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.     

Dynamic finite element method for generally laminated composite beams

A dynamic finite element method for free vibration analysis of generally laminated composite beams is introduced on the basis of first-order shear deformation theory. The influences of Poisson effect, couplings among extensional, bending and torsional deformations, shear deformation and rotary inertia are incorporated in the formulation. The dynamic stiffness matrix is formulated based on the exact solutions of the differential equations of motion governing the free vibration of generally laminated composite beam. The effects of Poisson effect, material anisotropy, slender ratio, shear deformation and boundary condition on the natural frequencies of the composite beams are studied in detail by particular carefully selected examples. The numerical results of natural frequencies and mode shapes are presented and, whenever possible, compared to those previously published solutions in order to demonstrate the correctness and accuracy of the present method.     

Flexural-torsional coupled vibration of slewing beams using various types of orthogonal polynomials

Dynamic behavior of flexural-torsional coupled vibration of rotating beams using the Rayleigh-Ritz method with orthogonal polynomials as basis functions is studied. Performance of various orthogonal polynomials is compared to each other in terms of their efficiency and accuracy in determining the required natural frequencies. Orthogonal polynomials and functions studied in the present work are : Legendre, Chebyshev, integrated Legendre, modified Duncan polynomials, the special trigonometric functions used in conjunction with Hermite cubics, and beam characteristic orthogonal polynomials. A total of 5 cases of beam boundary conditions and rotation are studied for their natural frequencies. The obtained natural frequencies and mode shapes are compared to those available in various references and the results for coupled flexural-torsional vibrations are especially compared to both previously available references and with those obtained using NASTRAN finite element package. Among all the examined orthogonal functions, Legendre orthogonal polynomials are the most efficient in overall CPU time, mainly because of ease in performing the integration required for determining the stiffness and mass matrices.     

A further study about the behaviour of foundation piles and beams in a Winkler–Pasternak soil

The natural vibrations and critical loads of foundation beams embedded in a soil simulated with two elastic parameters through the Winkler–Pasternak (WP) model are analysed. General end supports of the beam are considered by introducing elastic constraints to transversal displacements and rotations. The solution is tackled by means of a direct variational methodology previously developed by the authors who named it as whole element method. The solution is stated by means of extended trigonometric series. This method gives rise to theoretically exact natural frequencies and critical loads. A particular behaviour arises from the analysis of the lateral soil influence. It is found that the boundary conditions of the beam are influenced by the soil at the left and right sides of the beam. The possible alternatives are that the soil be cut or dragged by the non-fixed ends of the beam. In the standard WP model, the lateral soil influence is not considered. Natural frequencies and critical load numerical values are reported for beams and piles elastically supported and for various soil parameters. The results are found with arbitrary precision depending on the number of terms taken in the series. Some unexpected modes and eigenvalues are found when the different alternatives are studied. It should be noted that this special behaviour is present only when the Pasternak contribution is taken into account.     

利用传递矩阵法对阶梯悬臂梁的裂纹参数识别

基于Bernoulli-Euler梁振动理论,以等效弹簧模拟裂纹引起的局部软化效应,利用传递矩阵法推导阶梯悬臂梁振动频率的特征方程,对于含多个裂纹以及复杂边界条件的阶梯梁,仅需求解4×4的行列式即可获得相应的频率特征方程。直接利用该特征方程,提出两种有效估计裂纹参数的方法———等值线法和目标函数最小化法,并应用两段阶梯悬臂梁的数值算例说明方法的有效性。算例结果表明,只需结构前三阶频率即可识别裂纹位置和深度。应用“零设置”可减小计算频率与理论频率不相等对识别结果的影响。等值线法可以直观给出裂纹位置和裂纹深度参数,目标函数最小化法可给出最优的裂纹参数结果,并且该方法可推广应用到含多个裂纹复杂梁(如非完全固支、弹性支撑等)结构的裂纹参数识别中。     

Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions

An analytical approach for crack identification procedure in uniform beams with an open edge crack, based on bending vibration measurements, is developed in this research. The cracked beam is modeled as two segments connected by a rotational mass-less linear elastic spring with sectional flexibility, and each segment of the continuous beam is assumed to obey Timoshenko beam theory. The method is based on the assumption that the equivalent spring stiffness does not depend on the frequency of vibration, and may be obtained from fracture mechanics. Six various boundary conditions (i.e., simply supported, simple–clamped, clamped–clamped, simple–free shear, clamped–free shear, and cantilever beam) are considered in this research. Considering appropriate compatibility requirements at the cracked section and the corresponding boundary conditions, closed-form expressions for the characteristic equation of each of the six cracked beams are reached. The results provide simple expressions for the characteristic equations, which are functions of circular natural frequencies, crack location, and crack depth. Methods for solving forward solutions (i.e., determination of natural frequencies of beams knowing the crack parameters) are discussed and verified through a large number of finite-element analyses. By knowing the natural frequencies in bending vibrations, it is possible to study the inverse problem in which the crack location and the sectional flexibility may be determined using the characteristic equation. The crack depth is then computed using the relationship between the sectional flexibility and the crack depth. The proposed analytical method is also validated using numerical studies on cracked beam examples with different boundary conditions. There is quite encouraging agreement between the results of the present study and those numerically obtained by the finite-element method.     

