DQEM for free vibration analysis of Timoshenko beams on elastic foundations

 A differential quadrature element method (DQEM) based on first order shear deformation theory is developed for free vibration analysis of non-uniform beams on elastic foundations. By decomposing the system into a series of sub-domains or elements, any discontinuity in loading, geometry, material properties, and even elastic foundations can be considered conveniently. Using this method, the vibration analysis of general beam-like structures is to be studied. The governing equations of each element, natural compatibility conditions at the interface of two adjacent elements and the external boundary conditions are developed in a systematic manner, using Hamilton's principle. The present DQEM is to be implemented to Timoshenko beams resting on partially supported elastic foundations with various types of boundary conditions under the action of axial loading. The general versality, accuracy, and efficiency of the presented DQEM are demonstrated having solved different examples and compared to the exact or other numerical procedure solutions. Received: 11 October 2002/Accepted: 26 November 2002     

Dynamic response of shear-deformable axisymmetric orthotropic circular plate structures solved by the DQEM and EDQ based time integration schemes

The dynamic response of shear-deformable axisymmetric orthotropic circular plate structures is solved by using the DQEM to the spacial discretization and EDQ to the temporal discretization. In the DQEM discretization, DQ is used to define the discrete element model. Discrete dynamic equilibrium equations defined at interior nodes in all elements, transition conditions defined on the inter-element boundary of two adjacent elements and boundary conditions at the structural boundary form a dynamic equation system at a specified time stage. The dynamic equilibrium equation system is solved by the direct time integration schemes of time-element by time-element method and stages by stages method which are developed by using EDQ and DQ. Numerical results obtained by the developed numerical algorithms are presented. They demonstrate the developed numerical solution procedure.     

轴承刚度对船舶推进轴系振动传递路径影响分析

采用传递矩阵法,将船舶推进轴系简化为质量点单元、弹性支承单元和具有分布参数的梁单元。基于修正的Timoshenko梁理论,推导出推进轴系的场传递矩阵表达式。然后,引入相应的边界条件,形成方程组并实现不同轴承刚度下推进轴系轴承处的力和位移响应求解。最后,从能量的角度,对推进轴系各轴承传递路径处的功率流进行分析,并与有限元结果比较。结果表明:基于修正Timoshenko梁理论的传递矩阵法在计算推进轴系弯曲振动时是可行有效的;艉后轴承刚度对轴系振动传递影响最大,艉前轴承次之,推力轴承影响最小。     

Topology optimization of thin‐walled box beam structures based on the higher‐order beam theory

The investigation aims to formulate ground‐structure based topology optimization approach by using a higher‐order beam theory suitable for thin‐walled box beam structures. While earlier studies use the Timoshenko or Euler beams to form a ground‐structure, they are not suitable for a structure consisting of thin‐walled closed beams. The higher‐order beam theory takes into an additional account sectional deformations of a thin‐walled box beam such as warping and distortion. Therefore, a method to connect ground beams at a joint and a technique to represent different joint connectivity states should be investigated for streamlined topology optimization. Several numerical case studies involving different loading and boundary conditions are considered to show the effectiveness of employing a higher‐order beam theory for the ground‐structure based topology optimization of thin‐walled box beam structures. Through the numerical results, this work shows significant difference between optimized beam layouts based on the Timoshenko beam theory and those based on a more accurate higher‐order beam theory for a structure consisting of thin‐walled box beams. Copyright © 2015 John Wiley & Sons, Ltd.     

Transverse buckling of a rotating Timoshenko beam

Summary This work considers a group of problems associated with rotating Timoshenko beams. The beam is not assumed to be hubclamped, i.e. the axis of rotation does not necessarily pass through the beam's clamped end. Cases of physical interest involving off-clamped beams include wobbling rotors, impellor blades, and turbine blades.For clamped-free boundary conditions, we seek solutions of the governing equations which correspond to transverse buckling. For the rotor, it is known that Euler-Bernoulli beams do not have buckled modes. By contrast, the Timoshenko beam will have an infinite number of buckled modes. In the impellor blade case, both Euler-Bernoulli and Timoshenko beams will have an infinite number of buckled modes. However, the Timoshenko beam will buckle at a lower eigenrotation speed. This is also true for the case of a rotating Timoshenko beam with clamped-clamped boundary conditions, e.g. a turbine blade clamped at both the rim and hub of a rotating platform.Analytic results for both the clamped-free and clamped-clamped cases are augmented by results obtained from numerical solution of the corresponding boundary value problems.     

