功能梯度变截面梁自由振动和稳定性研究

基于忽略了梁截面剪切变形和转动惯量效应的Euler-Bernoulli梁理论,研究了轴向力作用下轴向功能梯度变截面梁的横向自由振动问题,将轴向功能梯度Euler-Bernoulli梁自由振动固有频率和临界荷载的计算转化为变系数常微分方程特征值问题。运用插值矩阵法可一次性计算出轴向功能梯度变截面梁各阶振动固有频率和临界荷载,分析了轴向荷载对轴向功能梯度Euler-Bernoulli梁自由振动固有频率的影响,即轴向压力使梁的第1阶固有频率降低,轴向拉力使梁的第1阶固有频率增大。在简支-简支梁(H-H)边界条件下、不同截面宽锥度系数c_b和截面高锥度系数c_h,且区间划分点数n为40时,本文计算结果与已有文献计算结果之间的最大相对误差不超过0.00768%;在简支-简支梁(H-H)、固端-自由梁(C-F)、固端-固端梁(C-C)这三种不同边界条件下,不同c_b和c_h,且n为40时,最大相对误差不超过0.101%,说明了本文方法的有效性和良好的计算精度。     

非对称混杂边界轴向运动Timoshenko梁橫向振动分析

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7imoshenko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.     

功能梯度材料梁自由振动的线性与非线性振动

基于数值模拟和理论分析两种方法，研究了功能梯度材料(functional gradient materials,FGM)梁自由振动下的线性与非线性振动问题。通过解析法求解了FGM梁在经典理论下以及一阶剪切理论下的力学行为，得到了FGM梁在简支和固端约束下的固有频率。理论分析了不同边界条件、不同梁理论下、梯度指数、长细比等因素对于FGM梁固有频率的影响；不论经典梁理论还是一阶剪切理论，随着梯度指数的增加，FGM梁的固有频率都随之减小。通过ABAQUS仿真模拟，得到FGM梁数值模拟下的非线性固有频率。将理论解与数值解进行对比，完善力学模型。在多种理论下，利用解析法和数值模拟的方法，给出FGM梁结构振动响应的线性与非线性解。     

多孔功能梯度材料Timoshenko梁的自由振动分析

基于Timoshenko梁理论研究多孔功能梯度材料梁(FGMs)的自由振动问题.首先,考虑多孔功能梯度材料梁的孔隙率模型,建立了两种类型的孔隙分布.其次,基于Timoshenko梁变形理论,给出位移场方程、几何方程和本构方程,利用Hamilton原理推导多孔功能梯度材料梁的自由振动控制微分方程,并进行无量纲化,然后应用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,得到含有固有频率的等价代数特征方程.最后,计算了固定-固定(C-C)、固定-简支(C-S)和简支-简支(S-S)三种不同边界下多孔功能梯度材料梁自由振动的无量纲固有频率.将其退化为均匀材料与已有文献数据结果对照,验证了正确性.讨论了孔隙率、细长比和梯度指数对多孔功能梯度材料梁无量纲固有频率的影响.     

弹性地基上转动FGM梁自由振动的DTM分析

基于Euler-Bernoulli梁理论,利用广义Hamilton原理推导得到弹性地基上转动功能梯度材料(FGM)梁横向自由振动的运动控制微分方程并进行无量纲化,采用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,计算了弹性地基上转动FGM梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种边界条件下横向自由振动的无量纲固有频率,再将控制微分方程退化到无转动和地基时的FGM梁,计算其不同梯度指数时第一阶无量纲固有频率值,并和已有文献的FEM和Lagrange乘子法计算结果进行比较,数值完全吻合。计算结果表明,三种边界条件下FGM梁的无量纲固有频率随无量纲转速和无量纲弹性地基模量的增大而增大;在一定无量纲转速和无量纲弹性地基模量下,FGM梁的无量纲固有频率随着FGM梯度指数的增大而减小;但在夹紧-简支和夹紧-自由边界条件下,一阶无量纲固有频率几乎不变。     

含多处裂纹梁的振动分析

基于传递矩阵方法,提出了一种计算含有任意处裂纹梁固有频率的新方法。将梁内裂纹模拟为无质量的弯曲弹簧,得到了梁的解析特征方程。通过数值模拟计算,讨论了裂纹数量,以及裂纹位置和裂纹深度对梁的固有频率的影响。通过与文献[4]的计算结果比较,验证了本文方法的有效性。     

基于数值分析的模态阻尼系数获取方法

结构模态阻尼系数是影响振动疲劳特性的主要因素,获取模态阻尼系数对于结构振动疲劳的分析和仿真计算有重要作用,对于揭示金属材料振动疲劳损伤形成机理有直接意义。本研究针对2024-O铝合金,进行了大量的元件级试件的振动疲劳仿真分析、试验研究以及数值分析计算,并提出了一种基于数值分析的快速获取结构模态阻尼系数的方法,适合于获取试验件振动疲劳试验过程中的模态阻尼系数变化趋势。研究表明:在不中断试件疲劳试验的情况下,本文方法可以快速准确地得到试验件在整个振动疲劳历程中的模态阻尼系数,固有频率的相对偏差小于0.06%,模态阻尼系数的相对偏差小于1%。为进一步揭示金属材料振动疲劳损伤的形成机理奠定了基础。     

