Natural frequencies of Euler-Bernoulli beam with open cracks on elastic foundations

A study of the natural vibrations of beam resting on elastic foundation with finite number of transverse open cracks is presented. Frequency equations are derived for beams with different end restraints. Euler-Bernoulli beam on Winkler foundation and Euler-Bernoulli beam on Pasternak foundation are investigated. The cracks are modeled by massless substitute spring. The effects of the crack location, size and its number and the foundation constants, on the natural frequencies of the beam, are investigated.     

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bernoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method (DTM) has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.     

A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect

A single-walled nanotube structure embedded in an elastic matrix is simulated by the nonlocal Euler-Bernoulli, Timoshenko, and higher order beams. The beams are assumed to be elastically supported and attached to continuous lateral and rotational springs to take into account the effects of the surrounding matrix. The discrete equations of motion associated with free transverse vibration of each model are established in the context of the nonlocal continuum mechanics of Eringen using Hamilton's principle and an efficient meshless method. The effects of slenderness ratio of the nanotube, small scale effect parameter, initial axial force and the stiffness of the surrounding matrix on the natural frequencies of various beam models are investigated for different boundary conditions. The capabilities of the proposed nonlocal beam models in capturing the natural frequencies of the nanotube are also addressed.     

纤维增强FGM梁的自由振动和临界屈曲载荷分析

基于经典梁理论（CBT）研究轴向力作用下纤维增强功能梯度材料（FGM）梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程,并应用微分变换法（DTM）对控制微分方程及边界条件进行变换,计算了纤维增强FGM梁在固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁,并与已有文献结果进行对比,验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

Postbuckling of shear deformable FGM cylindrical shells surrounded by an elastic medium

This paper presents a study on the postbuckling response of a shear deformable functionally graded cylindrical shell of finite length embedded in a large outer elastic medium and subjected to axial compressive loads in thermal environments. The surrounding elastic medium is modeled as a tensionless Pasternak foundation that reacts in compression only. The postbuckling analysis is based on a higher order shear deformation shell theory with von Kármán-Donnell-type of kinematic nonlinearity. The thermal effects due to heat conduction are also included and the material properties of functionally graded materials (FGMs) are assumed to be temperature-dependent. The nonlinear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the postbuckling response of the shells and an iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region between the shell and the elastic medium. Numerical solutions are presented in tabular and graphical forms to study the postbuckling behavior of FGM shells surrounded by an elastic medium of tensionless Pasternak foundation, from which the postbuckling results for FGM shells with conventional elastic foundations are also obtained for comparison purposes. The results reveal that the unilateral constraint has a significant effect on the postbuckling responses of shells subjected to axial compression in thermal environments when the foundation stiffness is sufficiently large.     

On the transverse vibrations of non-uniform beams with axial loads and elastically restrained ends

The dynamic behaviour of slender tapered beams is examined, in the presence of conservative axial loads, and lower and upper bounds on the free vibration frequencies are obtained. Two different approaches are employed, in order to obtain a narrow range to which the frequencies belong. In the first case a Rayleigh–Ritz method is used, with displacement trial functions given by linearly independent orthogonal polynomials. In the latter case the structure is reduced to rigid bars, connected together by means of elastic hinges, and lower bound to the true frequencies is obtained. It is well known that the Rayleigh–Ritz approach leads to upper bounds, and therefore a (narrow) range is obtained for the exact frequencies.The paper ends with some numerical examples which confirm the usefulness of the proposed methods, and are in good agreement with some previously known results.     

变截面智能压电梁的动态特性研究

建立变截面智能压电梁的动态模型,分析表面粘贴压电元件的均质梁的固有频率。在分析时考虑坟电元件、粘贴层、梁的剪切变形和转动惯量。研究表明,压电元件的刚度和惯量对梁的频率影响很大,变截面智能压电梁的一除非 固有频率总是高于等截面梁的固有频率。粘结层剪切 对一阶频率影响较 剪切变 转动惯量时,对梁的高阶频率影响较大。考虑梁的惊动国惯量和剪切谱形时,智能阶梯梁的频率比没有时的要低。     

An approximate analysis of the behaviour of laterally loaded, fully fixed beams and plates

An approximate method is developed for the elasto-plastic analysis of rotationally fixed and axially restrained rectangular beams and long plates subjected to lateral loading. The boundary conditions of the problem are satisfied exactly, but equilibrium is only satisfied at one point along the beam length. The load deflection behaviour can be determined by hand calculation. The behaviour of beams tested by others is analysed and the method is found to closely predict the behaviour observed experimentally. The effect of initial imperfection and axial movement at the beam ends is investigated. The possibility of beam instability is pointed out, and it is found that small movements at the beam ends can significantly affect its behaviour.     