层合梁的弯曲与自由振动分析

本文在铁摩辛柯梁理论的基础上, 应用迭合刚度的方法和Hamilton原理, 导出了适合于层合梁静力分析和动力分析的控制方程组(在单层情况下, 将退化成Timoshenko梁的方程)及边界条件。而且, 利用所获得的控制方程, 求得了层合梁一些问题的解析解及相应的数值结果。      

Further study on the refined theory of rectangle deep beams

Based on the refined theory for narrow rectangular deep beams, two different displacement boundary conditions of the fixed end of a cantilever beam are used to study the deformation of the beam. One is the conventional simplified displacement boundary condition, and the other is a new boundary condition determined by the least squares method. Three load cases are investigated, which are a transverse shear force at the free end of the beam, a uniformly distributed load at the top surface, and a linearly distributed load at the top surface, respectively. Solutions are given for both the refined theory and the Timoshenko beam theory and are compared with the known solutions from the elastic theory and results by the finite element method. It is shown that the solutions of the refined theory coincide with those of the elastic theory; the solutions from the Timoshenko theory by using the two different displacement boundary conditions are the same; the refined theory by using the new boundary condition provides better results than using the conventional boundary condition and also better than those of the Timoshenko beam theory.     

Differential quadrature element method for static analysis of shear deformable cross‐ply laminates

In this paper, the two‐dimensional differential quadrature element method (DQEM) is developed for the static analysis of symmetric cross‐ply laminates using the first‐order shear deformation plate theory. In this study, the laminated plate, which may contain different discontinuities in loading, geometry, material, and boundary conditions, is first divided into several simple plate elements and then the differential quadrature method (DQM) is applied to each simple element. Compatibility conditions are derived to connect the plate elements so that the overall matrix equation system for the whole plate is obtained and solved. The reliability of the DQEM for solving the titled problems is examined carefully through convergence and accuracy studies and finally some numerical test examples are given to demonstrate the applicability and flexibility of this method for practical use. The methodology presented here has overcome some critical drawbacks of the global DQM but is different from the Quadrature Element Method (QEM) since only one grid point is employed to represent the interface point. Copyright © 1999 John Wiley & Sons, Ltd.     

功能梯度材料Timoshenko型剪切梁的自由振动分析

基于Timoshenko梁理论,研究各向异性功能梯度材料梁的自由振动。假设材料参数沿梁厚度方向按同一函数规律变化,建立了功能梯度材料梁的振动方程,求得简支条件下其自振频率表达式。通过算例,给出指数函数梯度变化Timoshenko梁的自振频率和模态图,结果表明不同梯度变化对材料结构动力响应有较大影响。该方法为发展功能梯度材料梁的设计与数值计算提供了理论依据。     

DQEM for analyzing dynamic characteristics of layered fluid-saturated porous elastic media

Based on the Porous Media Theory presented by de Boer, the governing differential equations for a layered space-axisymmetrical fluid-saturated porous elastic body are firstly established, in which the suitable interface conditions between layers are presented. Then, a differential quadrature element method (DQEM) is developed, and the DQEM and the second-order backward difference scheme are applied to discretize the governing differential equations of the problem in the spatial and temporal domain, respectively. In order to show the validity of the present analysis, the dynamic response of a fluid-saturated porous medium is analyzed, and the obtained numerical results are directly compared with the existing analytical results. The effects of the numbers of the elements and grid points on the convergence of the numerical results are considered. Finally, the dynamic characteristics of a layered fluid-saturated elastic soil cylinder subjected to a water pressure or a dynamic loading are studied, and the effects of material parameters are considered in detail. From the above numerical results, it can be found that the DQEM has advantages, such as little amount in computation, good stability and convergence as well as high accuracy, so it is a very efficient method for solving the problems in soil mechanics, especially such problems with discontinuities.     