轴向受载的Euler-Bernoulli梁的双向弯曲扭转耦合振动研究

本文研究了轴向受载的Euler-Bernoulli梁的双向弯曲扭转耦合自由振动问题.选择梁横截面的剪切中心作为坐标原点,坐标轴平行于梁截面的几何轴;振动微分方程中有关梁截面几何特性的参数均采用相对于几何轴的参数.结合具体的边界条件求解自由振动微分方程组,辅以Mathematica软件计算梁振动的固有频率.针对具体的算例,给出了三种边界条件下梁弯扭耦合振动的固有频率的数值结果,并与Ansys软件的计算结果进行了比较,分析了误差来源以及轴向荷载对弯扭耦合自由振动的影响.数值结果验证了本文方法在其适用范围内的精确性和有效性.本文忽略了翘曲刚度的影响.     

基于高阶梁理论的功能梯度材料自由振动分析

本文基于一种新型的高阶梁理论,研究了功能梯度材料梁的自由振动问题。首先对该新型高阶梁理论进行了介绍,然后对该理论进行了有限元实现,并利用Hamilton原理推导得到了离散的动力学平衡方程,构造了2节点8自由度的C1型高阶梁单元。参照文献作了均质悬臂梁的模态分析,验证了该梁单元的精度。然后利用该单元进行功能梯度梁的模态分析,并构造了一种材料相关性很弱的无量纲固有频率。由该无量纲固有频率引入了功能梯度梁与均质梁固有频率之间的转换关系,并通过算例分析了该转换关系的适用条件。     

变截面梁横向振动固有频率数值计算

根据边界条件对变截面梁横向振动四阶变系数微分方程降阶, 形成关于挠度和弯矩的二 阶非显式递推变系数微分方程组; 利用有限差分法, 研究了变截面简支梁横向振动固有频率 的数值计算方法及其精度. 理论分析和正交计算的算例表明: 数值计算算法简单, 计算精度 取决于计算步长的数目和梁横截面竖向渐变率, 与梁宽和梁长无关; 对于给定的计算步长或 数目, 可以估算数值计算的精度; 对于给定的精度要求, 可以确定合理的计算步长或数目.     

Natural frequencies of nonlinear vibration of axially moving beams

Axially moving beam-typed structures are of technical importance and present in a wide class of engineering problem. In the present paper, natural frequencies of nonlinear planar vibration of axially moving beams are numerically investigated via the fast Fourier transform (FFT). The FFT is a computational tool for efficiently calculating the discrete Fourier transform of a series of data samples by means of digital computers. The governing equations of coupled planar of an axially moving beam are reduced to two nonlinear models of transverse vibration. Numerical schemes are respectively presented for the governing equations via the finite difference method under the simple support boundary condition. In this paper, time series of the discrete Fourier transform is defined as numerically solutions of three nonlinear governing equations, respectively. The standard FFT scheme is used to investigate the natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results are compared with the first two natural frequencies of linear free transverse vibration of an axially moving beam. And results indicate that the effect of the nonlinear coefficient on the first natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results also illustrate the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters.     

Effect of rotary inertia and shear on vibration and buckling of a double beam system under compressive axial loading

Free transverse vibration and buckling of a double-beam continuously joined by a Winkler elastic layer under compressive axial loading with the influence of rotary inertia and shear are considered in this paper. The motion of the system is described by a homogeneous set of two partial differential equations, which is solved by using the classical Bernoulli?CFourier method. The boundary value and initial value problems are solved. The natural frequencies and associated amplitude ratios of an elastically connected double-beam complex system and the analytical solution of the critical buckling load are determined. The presented theoretical analysis is illustrated by a numerical example, in which the effect of physical parameters characterizing the vibrating system on the natural frequency, the associated amplitude ratios and the critical buckling load are discussed.     