The free vibrations of a thin circular finite rotating cylinder

The natural frequencies of a finite circular thin cylinder are obtained by employing an exponential matrix expansion of the so-called “fundamental matrix”. It is shown that the method is general enough and able to handle any system of linear differential equations of constant coefficients together with arbitrary boundary conditions. Results are given for rotating cylinders with clamped and free edges. The vibration frequencies of a stationary finite cylinder, previously obtained by other methods of solution, are used as a check on the present method with the special case of zero spinning velocity.     

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.     

In-plane free vibration analysis of curved Timoshenko beams by the pseudospectral method

The pseudospectral method is applied to the analysis of in-plane tree vibration of circularly curved Timoshenko beams. The analysis is based on the Chebyshev polynomials and the basis functions are chosen to satisfy the boundary conditions. Natural frequencies are calculated for curved beams of rectangular and circular cross sections under hinged-hinged, clampedclamped and hinged-clamped end conditions and the results are compared with those by transfer matrix method. The present method gives good accuracy with only a limited number of collocation points.     

求解任意梁的普遍化方法

介绍了一种求解任意弹性梁的新方法。该方法利用奇异函数与拉普拉斯变换相结合的方法导出弹性梁弯曲变形的普遍表达式,并利用边界条件和约束处的变形协调条件确定约束反力和变形常数(左端面的挠度和转角),对由固定和活动铰链支座、径向和角度弹性支承以及固定端等支承形式任意组合而成的,具有任意支承沉降的,承受任意载荷(集中力、集中力偶和均布力)的,具有任意阶梯形状的静定或超静定弹性梁具有普遍的适用性。该方法可以方便、准确地确定任意梁在支座处的约束反力以及任一截面的挠度和转角等参数,可用于复杂梁的计算机分析、优化设计和计算机辅助设计。     

Vibrations of rectangular, stepped thickness plates with edges elastically restrained against rotation

The title problem is solved using simple polynomial coordinate functions which identically satisfy the boundary conditions. The Rayleigh-Ritz method is used to evaluate the natural frequencies. Numerical results are presented for several values of the parameter length to width ratio and a particular type of thickness variation.It is shown that the approximate approach presented herein is valid for a large combination of boundary conditions.     

Analysis of cold rigid-plastic axisymmetric forging problem by radial basis function collocation method

In the present work, an axi-symmetric cold forging problem is analyzed using radial basis function collocation method. The material is assumed to be rigid-plastic strain hardening. At each increment of the punch displacement, the problem is solved using an Eulerian control volume approach. The mixed pressure-velocity formulation is adopted, in which the hydrostatic stress and velocities are approximated by linear combinations of multiquadrics radial basis functions, the coefficients of which are obtained by satisfying the continuity and equilibrium equations at certain points called collocation points. The resulting non-linear equations are solved using a trust region method available in MATLAB, which is based on interior-reflective Newton method. Because of the nature of the equations, hydrostatic stress values contain spurious terms. To eliminate them, boundary conditions on hydrostatic stress are required, which are not known initially. Therefore the problem is solved in two stages. In the first stage, the problem is solved without any boundary condition for the hydrostatic stress and the forging load is computed by dividing the total power by the punch velocity. The hydrostatic stress at the punch-workpiece interface is obtained from the known forging load. In the second stage, the problem is solved again by putting the additional hydrostatic stress boundary conditions. Computational performance of the proposed method is studied by carrying out parametric study.     

On the dynamic response of beams with multiple geometric or material discontinuities

An analytic framework is developed for determining closed form expressions for the natural frequencies, mode shapes, and frequency response function for Euler–Bernoulli beams with any number of geometric or material discontinuities. The procedure uses a convenient matrix formulation to generalize the single discontinuity beam problem to beams with multiple step changes. Specifically, the multiple discontinuity beam problem is solved by analyzing the total structure as a series of distinct Euler–Bernoulli elements with continuity and compatibility enforced at separation locations. The method yields each respective section's eigenmode which may then be superpositioned to give the entire beam's mode shape and derivation of the frequency response function follows. Although the Euler–Bernoulli beam problem is demonstrated, any one-dimensional continuous structure is amenable to the prescribed analysis. Theoretical predictions are experimentally validated as well.     

Natural frequencies,modes and critical speeds of axially moving Timoshenko beams with different boundary conditions

In this paper, natural frequencies, modes and critical speeds of axially moving beams on different supports are analyzed based on Timoshenko model. The governing differential equation of motion is derived from Newton's second law. The expressions for various boundary conditions are established based on the balance of forces. The complex mode approach is performed. The transverse vibration modes and the natural frequencies are investigated for the beams on different supports. The effects of some parameters, such as axially moving speed, the moment of inertia, and the shear deformation, are examined, respectively, as other parameters are fixed. Some numerical examples are presented to demonstrate the comparisons of natural frequencies for four beam models, namely, Timoshenko model, Rayleigh model, Shear model and Euler–Bernoulli model. Finally, the critical speeds for different boundary conditions are determined and numerically investigated.     