Prediction of vibration of rotating damped beams with arbitrary pretwist

A solution procedure for the bending–bending vibration of a rotating damped beam with arbitrary pretwist and an elastically restrained root is derived. The viscous damping is assumed to be proportional to the distributed mass. The general complex system is divided into two subsystems. The physical meanings of the subsystems are studied. The exact complex frequency relations between two viscously damped beams with arbitrary pretwist and elastic root are revealed. The underdamping, critical damping and overdamping systems are analyzed. Moreover, the influence of the parameters on the decay rate, the natural frequencies, the critical damping, and the phenomenon of divergence instability are investigated.     

Torsional buckling of a DWCNT embedded on winkler and pasternak foundations using nonlocal theory

The small-scale effect on the torsional buckling of a double-walled carbon nanotube (DWCNT) embedded on Winkler and Pasternak foundations is investigated in this study using the theory of nonlocal elasticity. The effects of the surrounding elastic medium, such as the spring constant of the Winkler type and the shear constant of the Pasternak type, as well as the van der Waals (vdW) forces between the inner and the outer nanotubes are taken into account. Finally, based on the theory of nonlocal elasticity and by employing the continuum models, an elastic double-shell model is presented for the nonlocal torsional buckling load of a DWCNT. It is seen from the results that the shear constant of the Pasternak type increases the nonlocal critical torsional buckling load, while the difference between the presence and the absence of the shear constant of the Pasternak type becomes large. It is shown that the nonlocal critical buckling load is lower than the local critical buckling load. Moreover, a simplified analysis is carried out to estimate the nonlocal critical torque for the torsional buckling of a DWCNT.     

[BIM模型与智能设计]中低速磁浮系统起浮阶段的振动特性分析

为了研究电磁悬浮（EMS）中低速磁浮系统在起浮阶段的悬浮稳定性问题,建立了“车辆-电磁-轨道梁”耦合系统动力学模型。利用仿真模型分析了EMS中低速磁浮系统在起浮阶段的振动响应特性,对比分析了弹性梁与刚性梁的振动差别,仿真分析了单级悬浮控制与双级悬浮控制的效果差异。结果表明：EMS中低速磁浮列车在弹性轨道梁上起浮时,容易发生悬浮失稳的异常振动现象,原因是单级悬浮控制模式的悬浮架自激振动频率激发了弹性轨道梁的固有频率,从而诱发了车轨剧烈耦合共振;提高轨道梁刚度或采用双级悬浮控制器模式,可以缓解EMS中低速磁浮系统起浮阶段异常振动引起的悬浮失稳现象。     

A parametric study of the clustering of modes in the vibration of partially embedded beams

The phenomenon of modal clustering is investigated for any beam which is partially embedded in an elastic Winkler foundation. An exact closed-form solution to the governing equations enables an understanding of the phenomenon to be developed. A specially written computer algorithm allows a parametric study to be undertaken from which a series of charts are developed which facilitate the prediction of the existence of clustered modes for a wide range of problems. Trends in natural frequencies are explained and mode shapes are examined where clustering exists.     

Effects of shear deformation and rotary inertia on the natural frequencies of axially loaded beams

The purpose of this study is to investigate how the axial load in beams influences the relationships between the natural frequencies and the effects of shear deformation and rotary inertia. Four beam theories are considered in this study. Finite element equations of motion for the beams under a tensile load are formulated to allow the application of various axial loads as well as to impose any type of boundary conditions. The results demonstrate that the stiffening effect by a tensile load may not reduce the frequency error of the Euler beam theory, unlike the results reported in other studies.     

Dynamic identification of a reinforced concrete damaged bridge

The results of a series of harmonically forced tests carried out on a reinforced concrete single-span bridge subjected to increasing levels of damage are interpreted in this paper. The deck structure of the bridge consists of a slab and three simply supported beams. The damage is represented by a series of notches made on a lateral beam to simulate the effect of incremental concentrated damage. The variation of lower natural frequencies shows an anomalous increase in the transition from one intermediate damage configuration to the next ones. Vibration mode shapes show an appreciable asymmetry in the reference configuration, despite the nominal symmetry of the bridge. A justification of this unexpected dynamic behavior is presented in this paper. The analysis is based on progressive identification of an accurate finite element model of the reference configuration and on reconstruction of damage evolution from natural frequency and vibration mode measurements. Changes in modal curvature of the first two vibration modes evaluated along the main beams are successfully used to identify the location of the damage.     