轴向功能梯度变截面梁的自由振动研究

摘 要：本文引入一种新的、简单易行的近似方法,求解轴向非均匀变截面梁的自由振动固有频率。将位移展开成切比雪夫多项式,从而变系数控制微分方程转化为含未知系数的齐次线性方程组。利用非零解的存在条件,进而得到含固有频率的特征方程。通过和特定梯度下已有的精确解进行比较,验证了该方法的精度和有效性,并分析了梯度参数、支承条件等对固有频率的影响。     

Nonlinear dynamic behavior analysis of an elastically restrained double-beam connected through a mass-spring system that is nonlinear

Some complex engineering structures can be modeled as multiple beams connected through coupling elements. When the coupling element is elastic, it can be simplified as a mass-spring system. The existing studies mainly concentrated on the double-beam coupled through elastic connectors, where the connector is simplified as the equivalent linear stiffness element or linear mass-spring system. Furthermore, many researches ignore rotational boundary restraints in analyzing dynamic behavior of the double-beam connected through elastic connectors, limiting their engineering generality. Considering the above limitations, this study attempts to employ the cubic nonlinear stiffness in the coupling mass-spring system and study the potential application of the mass-spring system that is nonlinear on the vibration control of the double-beam system. Using the variational method and the generalized Hamiltonian method build the corresponding system’s governing functions. Applying the Galerkin truncation method (GTM) obtains the dynamic behavior of the double-beam connected through a mass-spring system that is nonlinear. According to this study, the change of the mass-spring system that is nonlinear significantly influences the dynamic behavior of the double-beam system, where the complex dynamic behavior occurs under certain parameters of the mass-spring system that is nonlinear. Suitable parameters of the mass-spring system that is nonlinear are good at the vibration suppression at the boundary of the vibration system. Furthermore, the mass-spring system that is nonlinear can change the characteristics of the double-beam system’s kinetic energy transfer. For the vibration model established in this work, a quasi-periodic vibration state can be regarded as a sign of the occurrence of the targeted energy transfer of the double-beam connected through a mass-spring system that is nonlinear.

     

一维六方准晶层合简支梁自由振动与屈曲的精确解

伪Stroh型公式能够将多场耦合材料的控制方程转化为线性特征系统来求解，从而获得多层结构简支边界条件的精确解.本文利用伪Stroh型公式，研究一维六方准晶层合简支梁的自由振动和屈曲问题，通过传递矩阵法，获得准晶层合梁自由振动固有频率与临界屈曲载荷的精确解.通过与已有梁的剪切变形理论结果比较，验证了本文伪Stroh型公式的正确性和有效性.通过数值算例，分析由两种不同准晶材料组成的三明治层合梁的叠层方式、高跨比、层厚比及层数对梁的固有频率、临界屈曲载荷及其模态的影响规律.结果表明，叠层顺序和梁的高跨比、层厚比对准晶层合梁的自由振动固有频率和临界屈曲载荷有很大影响，可通过调整梁的几何尺寸和叠层顺序得到准晶层合梁的最佳固有频率和临界屈曲载荷.本文给出的精确解可为工程上研究准晶梁的各种数值解法和实验方法提供理论参考.     

Crack detection of a double-beam carrying a concentrated mass

This paper presents the influence of a concentrated mass location on the natural frequencies of a cracked double-beam. The double-beam consists of two different beams connected by an elastic medium. The concentrated mass is located on the main beam. The relationship between the natural frequency and the location of concentrated mass is established and called “Frequency–Mass Location” (FML). The numerical simulations show that when there is a crack, the frequency of the double-beam changes irregularly when the concentrated mass is attacked at the crack position. This irregular change can be amplified by the wavelet transform and this is useful for crack detection: the crack location can be detected by the location of peaks in the wavelet transform of the FML. Finite element model for the cracked double-beam carrying a concentrated mass is presented and numerical simulations are also provided.     

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.     

Chebyshev collocation approach for vibration analysis of functionally graded porous beams based on third-order shear deformation theory

In this paper, vibration analysis of functionally graded porous beams is carried out using the third-order shear deformation theory. The beams have uniform and non-uniform porosity distributions across their thickness and both ends are supported by rotational and translational springs. The material properties of the beams such as elastic moduli and mass density can be related to the porosity and mass coefficient utilizing the typical mechanical features of open-cell metal foams. The Chebyshev collocation method is applied to solve the governing equations derived from Hamilton’s principle, which is used in order to obtain the accurate natural frequencies for the vibration problem of beams with various general and elastic boundary conditions. Based on the numerical experiments, it is revealed that the natural frequencies of the beams with asymmetric and non-uniform porosity distributions are higher than those of other beams with uniform and symmetric porosity distributions.     

Supercritical vibration of nonlinear coupled moving beams based on discrete Fourier transform

Natural frequencies of nonlinear coupled planar vibration are investigated for axially moving beams in the supercritical transport speed ranges. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. The finite difference scheme is developed to calculate the non-trivial static equilibrium. The equations are cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. Under fixed boundary conditions, time series are calculated via the finite difference method. Based on the time series, the natural frequencies of nonlinear planar vibration, which are determined via discrete Fourier transform (DFT), are compared with the results of the Galerkin method for the corresponding governing equations without nonlinear parts. The effects of material parameters and vibration amplitude on the natural frequencies are investigated through parametric studies. The model of coupled planar vibration can reduce to two nonlinear models of transverse vibration. For the transverse integro-partial-differential equation, the equilibrium solutions are performed analytically under the fixed boundary conditions. Numerical examples indicate that the integro-partial-differential equation yields natural frequencies closer to those of the coupled planar equation.