Transverse free vibration of non uniform rotating Timoshenko beams with elastically clamped boundary conditions

In the present paper the Differential Quadrature Method, DQM, and the domain decomposition are used to carry out the free transverse vibration analysis of non-uniform multi-span rotating Timoshenko beams with perfect and not perfect boundary conditions. The cross section could vary in a continuous or discontinuous fashion along the beam length. The material of the beam could be different in each beam span. The influence of elastically clamped boundary conditions at hub end are studied and discussed. The effect of an arbitrary hub radius is considered. The governing differential equations of motion for rotating Timoshenko beams come from the derivation of Hamilton’s principle. The first six natural frequencies of vibration are obtained for many particular situations and for some of them the mode shapes are also available. The examples of applications of the method indicated its effectiveness. The results for particular cases are in excellent agreement with published results and results obtained by means of the finite element method.     

Non-uniform rational B-spline based free vibration analysis of axially functionally graded tapered Timoshenko curved beams

The free vibration of axially functionally graded (FG) tapered Timoshenko curved beams is studied with the numerical approach. By using the non-uniform rational B-spline (NURBS) basis functions, the exact geometry and the generalized displacement field are formulated. Variable geometric parameters and material properties, including the curvature, cross-sectional area, area moment of inertia, mass density, and Young’s modulus, are expanded as functions of the coordinate in a parametric domain. Based on Hamilton’s principle, the weak formulation is derived by applying a refined constitutive relation which considers the thickness effect. Natural frequencies and mode shapes are obtained from the eigenvalue equation. Circular, elliptic, and parabolic curved beams are considered in numerical examples. The obtained results are in good agreement with those in the existing studies and those calculated by the finite element software ANSYS. Moreover, the effects of the material gradient, taper ratio, slenderness ratio, and heightspan ratio on vibration behaviors are discussed.     

Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded(AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary conditions.Furthermore, assuming polynomial distributions of Young's modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.     

Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions

Axially moving beams are often discussed with several classic boundary conditions, such as simply-supported ends, fixed ends, and free ends. Here, axially moving beams with generalized boundary conditions are discussed for the first time. The beam is supported by torsional springs and vertical springs at both ends. By modifying the stiffness of the springs, generalized boundaries can replace those classical boundaries.Dynamic stiffness matrices are, respectively, established for axially moving Timoshenko beams and Euler-Bernoulli(EB) beams with generalized boundaries. In order to verify the applicability of the EB model, the natural frequencies of the axially moving Timoshenko beam and EB beam are compared. Furthermore, the effects of constrained spring stiffness on the vibration frequencies of the axially moving beam are studied. Interestingly, it can be found that the critical speed of the axially moving beam does not change with the vertical spring stiffness. In addition, both the moving speed and elastic boundaries make the Timoshenko beam theory more needed. The validity of the dynamic stiffness method is demonstrated by using numerical simulation.     

Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis

Advancements in manufacturing technology, including the rapid development of additive manufacturing (AM), allow the fabrication of complex functionally graded material (FGM) sectioned beams. Portions of these beams may be made from different materials with possibly different gradients of material properties. The present work proposes models to investigate the free vibration of FGM sectioned beams based on onedimensional (1D) finite element analysis. For this purpose, a sample beam is divided into discrete elements, and the total energy stored in each element during vibration is computed by considering either the Timoshenko or Euler-Bernoulli beam theory. Then, Hamilton's principle is used to derive the equations of motion for the beam. The effects of material properties and dimensions of FGM sections on the beam's natural frequencies and their corresponding mode shapes are then investigated based on a dynamic Timoshenko model (TM). The presented model is validated by comparison with three-dimensional (3D) finite element simulations of the first three mode shapes of the beam.     

