amplitude, slenderness ratio and end supports on the nonlinear free vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear free vibration characteristics of FG-CNTRC beams.     

Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams

The surface and nonlocal effects on the nonlinear flexural free vibrations of elastically supported non-uniform cross section nanobeams are studied simultaneously. The formulations are derived based on both Euler–Bernoulli beam theory (EBT) and Timoshenko beam theory (TBT) independently using Hamilton’s principle in conjunction with Eringen’s nonlocal elasticity theory. Green’s strain tensor together with von Kármán assumptions are employed to model the geometrical nonlinearity. The differential quadrature method (DQM) as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeams subjected to different boundary conditions. After demonstrating the fast rate of convergence of the method, it is shown that the results are in excellent agreement with the previous studies in the limit cases. The influences of surface free energy, nonlocal parameter, length of non-uniform nanobeams, variation of nanobeam width and elastic medium parameters on the nonlinear free vibrations are investigated.     

Free vibration analysis of pre-twisted rotating FGM beams

The natural frequencies of vibration of a rotating pre-twisted functionally graded cantilever beam are investigated. Rotating cantilever beam with pre-twist made of a functionally gradient material (FGM) consisting of metal and ceramic is considered for the study. The material properties of the FGM beam symmetrically vary continuously in thickness direction from core at mid section to the outer surfaces according to a power-law form. Equations of motion for free vibration are derived using Lagrange’s equation and the natural frequencies are determined using Rayleigh–Ritz method. The effect of parameters such as the pre-twist angle, power law index, hub radius and rotational speed on the natural frequencies of rotating functionally graded pre-twisted cantilever beams are examined through numerical studies and comparison is made with the numerical results obtained using other methods reported in literature. The effect of coupling between chordwise and flapwise bending modes on the natural frequencies has also been investigated.     

Vibration analysis of CNT-reinforced functionally graded rotating cylindrical panels using the element-free kp-Ritz method

In this paper, analysis of free vibration of carbon nanotube (CNT) reinforced functionally graded rotating cylindrical panels is presented. The analysis is performed by using the element-free kernel particle Ritz method or in short the kp-Ritz method. The rotating cylindrical panels are reinforced by single-walled carbon nanotubes (SWCNTs) with different types of distributions along thickness direction of the panels. Extended rule of mixture is selected to estimate the effective material properties of the resulting nanocomposite rotating panels. Two-dimensional displacement fields of the plates are approximated by a set of mesh-free kernel particle functions. The discretized governing eigen-equations are developed via the Ritz procedure. This kp-Ritz method enforces essential boundary conditions through the full transformation method. Detailed parametric studies have been carried out to reveal the influences of volume fraction of carbon nanotubes, edge-to-radius ratio and rotation speed on the frequency characteristics, with mode shape visualization provided. In addition, effects of different boundary conditions and types of distributions of carbon nanotubes are examined in detail.     

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.     

考虑底板与池壁相互作用和池壁剪切变形影响的圆形水池动力特性计算

考虑圆形水池池壁剪切变形的影响、底板对池壁的径向约束作用和转动约束作用,将圆形水池底板与池壁的相互作用简化成端部受切向弹性约束和转动弹性约束下的弹性地基Timoshenko梁,基于Timoshenko梁振动的修正理论,导出了底部环向简支、顶部分别为自由、铰支和固支三种边界条件下的振动频率超越方程;根据池壁和弹性地基梁微分方程的类比性,阐述了利用ANSYS建立圆形水池振动模态分析的有限元方法。利用二分法对底板环形简支的圆形水池的振动频率进行了计算,分析了顶部不同边界条件、池高、池的半径和底板对池壁弯曲约束刚度对池壁振动频率的影响。得到了圆形水池轴对称振动可采用弹簧-质量模型进行基频估算、该文所建立的分析方法只能分析圆形水池的轴对称振动模态、圆形水池底板与池壁相互作用对基频影响不明显、壁厚的剪切变形和转动惯量对高阶振动影响大等结论。     

Free vibration analyses of multiple delaminated angle-ply composite conical shells – A finite element approach

In this paper, the finite element method is employed to investigate the effects of delamination on free vibration characteristics of graphite–epoxy pretwisted shallow angle-ply composite conical shells. The generalized dynamic equilibrium equation is derived from Lagrange’s equation of motion neglecting Coriolis effect for moderate rotational speeds. The theoretical formulation is based on the Mindlin’s theory and the multi-point constraint algorithm is considered for an eight noded isoparametric plate bending element. The standard eigenvalue problem is solved by applying the QR iteration algorithm. The mode shapes are also depicted for a typical laminate configuration. Non-dimensional natural frequencies obtained are the first known results for the type of analyses carried out here.