Vibrations of non-uniform functionally graded MWCNTs-polystyrene nanocomposite beams under action of moving load

Free and forced vibrations of non-uniform functionally graded multi-walled carbon nanotubes (MWCNTs)-polystyrene nanocomposite beams are investigated via Timoshenko beam theory. Different MWCNTs distributions in the thickness direction are introduced to improve fundamental natural frequency and dynamic behavior of non-uniform polymer composite beam under action of moving load. So, linear distribution patterns of carbon nanotubes (CNTs) in the thickness direction which can readily be achieved in practice are studied. The effects of shear deformation, rotary inertia, non-uniformity of the cross-section are also considered in the formulation. The finite element method is employed to obtain a numerical approximation of the motion equation. The non-uniform beam is approximated by another beam consisting of n elements with piecewise constant thickness so that the volume remains constant for each element. The effects of non-uniformity parameters, material distributions, velocity of the moving load and boundary conditions on the dynamic behavior are investigated. It is found that the symmetrical linear distribution of MWCNTs results in an increase in the fundamental natural frequency of nanocomposite beams which are higher than those of beams with uniform and unsymmetrical MWCNTs distributions.     

轴向受载粘弹性Ｔｉｍｏｓｈｅｎｋｏ梁横向自由振动

用复模态分析方法研究了在两端简支和固支边界下,轴向受载粘弹性Ｔｉｍｏｓｈｅｎｋｏ梁自由振动的固有频率、衰减系数和模态函数的特性,给出了模态函数的正交性条件。通过数值算例,讨论了长宽比、轴力和粘弹性系数对梁的固有频率、衰减系数和模态函数的影响。数值结果表明增加轴力使梁的固有频率变大、模态函数幅值变小,但衰减系数变化不明显;粘弹性系数的增大对梁的固有频率与模态函数幅值影响较小,而衰减系数明显增大。     

Free out-of-plane vibration of cracked curved beams on elastic foundation by estimating the stress intensity factor

Abstract

This paper is concerned with evaluating stress intensity factors (SIFs), for a cracked curved beam of rectangular cross section, applying an approach which allows us to estimate the strain energy release rate. The beam is located on an elastic foundation. The out-of-plane vibration of the beam is investigated. This approach requires an additional factor namely correction factor, on the basis of the energy release zone slope to approximate the SIFs. The initial curvature of the beam, however, adds some complication in using this factor. The second part of this study is investigating a numerical approach, namely differential quadrature element method (DQEM), to gain the natural frequencies of the cracked beam. This method is applied to show the application of the SIFs to calculate the compliance of the cracked section for modeling the crack. The other method which is used to obtain the natural frequencies is the finite element method (FEM). The results of these two methods are found to be in good agreement, which shows the precision of the stress intensity factors of the cracked beam.     

Mixed state-vector finite element analysis for a higher-order box beam theory

If thin-walled closed beams are analyzed by the standard Timoshenko beam elements, their structural behavior, especially near boundaries, cannot be accurately predicted because of the incapability of the Timoskenko theory to predict the sectional warping and distortional deformations. If a higher-order thin-walled box beam theory is used, on the other hand, accurate results comparable to those obtained by plate finite elements can be obtained. However, currently available two-node displacement based higher-order beam elements are not efficient in capturing exponential solution behavior near boundaries. Based on this motivation, we consider developing higher-order mixed finite elements. Instead of using the standard mixed formulation, we propose to develop the mixed formulation based on the state-vector form so that only the field variables that can be prescribed on the boundary are interpolated for finite element analysis. By this formulation, less field variables are used than by the standard mixed formulation, and the interpolated field variables have the physical meaning as the boundary work conjugates. To facilitate the discretization, two-node elements are considered. The effects of interpolation orders for the generalized stresses and displacements on the solution behavior are investigated along with numerical examples.     

