Vibration analysis of shape-optimized rotating cantilever beams

The shape optimization of rotating beams is carried out in order to minimize the vibrations. The objective is the maximization of the fundamental frequency with constraints on the beam mass and static tip deflections. The finite-element method (FEM) is used to model the rotating beam and sequential quadratic programming (SQP) is used for the optimization. The effects of beam frequency maximi-zation and hub–beam frequency maximization on the optimized shapes and the dynamic characteristics of these shapes are studied. The beam shapes are optimized for different speeds. The natural frequencies and time responses of these optimized shapes are studied using numerical simulation. Based on the numerical study, suggestions for formulating an appropriate optimization problem in a given context are made.     

Vibration analysis of orthotropic cantilever cylindrical shells with axial thickness variation

A semi-analytical finite element method is used to determine the free vibration characteristics of thin orthotropic cantilever circular cylindrical shells. Love's first approximation shell theory is used for the formulation. The effect of the variation of thickness along the axial direction on natural frequencies, especially on the lowest frequency, has been studied with a constraint on the total mass of the shell for a particular length to middle surface radius ratio. Studies have been conducted for different degrees of orthotropy and various values of length to radius ratio.     

Vibration analysis of orthotropic rectangular plates on elastic foundations

The vibration responses of orthotropic plates on nonlinear elastic foundations are numerically modeled using the differential quadrature method. The differential quadrature technique is utilized to transform partial differential equations into a discrete eigenvalue problem. Numerical results and those from literature closely correspond to each other. Numerical results demonstrate that elastically restrained stiffness, plate aspect ratio and foundation stiffness significantly impact the dynamic behavior of orthotropic plates.     

Vibration analysis of sandwich plates with lattice truss core

An effective methodology is developed to investigate the vibration of the sandwich plate with pyramidal lattice core. Equation of motion of lattice sandwich plate is established by Hamilton's principle. Displacement fields are expressed with a simple method, and the natural frequencies of the lattice sandwich plate are conveniently calculated. The correctness of the analytical method is verified by comparing the present results with published literatures. The effects of structural and material parameters on the vibration characteristics of lattice sandwich plate are analyzed. The present method will be useful for vibration analysis and design of lattice sandwich plates.     

Vibration analysis of a rotating flexible shaft–disk system

Free vibrations of a spinning disk–shaft system are analysed using the finite-element method. The spinning disk is described by the Kirchhoff plate theory. The shaft is modelled by a rotating beam. Using Lagrange’s principle and including the rigid-body translation and tilting motion, equations of motion of the spinning flexible disk and shaft are derived consistently to satisfying the geometric compatibility conditions on the internal boundaries among the substructures. The finite-element method is then used to discretize the derived governing equations. The method is applied to the shaft–disk spinning system. The sensitivity to the running speed as well as the effect of both disk flexibility and boundary condition on the natural frequencies of the spinning system are numerically investigated.     

Vibration of pretwisted cantilever trapezoidal symmetric laminates

Summary A first known investigation on the vibratory characteristics of pretwisted composite symmetric laminates with trapezoidal planform is presented. A governing eigenvalue equation is derived based on the Ritz minimization procedure. This formulation shows that bending and stretching effects of these symmetric laminates are coupled by the presence of twisting curvature. For the solution method, a set of orthogonally generated shape functions is employed to approximate the transverse and in-plane displacements. These functions are generated through a proposed recurrence formula. During the orthogonalization process, a basic function is introduced to ensure the satisfaction of the essential boundary conditions. This proposed method is applied to determine the vibration response of the titled problem. The effects of angles of twist and laminated layers upon the vibration frequencies are examined. Convergence tests for selected examples are presented in which the accuracy of the results is established. It has been found that an increase in the angle of twist also strengthens the torsional stiffness of the laminated plates and therefore results in higher twisting frequencies. The present solutions, where possible, are verified by comparison with data of simplified examples that are available in the literature.     

On buckling of crosswise rigid elastic cantilever plates

Summary The classical two-dimensional equations for the buckling of thin elastic anisotropic plates are reduced, on the basis of an assumption of crosswise rigidity, to a system of one-dimensional equations. The reduction pre-supposes that the crosswise dimension of the plate is small compared to it's spanwise dimension and leads, effectively, to a system of beam buckling equations which automatically associates a warping stiffness effect with the classical beam bending and twisting stiffness effects. In the event that inertia load terms are to be considered the system is of the eight order with respect to the spanwise space coordinate. In the absence of such load terms the system can be reduced to the sixth order, with further reductions possible for suitably specialized load conditions.     

Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam

In this article, we present the nonlocal, nonlinear finite element formulations for the case of nonuniform rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress effects are also taken into consideration. Nonlocal stress resultants are obtained by employing Eringen’s nonlocal differential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the effect of nonlocal parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the effects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied.     