利用局部振动频率识别框架结构构件刚度参数

鉴于依据结构整体动力反应很难准确识别结构局部构件的物理参数,提出了结构局部激振检测方法,既对结构局部进行激振,依据结构局部激振反应提取其固有频率,再利用此固有频率运用方法识别结构局部构件的物理参数。本文将框架结构中的梁柱简化为两端有弹性约束、具有等刚度和等分布质量的杆件模型,推导了这一模型的频率特征方程,并进一步分析了两端弹性约束对其振动固有频率的影响。仿真算例表明：在固有频率误差较小的情况下,本文方法可准确地给出构件抗弯刚度和边界条件。     

Large amplitude free vibration analysis of Timoshenko beams using a relatively simple finite element formulation

Free vibration analysis of uniform isotropic Timoshenko beams with geometric nonlinearity is investigated through a relatively simple finite element formulation, applicable to homogenous cubic nonlinear temporal equation (homogenous Duffing equation). Geometric nonlinearity is considered using von-Karman strain displacement relations. The finite element formulation begins with the assumption of the simple harmonic motion and is subsequently corrected using the harmonic balance method. Empirical formulas for the non-linear to linear radian frequency ratios, for the boundary conditions considered, are presented using the least square fit from the solutions of the same obtained for various central amplitude ratios. Numerical results using the empirical formulas compare very well with the results available from the literature for the classical boundary conditions such as the hinged–hinged, clamped–clamped and clamped–hinged beams. Numerical results for the beams with non-classical boundary conditions such as the hinged-guided and clamped-guided, hitherto not studied, are also presented.     

Exact dynamic stiffness matrix of a Timoshenko three-beam system

An exact dynamic stiffness matrix is established for an elastically connected three-beam system, which is composed of three parallel beams of uniform properties with uniformly distributed-connecting springs among them. The formulation includes the effects of shear deformation and rotary inertia of the beams. The dynamic stiffness matrix is derived by rigorous use of the analytical solutions of the governing differential equations of motion of the three-beam system in free vibration. The use of the dynamic stiffness matrix to study the free vibration characteristics of the three-beam system is demonstrated by applying the Muller root search algorithm. Numerical results for the natural frequencies and mode shapes of the illustrative examples are discussed for 10 interesting boundary conditions and three different stiffness constants of springs.     

任意连续梁内力包络图的自动绘制法

根据内力包络图设计梁的断面,可以保证满足安全和经济两方面的要求。但是按传统方法绘制梁的内力包络图手续十分繁琐,对于连续梁之类的多跨梁尤其如此。介绍一种利用奇异函数对连续梁进行力学分析的新方法及其智能分析软件。该方法利用奇异函数与拉普拉斯变换相结合的方法导出连续梁变形和内力的普遍表达式,利用变形和内力的边界条件确定约束反力,然后再在内力普遍表达式的基础上建立一种内力包络图的自动绘制方法。该方法和程序对承受任意恒载和活载、具有任意跨数的连续梁具有普遍的适用性。     

Numerical differential quadrature method for Reissner/Mindlin plates on two-parameter foundations

Approximate solutions for the bending of moderately thick rectangular plates on two-parameter elastic foundations (Pasternak-type) as described by Mindlin's theory are presented. The plates are subjected to an arbitrary combination of clamped and simply-supported boundary conditions. An efficient computational technique, the differential quadrature (DQ) method, is employed to transform the governing differential equations and boundary conditions into a set of linear algebraic equations for approximate solutions. These resulting algebraic equations are solved numerically. In this study, the accuracy of the DQ method is established by direct comparison with results in the existing literature. The convergence properties of the method are illustrated for different combinations of boundary conditions. The deflections, moments and shear forces at selected locations are tabulated in detail for different elastic foundations. The efficiency and simplicity of the solution method are highlighted.     

Analysis of the vibration behavior of FGM cylindrical shells including internal pressure and ring support effects based on Love-Kirchhoff theory with various boundary conditions

This paper presents the study on vibration behavior of functionally graded material (FGM) cylindrical shell with the effects of internal pressure and ring support. The FGM properties are graded along the thickness direction of the shell. The FGM shell equations with internal pressure and ring support are established based on strain-displacement relationship using Love-Kirchhoff shell theory. The governing equations of motion were solved by using energy functional and by applying Ritz method. The boundary conditions represented by end conditions of the FGM cylindrical shell are simply supported-simply supported (SS-SS), clamped-clamped (C-C), free-free (F-F), clamped-free (C-F), clamped-simply supported (C-SS), free-simply supported (F-SS), free-sliding (F-SL) and clamped-sliding (C-SL). To check the validity and accuracy of the present method, the results obtained are compared with those available in the literature. The influence of internal pressure, ring support position and the effect of the different boundary conditions on natural frequencies characteristics are studied. These results presented can be used as important benchmark for researchers to validate their numerical methods when studying natural frequencies of shells with internal pressure and ring support.