Vibration Analysis Of Timoshenko Beams Research Articles

Thermally induced vibration analysis of Timoshenko beams based on the micropolar thermoelasticity

Thermally induced vibration analysis of Timoshenko beams based on the micropolar thermoelasticity

Nonlinear free vibration analysis of Timoshenko beams with porous functionally graded materials

基于Timoshenko梁变形理论研究多孔功能梯度材料梁的非线性自由振动问题。针对多孔功能梯度材料梁的孔隙均匀分布和孔隙线性分布2种形式, 根据广义Hamilton原理推导多孔功能梯度材料Timoshenko梁的非线性自由振动的控制微分方程组并对方程组进行无量纲化。采用微分变换法(DTM)对各种边界条件下的控制微分方程组进行变换, 得到等价代数特征方程。计算了多孔功能梯度材料Timoshenko梁在固支-固支(C-C)、固支-简支(C-S)、简支-简支(S-S)和固支-自由(C-F)4种边界条件下非线性横向自由振动的无量纲固有频率比值。将其退化为无孔隙功能梯度材料Timoshenko梁的非线性自由振动后, 所得非线性无量纲固有频率比值与已有文献的计算结果进行对照, 验证了文中方法的有效性和正确性, 讨论了边界条件、孔隙率、细长比和梯度指数对多孔功能梯度材料Timoshenko梁非线性无量纲固有频率比值的影响。

Free vibration analysis of Timoshenko beams reinforced by BNNTs and a comparison with CNT-reinforced composite

This paper studies free vibration of composite beams reinforced by boron nitride nanotubes (BNNTs), lying on an elastic foundation. The beam matrix is taken as poly methyl methacrylate. The nanotubes have different layouts in the matrix (aligned and randomly oriented) and are also distributed through the beam thickness as one uniform and three different functionally graded distributions. A micromechanical model is applied to estimate the properties of BNNT-reinforced composite (BNNTRC). Equations of motion are achieved by means of Hamilton’s principle as well as Timoshenko beam theory. Natural frequencies are gained by introducing generalized differential quadrature method to the equations. Investigating the influence of different parameters as diverse as nanotube volume fraction, nanotubes distribution type, the elastic foundation, boundary conditions and the beam’s slenderness ratio on the natural frequency, reveals that any changes in aforementioned parameters has a significant impact on the natural frequency. Furthermore, the results for aligned BNNTRC beam are measured against the results of similar CNT-reinforced beam in the literature, illustrating that natural frequency of BNNTRC beam is lower than CNTRC beam. Finally, the comparison between natural frequencies of two BNNTRC beams with two different alignments of BNNTs, illustrates their different behavior due to different factors.

A unified procedure for the vibration analysis of elastically restrained Timoshenko beams with variable cross sections

A generalized analytical method is developed for the vibration analysis of Timoshenko beams with elastically restrained ends. For a beam with any variable cross section along the lengthwise direction, the finite element method is the only unified approach to handle those kinds of problems, since the analytical solutions could not be obtained by the governing equations when the cross section area and the second moment of area changing variably lengthwise. In this article, a unified approach is proposed to study the Timoshenko beam with any variable cross sections. The cross section area and second moment of area of the beam are both expanded into cosine series, which are mathematically capable of representing any variable cross section. The translational displacement and rotation of cross section are expressed in the Fourier series by adding some admissible functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. By using Hamilton's principle, the eigenvalues and the coefficients of the Fourier series are both obtained. Some examples are presented to illustrate the excellent accuracy of this method. Analytical solutions of the vibration of the beam are achieved for different combinations of boundary conditions including classical and elastically restrained ones. The derived results can be used as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of Timoshenko beams with any variable cross section.

Static and free vibration analysis of Timoshenko beam based on combined peridynamic-classical theory besides FEM formulation

In this paper, combination of classical and peridynamic theories is implemented to study static and free vibrational behavior of a Timoshenko beam. Employing Hamilton’s principle, the governing integro-differential equations are developed. Due to problematic analytical solution, FEM formulations are constructed and necessary computer codes are developed. Static and vibrational problems are numerically studied to reveal influential parameters. Mesh sensitivity analysis and comparison with relevant results reported in the open literature are done for verification of the methodology and ensuring reliable results. Effects of peridynamic parameters, boundary conditions, and the beam size are studied. Regarding each influencing factor, relevant illustration is presented and the results are discussed from the physical point of view.

An accurate solution method for the vibration analysis of Timoshenko beams with general elastic supports

In this investigation, an accurate solution method is presented for the free vibrations of Timoshenko beams with general elastic restraints at the end points, a class of problems which are rarely attempted in the literatures. Unlike in most existing studies where solutions are often developed for a particular type of boundary conditions, the current method can be generally applied to a wide range of boundary conditions with no need of modifying solution algorithms and procedures. Under the current framework, the displacement and rotation functions are generally sought, regardless of boundary conditions, as an improved trigonometric series in which several supplementary functions introduced to remove the potential discontinuities with the displacement components and its derivatives at the end points and accelerated the series expansion. Mathematically, the current Fourier series expansion is an exact solution for a class of problems with the Timoshenko beam such that both the governing equations and the boundary conditions simultaneously satisfy any specified degree of accuracy. The effectiveness and reliability of the presented solution are demonstrated by comparing the present results with those results published in literatures and finite element method data, and numerous new results for beams with elastic boundary restraints is presented, which may serve as benchmark solution for future researches.