An automatic root searching myklestad procedure for vibration analysisEin Automatisches Myklestadsches Wurzelverfahren einer

This paper presents an extension to Mykelstad's method that enables the natural frequencies of systems represented by an idealised lumped mass vibrating beam to be obtained efficiently and automatically, with the exception of these values lying within specified frequency tolerance bandwidths. The method is described and illustrated by determining the first 5 lateral natural frequencies of beams with 2 different types of end-conditions.     

Laterally restrained beam columns with uniform biaxial loading

The analytical and experimental results for sixteen beam columns with uniform biaxial loading and an enforced axis of rotation are presented. The theoretical behaviour is predicted using a finite beam column element computer program which allows for non-linear elastic effects and the formation of complex plastic hinges. The iteration procedure increases the external loads incrementally, thus allowing the complete load-deflexion behaviour up to failure to be examined. A theoretical investigation is made of the effect of flange plates at the beam column ends, the number of lateral supports and geometrical imperfections.     

Critical speeds and the response of a tensioned beam on an elastic foundation to repetitive moving loads

A finite-length tensioned beam on a damped elastic foundation is acted upon by an infinite series of equally spaced and steadily moving concentrated transverse loads. The deflection response of the beam is obtained by an expansion in terms of the normal modes of vibration. Numerical results are determined for various values of the load-spacing, beam tension, foundation stiffness and damping, and for a range of load-speeds. It is found that the critical velocities for repetitive loading exist at significantly lower speeds than would be expected based upon the well-known critical speed for a single moving load. An interpretation in terms of forced vibration response is given.     

弹性地基上转动功能梯度材料Timoshenko梁自由振动的微分变换法求解

基于Timoshenko梁理论研究弹性地基上转动功能梯度材料（FGM）梁的自由振动。首先确定功能梯度材料Timoshenko梁的物理中面,利用广义Hamilton原理推导出该梁在弹性地基上转动时横向自由振动的两个控制微分方程。其次采用微分变换法（DTM）对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功能梯度材料Timoshenko梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种不同边界条件下横向自由振动的量纲一固有频率,与已有文献的计算结果进行比较,退化后结果一致。最后讨论了不同边界条件、转速、弹性地基模量和梯度指数对功能梯度材料Timoshenko梁自振频率的影响。结果表明：功能梯度材料Timoshenko梁的量纲一固有频率随量纲一转速和量纲一弹性地基模量的增大而增大;在量纲一转速和量纲一弹性地基模量一定的情况下,梁的量纲一固有频率随着功能梯度材料梯度指数的增大而减小。     

Explicit formulation for natural frequencies of double-beam system with arbitrary boundary conditions

In this paper, free transverse vibration of two parallel beams connected through Winkler type elastic layer is investigated. Euler-Bernoulli beam hypothesis has been applied and it is assumed that boundary conditions of upper and lower beams are similar while arbitrary without any limitation even for non-ideal boundary conditions. Material properties and cross-section geometry of beams could be different from each other. The motion of the system is described by a homogeneous set of two partial differential equations, which is solved by using the classical Bernoulli-Fourier method. Explicit expressions are derived for the natural frequencies. In order to verify accuracy of results, the problem once again solved using modified Adomian decomposition method. Comparison between results indicates excellent accuracy of proposed formulation for any arbitrary boundary conditions. Derived explicit formulation is simplest method to determine natural frequencies of double-beam systems with high level of accuracy in comparison with other methods in literature.     

考虑翘曲影响的Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀薄壁梁单元弯扭耦合振动的运动偏微分方程，推导其自由振动时的精确动态传递矩阵。采用考虑翘曲影响的Bernoulli-Euler梁理论，且假定薄壁梁单元的横截面是单对称的。动态传递矩阵可以用于计算薄壁梁集合体的精确固有频率和模态形状。针对两个薄壁梁算例，采用自动Muller法和结合频率扫描法的二分法求解频率特征方程，并讨论翘曲刚度对弯扭耦合：Bernoulli-Euler薄壁梁固有频率的影响。数值结果验证了本文方法的精确性和有效性，并指出翘曲刚度可以显著改变薄壁开口截面梁的固有频率。