Vibration analysis of piezoelectric laminated slightly curved beams using distributed transfer function method

Piezoelectric laminated slightly curved beams (PLSCB) is currently one of the most popular actuators used in smart structure applications due to the fact that these actuators are small, lightweight, quick response and relatively high force output. This paper presents an analytical model of PLSCB, which includes the computation of natural frequencies, mode shapes and transfer function formulation using the distributed transfer function method (DTFM). By setting the radius of curvature of the proposed model to infinity, a piezoelectric laminated straight beams (PLSB) model can be obtained. The DTFM is applied and extended to carry out the transfer function formulation of the PLSCB and PLSB models. This method will be used to solve for the natural frequencies, mode shapes and transfer functions of the PLSCB and PLSB models in exact and closed form solution without using truncated series of particular comparison or admissible functions. The natural frequencies of the cantilevered PLSCB and PLSB are calculated by the DTFM and the Rayleigh–Ritz method. The analysis indicates that the stretching–bending coupling due to curvature has a considerable effect on the frequency parameters. Increasing the radius of curvature of the PLSCB has its largest effect on the natural frequencies. But the inhomogeneity of the boundary conditions does not have any effects on the natural frequencies or system spectrum due to the both receptance and boundary transfer functions have the same characteristic equations. The method can also be generalized to the vibration analysis of non-piezoelectric composite beams with arbitrary boundary conditions.     

Analysis of curved beams using a new differential transformation based curved beam element

Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved beam element including the effects of shear deformation considering axially functionally graded material. The proposed shape functions depend on the variations in cross-sectional area, moment of inertia, curvature and material properties along the axis of the curved beam element. The static and free vibration of axially functionally graded tapered curved beams including shear deformation and rotary inertia are studied through solving several examples. Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal beams (both forms—prime and quadratic) with hinged-hinged, hinged-clamped and clamped-clamped and clamped-free end restraints. Three general taper types (depth taper, breadth taper and square taper) for rectangular cross section are studied. Out of plane vibration is studied and the lowest natural frequencies are calculated and compared with the published results. Out of plane buckling is investigated for circular beams due to radial load.     

Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli–Euler beams

The dynamic transfer matrix is formulated for a straight uniform and axially loaded thin-walled Bernoulli–Euler beam element whose elastic and inertia axes are not coincident by directly solving the governing differential equations of motion of the beam element. Bernoulli–Euler beam theory is used, and the cross section of the beam does not have any symmetrical axes. The bending vibrations in two perpendicular directions are coupled with torsional vibration and the effect of warping stiffness is included. The dynamic transfer matrix method is used for calculation of exact natural frequencies and mode shapes of the nonsymmetrical thin-walled beams. Numerical results are given for a specific example of thin-walled beam under a variety of end conditions, and exact numerical solutions are tabulated for natural frequencies and solutions calculated by the other method are also tabulated for comparison. The effects of axial force and warping stiffness are also discussed.     

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。     

An evolutionary search technique to determine natural frequencies and mode shapes of composite Timoshenko beams

Natural frequencies and mode shapes of composite Timoshenko beams are determined by a diversity guided evolutionary algorithm (DGEA) with different boundary conditions. After applying boundary conditions, frequency equation is obtained in determinant form. Then, natural frequencies and consequently mode shapes are obtained using DGEA where the absolute value of determinant is the subject of optimization. Advantages of employing DGEA are: first, all natural frequencies are produced in a simple run, second, its simplicity for implementation and third, the procedure is not computationally prohibitive. Results clearly show the applicability of the proposed method for obtaining natural frequencies and mode shapes.     

Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach

The free vibration of axially functionally graded (FG) non-uniform beams with different boundary conditions is studied using Differential Transformation (DT) based Dynamic Stiffness approach. This method is capable of modeling any beam (Timoshenko or Euler, centrifugally stiffened or not) whose cross sectional area, moment of Inertia and material properties vary along the beam. The effectiveness of the method is confirmed by comparing the present results with existing closed form solutions and numerical results. In FG beams, flexural rigidity and mass density may take majority of functions including polynomials, trigonometric and exponential functions (converted to polynomial expressions). DT based Dynamic stiffness approach is proved to be a versatile and simple approach compared to many other methods already proposed.     