Continuous model for flexural vibration analysis of Timoshenko beams with a vertical edge crack

In this paper, a continuous model for flexural vibration of beams with a vertical edge crack including the effects of shear deformation and rotary inertia is presented. The crack is assumed to be an open-edge crack perpendicular to the neutral plane. A quasi-linear displacement filed is suggested for the beam, and the strain and stress fields are calculated. The governing equation of motion for the beam has been obtained using Hamilton principle. The equation of motion is solved with a modified weighted residual method, and the natural frequencies and mode shapes are obtained. The results are compared to the results of similar model with Euler–Bernoulli assumptions and finite element model to confirm the advantages of the proposed model in the case of short beams.

Thermal post-buckling & large amplitude free vibration analysis of Timoshenko beams: Simple closed-form solutions

In present work, thermal post-buckling and large amplitude free vibration (including the effect of rotary inertia) behavior of prismatic, shear flexible Timoshenko beams is expressed in the form of simple closed-form solutions by making use of the Rayleigh–Ritz method. The total cross-section rotation field (θ) variable of the shear flexible beam based on Timoshenko beam theory is expressed as function of lateral displacement field (w) variable of the beam by appropriately making use of the homogeneous form of governing static differential equations of the Timoshenko beam. Numerical accuracy of the proposed closed-form solutions obtained from the Rayleigh–Ritz method are compared to the available literature values which indicates simplicity, accuracy and robustness of the proposed approach.

Vibrations of Timoshenko beams with isogeometric approach

A study on the free vibration analysis of Timoshenko beams is presented here. In order to determine natural frequencies of beams, a thick beam element is developed by using isogeometric approach based on Timoshenko beam theory which allows the transverse shear deformation and rotatory inertia effect. Three refinement schemes such as h-, p- and k-refinement are used in the analysis and the identification of shear locking is also conducted by using numerical examples. From numerical results, the present element can produce very accurate values of natural frequencies and the mode shapes due to exact definition of the geometry. With higher order basis functions, there is no shear locking phenomenon in very thin beam situations. Finally, the benchmark tests described in this study are provided as future reference solutions for Timoshenko beam vibration problem.

Finite Element Formulation for Stability and Free Vibration Analysis of Timoshenko Beam

A two-node element is suggested for analyzing the stability and free vibration of Timoshenko beam. Cubic displacement polynomial and quadratic rotational fields are selected for this element. Moreover, it is assumed that shear strain of the element has the constant value. Interpolation functions for displacement field and beam rotation are exactly calculated by employing total beam energy and its stationing to shear strain. By exploiting these interpolation functions, beam elements' stiffness matrix is also examined. Furthermore, geometric stiffness matrix and mass matrix of the proposed element are calculated by writing governing equation on stability and beam free vibration. At last, accuracy and efficiency of proposed element are evaluated through numerical tests. These tests show high accuracy of the element in analyzing beam stability and finding its critical load and free vibration analysis.

Vibration of Timoshenko Beams Using Non-classical Elasticity Theories

This paper presents a comparison among classical elasticity, nonlocal elasticity, and modified couple stress theories for free vibration analysis of Timoshenko beams. A study of the influence of rotary inertia and nonlocal parameters on fundamental and higher natural frequencies is carried out. The nonlocal natural frequencies are found to be lower than the classical ones, while the natural frequencies estimated by the modified couple stress theory are higher. The modified couple stress theory results depend on the beam cross-sectional size while those of the nonlocal theory do not. Convergence of both non-classical theories to the classical theory is observed as the beam global dimension increases.

Vibration of Timoshenko Beams Using Non-classical Elasticity Theories

This paper presents a comparison among classical elasticity, nonlocal elasticity, and modified couple stress theories for free vibration analysis of Timoshenko beams. A study of the influence of rotary inertia and nonlocal parameters on fundamental and higher natural frequencies is carried out. The nonlocal natural frequencies are found to be lower than the classical ones, while the natural frequencies estimated by the modified couple stress theory are higher. The modified couple stress theory results depend on the beam cross-sectional size while those of the nonlocal theory do not. Convergence of both non-classical theories to the classical theory is observed as the beam global dimension increases.

Free vibration analyses of Timoshenko beams with free edges by using the discrete singular convolution

In this paper, the free vibration analysis of Timoshenko beams with various combinations of boundary conditions are performed by using the discrete singular convolution (DSC). Since Timoshenko beams with clamped-free, pinned-free, slide-free, and free-free boundaries have not been successfully solved by using the DSC thus far, the main objective of the present paper is focused on the way to apply the free boundary conditions in using the DSC for the free vibration analysis of Timoshenko beams. A simple method to apply the free boundary conditions is proposed and verified. Results for various ratios of thickness-to-length (h/L) are presented and compared with either existing solutions in the literature or the results obtained by the differential quadrature method (DQM). It is shown that accurate results can be obtained by using the DSC together with the proposed method for applying the free boundary conditions. The research extends the application range of the DSC method.