求解圆柱壳稳态谐响应的微分求积单元法

本文以Donnell经典壳体振动微分方程为基础,研究微分求积单元法(DQEM)在圆柱壳稳态谐响应计算中的应用。研究结果表明：微分求积单元法可较为方便的处理多种边界条件;与有限元法相比,微分求积单元法直接面向问题的微分方程,可用较少的节点获得较高的计算精度,计算效率较高。本文结果可为微分求积单元法在结构动力响应问题求解中的应用提供参考。     

Timoshenko梁功率流主动控制研究

为了研究扰动影响下梁式结构的动力学响应与主动控制,首先基于Timoshenko梁理论,采用行波方法建立了悬臂梁结构的动力学模型并获得了其在扰动下的精确动力学响应,进一步得到结构中传播的功率流,并以此为目标函数,优化得到了最优控制力的大小与相位,然后对结构施加最优控制力,实现了Timoshenko梁结构的功率流主动控制。对Timoshenko梁结构动力学响应与功率流主动控制方法进行了数值计算,并与Euler-Bernoulli梁理论计算结果进行了对比分析。结果表明：采用行波方法计算梁结构的动力学响应准确可靠;Timoshenko梁模型较Euler-Bernoulli梁模型在中、高频段更为精确,且更接近工程实际;通过数值计算与分析验证了基于行波方法功率流主动控制的正确性与有效性,并且功率流主动控制可以明显降低梁式结构全频域内的抖动。     

Shape control and damage identification of beams using piezoelectric actuation and genetic optimization

This paper deals with the shape control of beams under general loading conditions, using piezoelectric patch actuators that are surface bonded onto beams to provide the control forces. The mathematical formulation of the model is based on the shear deformation beam theory (Timoshenko theory) and the linear theory of piezoelectricity. The numerical solution of the model is based on the development of superconvergent (locking-free) finite elements using the form of the exact solution of the Timoshenko beam theory and Hamilton’s principle. The optimal values for the locations of the piezo-actuators are determined and optimal voltages for shape control are obtained for cantilever beams by using a genetic optimization procedure. Finally, a simplified related damage identification problem is formulated and solved using static data and genetic optimization.     

Winkler地基上修正Timoshenko梁振动分析

利用Bernoulli-Euler梁理论建立的弹性地基梁模型应用广泛,但其在高阶频率及深梁计算中误差较大,利用修正的Timoshenko梁理论建立新的弹性地基梁振动微分方程,由于其在Timoshenko梁的基础上考虑了剪切变形所引起的转动惯量,因而具有更好的精确度。利用ANAYS beam54梁单元进行振动模态的有限元计算,所求结果与理论基本无误差,从而验证了该理论的正确性。基于修正Timoshenko梁振动理论推导出了弹性地基梁双端自由-自由、简支-简支、简支-自由、固支-固支等多种边界条件下的频率超越方程及模态函数。分析了弹性地基梁在不同理论下不同约束条件及不同高跨比情况下的计算结果,从而论证了该理论计算弹性地基梁的适用性。分析了不同弹性地基梁理论下波速、群速度与波数的关系。得到了约束条件和梁长对振动模态及地基刚度对振动频率有重要影响等结论。     

Frequency equation and mode shape formulae for composite Timoshenko beams

Exact expressions for the frequency equation and mode shapes of composite Timoshenko beams with cantilever end conditions are derived in explicit analytical form by using symbolic computation. The effect of material coupling between the bending and torsional modes of deformation together with the effects of shear deformation and rotatory inertia is taken into account when formulating the theory (and thus it applies to a composite Timoshenko beam). The governing differential equations for the composite Timoshenko beam in free vibration are solved analytically for bending displacements, bending rotation and torsional rotations. The application of boundary conditions for displacement and forces for cantilever end condition of the beam yields the frequency equation in determinantal form. The determinant is expanded algebraically, and simplified in an explicit form by extensive use of symbolic computation. The expressions for the mode shapes are also derived in explicit form using symbolic computation. The method is demonstrated by an illustrative example of a composite Timoshenko beam for which some published results are available.     

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。