复合材料叠层梁和金属梁的固有振动特性

对根据三种梁理论得到的金属梁和复合材料叠层梁的固有振动特性进行了对比性的研究对常用的三种梁理论在弹性碰撞分析中的应用进行了分析和比较     

Free vibration of axially loaded thin-walled composite Timoshenko beams

Based on shear-deformable beam theory, free vibration of thin-walled composite Timoshenko beams with arbitrary layups under a constant axial force is presented. This model accounts for all the structural coupling coming from material anisotropy. Governing equations for flexural-torsional-shearing coupled vibrations are derived from Hamilton’s principle. The resulting coupling is referred to as sixfold coupled vibrations. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for thin-walled composite beams to investigate the effects of shear deformation, axial force, fiber angle, modulus ratio on the natural frequencies, corresponding vibration mode shapes and load–frequency interaction curves.     

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.     

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.     

Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory

The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.     

Free flapewise vibration analysis of rotating double-tapered Timoshenko beams

Free vibration analysis of a rotating double-tapered Timoshenko beam undergoing flapwise transverse vibration is presented. Using an assumed mode method, the governing equations of motion are derived from the kinetic and potential energy expressions which are derived from a set of hybrid deformation variables. These equations of motion are then transformed into dimensionless forms using a set of dimensionless parameters, such as the hub radius ratio, the dimensionless angular speed ratio, the slenderness ratio, and the height and width taper ratios, etc. The natural frequencies and mode shapes are then determined from these dimensionless equations of motion. The effects of the dimensionless parameters on the natural frequencies and modal characteristics of a rotating double-tapered Timoshenko beam are numerically studied through numerical examples. The tuned angular speed of the rotating double-tapered Timoshenko beam is then investigated.     

Vibration analysis of a planar multilink manipulator using a reduced-order matrix formulation

The main scope of this paper is both to provide an efficient formulation for deriving natural frequencies and mode shapes of a multilink flexible manipulator in an arbitrary posture and discuss the importance of introducing axial deformations along with transversal ones in the same postures. The free vibration problem is solved through a global transfer matrix formulation, which is obtained by opportunely assembling the transfer matrices of links and joints associated with the desired configuration of the robot. The formulation has been chosen in order to keep the same matrix order of the problem, regardless of the number of links of the manipulator. Manipulator links are treated as Timoshenko beams by accounting for both axial loading capability and mass/inertial effects due to the joint actuators. Numerical comparisons between the solutions derived herein and those obtained through existing formulations highlight the importance of the proposed model within the frame of multilink flexible manipulators. A forced vibration analysis is finally presented for a two-link manipulator.     

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。     

黏弹性阻尼作用下轴向运动Timoshenko梁振动特性的研究

黏弹性阻尼一直是轴向运动系统的研究热点之一.以往研究轴向运动系统大都没有考虑黏弹性阻尼的影响.但在工程实际中, 存在黏弹性阻尼的轴向运动体系更为普遍.本文研究了黏弹性阻尼作用下轴向运动Timoshenko梁的振动特性.首先, 采用广义Hamilton原理给出了轴向运动黏弹性Timoshenko梁的动力学方程组和相应的简支边界条件.其次, 应用直接多尺度法得到了轴速和相关参数的对应关系, 给出了前两阶固有频率和衰减系数在黏弹性作用下的近似解析解.最后, 采用微分求积法分析了在有无黏弹性作用下前两阶固有频率和衰减系数随轴速的变化; 给出了前两阶固有频率和衰减系数在黏弹性作用下的近似数值解, 验证了近似解析解的有效性.结果表明: 随着轴速的增大, 梁的固有频率逐渐减小.梁的固有频率和衰减系数随着黏弹性系数的增大而逐渐减小, 其中衰减系数与黏弹性系数成正比关系, 黏弹性系数对第一阶衰减系数和固有频率的影响很小, 对第二阶衰减系数和固有频率的影响较大.