New formulation for vibration analysis of Timoshenko beam with double-sided cracks

It is the intention of this study to synthesize the effects of double-edge cracks on the dynamic characteristics of a beam. The stiffness matrix is first determined for a Timoshenko beam containing two same-line edge cracks. The presented model is then developed for elements with two parallel double-sided cracks, considering the interaction between the stress fields of adjacent cracks. Finally, a finite element code is implemented, to examine the influence of depth and location of double cracks, on the natural frequencies of the damaged system.

Free vibration analysis of Timoshenko beams by DSC method

AbstractFree vibration analysis of Timoshenko beams has been presented. Discrete singular convolution method is used for numerical solution of equation of motion of Timoshenko beam. Clamped, pinned and sliding boundary conditions and their combinations are taken into account. Typical results are presented for different parameters and boundary conditions. Numerical results are presented and compared with that available in the literature. It is shown that very good results are obtained. This method is very effective for the study of vibration problems of Timoshenko beam. Copyright © 2009 John Wiley & Sons, Ltd.

Vibration Analysis of Timoshenko Beams on a Nonlinear Elastic Foundation

The vibrations of beams on a nonlinear elastic foundation were analyzed considering the effects of transverse shear deformation and the rotational inertia of beams. A weak form quadrature element method (QEM) is used for the vibration analysis. The fundamental frequencies of beams are presented for various slenderness ratios and nonlinear foundation parameters for both slender and short beams. The results for slender beams compare well with finite element results. The analysis shows that the transverse shear deformation and the nonlinear foundation parameter significantly affect the fundamental frequency of the beams.

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part II: applications and validation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. In Part I the governing equations of the aforementioned problem have been derived, leading to the formulation of five boundary value problems with respect to the transverse displacements, to the axial displacement and to two stress functions. These problems are numerically solved using the Analog Equation Method, a BEM based method. In this Part II, numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. Thus, the results obtained from the proposed method are presented as compared with those from both analytical and numerical research efforts from the literature. More specifically, the shear deformation effect in nonlinear free vibration analysis, the influence of geometric nonlinearities in forced vibration analysis, the shear deformation effect in nonlinear forced vibration analysis, the nonlinear dynamic analysis of Timoshenko beams subjected to arbitrary axial and transverse in both directions loading, the free vibration analysis of Timoshenko beams with very flexible boundary conditions and the stability under axial loading (Mathieu problem) are presented and discussed through examples of practical interest.

Vibration analysis of Timoshenko beams under uniform partially distributed moving masses

The current article presents an analytical approach, as well as a calculation method for determining the dynamic response of Timoshenko beams under uniform partially distributed moving masses, using the so-called discrete element technique (DET). It is shown that the proposed methodology offers a compact and computationally efficient way of conducting parametric studies to evaluate the dynamic response of the beam-like structures with arbitrary boundary conditions such as railway and road bridges. First, a formulation is presented in a matrix form for beams under partially distributed masses. The results have been validated against analytical formulations, finite difference, and finite element results. A number of parametric studies were conducted to assess the effects of the moving mass velocity and moving mass length on the beam deflection. A major aspect of the work is the study of successive travelling masses for which the transient effects are important. Both analytical and DET results showed that the critical speeds were affected for successive moving masses. Finally, the effects of rotary inertia were studied for a range of inertia values to quantify the error arising from using the Bernoulli-Euler formulation for beams with large inertias.

Higher-Order Nonlinear Vibration Analysis of Timoshenko Beams by the Spline-Based Differential Quadrature Method

Higher-order nonlinear vibrations of Timoshenko beams with immovable ends are studied. The nonlinear effects of axial deformation, bending curvature and transverse shear strains are considered. The nonlinear governing differential equations are solved using a spline-based differential quadrature method (SDQM), which is constructed based on quartic B-splines. Ratios of the nonlinear to the linear frequencies are extracted and their variations with the ratio of amplitude to radius of gyration are examined. In contrast to the well-recognized finding for the nonlinear fundamental frequency of beams, some higher-order nonlinear frequencies decrease with the increase of ratio of amplitude to radius of gyration.

Wave Reflection and Transmission in Timoshenko Beams and Wave Analysis of Timoshenko Beam Structures

This paper concerns wave reflection, transmission, and propagation in Timoshenko beams together with wave analysis of vibrations in Timoshenko beam structures. The transmission and reflection matrices for various discontinuities on a Timoshenko beam are derived. Such discontinuities include general point supports, boundaries, and changes in section. The matrix relations between the injected waves and externally applied forces and moments are also derived. These matrices can be combined to provide a concise and systematic approach to vibration analysis of Timoshenko beams or complex structures consisting of Timoshenko beam components. The approach is illustrated with several numerical examples.