Experimental measurement and numerical analysis on resonant characteristics of cantilever plates for piezoceramic bimorphs

Three experimental techniques are used in this study to access the resonant characteristics of piezoceramic bimorphs in parallel and series connections. These experimental methods, including the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), laser Doppler vibrometer-dynamic signal analyzer (LDV-DSA), and impedance analysis, are based on the measurement of full-field displacement, point-wise displacement, and electric impedance, respectively. Because the clear fringe patterns will be shown only at resonant frequencies, both the resonant frequencies and the corresponding vibration mode shapes are successfully obtained at the same time by the AF-ESPI method. LDV-DSA is used to determine the resonant frequencies of the vibration mode for out-of-plane motion. The impedance analysis is used to measure the resonant and antiresonant frequencies for in-plane motion. Although the out-of-plane mode is the dominant motion of piezoceramic bimorphs, it is found in this study that the amount of displacement for the in-plane motion in parallel connection is large enough to be measured by AF-ESPI and impedance. It is interesting to note that resonant frequencies of the specimen in parallel connection for the out-of-plane motion determined by LDV-DSA are the same as that for the in-plane motion obtained by impedance. Furthermore, both in-plane and out-of-plane mode shapes for the specimen in parallel connection are obtained in the same resonant frequency from the AF-ESPI method. It is concluded in this study that the particle motions of piezoceramic bimorphs for parallel connection in resonance are essentially three-dimensional. However, it is found that only out-of-plane vibration modes can be excited for the specimen in series connection. Numerical computations based on the finite-element method are presented, and the theoretical predicted results are compared with the experimental measurements. Good agreements between the experimental measured data and numerical calculated results are found for resonant frequencies and mode shapes of the piezoceramic bimorph.     

基于Bessel和Meijer-G函数的楔形和锥形悬臂梁振动分析

基于欧拉-伯努利梁理论,利用Lagrange法建立了楔形和锥形截面梁在外激作用下的非线性微分方程.提出了一种基于Bessel函数和Meijer-G函数线性组合的无需迭代及近似截断的振型函数,且该振型函数不依赖于楔形和锥形变截面梁的弯曲振动的运动方程是否为标准的Bessel形式,该方法能快速求解线性基频和模态函数.随后将...     

Vibration analysis of rotating turbomachinery blades by an improved finite difference method

The problem of calculating the natural frequencies and mode shapes of rotating blades is solved by an improved finite difference procedure based on second-order central differences. Lead-lag, flapping and coupled bending–torsional vibration cases of untwisted blades are considered. Results obtained by using the present improved theory have been observed to be close lower bound solutions. The convergence has been found to be rapid in comparison with the classical first-order finite difference method. While the computational space and time required by the present approach is observed to be almost the same as that required by the first-order theory for a given mesh size, accuracies of practical interest can be obtained by using the improved finite difference procedure with a relatively smaller matrix size, in contrast to the classical finite difference procedure which requires either a larger matrix or an extrapolation procedure for improvement in accuracy.     

Vibration of stiffened skew Mindlin plates

Summary This paper presents a formulation for the free vibration analysis of skew Mindlin plates with intermediate parallel stiffeners attached in two directions. The Mindlin theory is used to account for the effects of transverse shear deformations and rotary inertia of the plate while the Engesser theory associated with the consideration of torsion is employed for stiffeners. Based on these two theories, the energy functionals for the plate system have been derived. To obtain the vibration frequencies, these energy functionals are minimized with the shape functions assumed in a set of two-dimensional mathematically complete polynomials. This procedure has been implemented numerically to compute the vibration solutions. Convergence studies have been performed to verify the accuracy of this method. Sets of first known results have been presented for several stiffened plate structures.     

Vibration of rectangular plates with edge-beams

Summary The present paper deals with vibration of thin, isotropic, rectangular plates stiffened along two parallel edges and simply supported along the other two. An exact solution is obtained by reformulating the initial eigenfrequency problem as one of determining the wave-number codirectionally with simply-supported edges, with three different forms of eigenfrequency wave-number relationships. Examples are worked out, indicating the effect of stiffener parameters.

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Schwingungen rechteckiger Platten mit Randbalkenversteifungen

Zusammenfassung Die vorliegende Arbeit behandelt Schwingungen dünner, isotroper, rechteckiger Platten, die längs zweier paralleler Kanten versteift sind und längs der beiden anderen frei drehbar gelagert werden. Es wird eine exakte Lösung angegeben über die Neuformulierung des ursprünglichen Eigenfrequenzproblems als eines der Bestimmung der Wellenzahl in Richtung der frei drehbaren Ränder mit drei verschiedenen Beziehungen zwischen Eigenfrequenzen und der Wellenzahl. Beispiele werden angegeben, die den Effekt der Steifigkeitsparameter aufzeigen.

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Vibration analysis of laminated composite plates: TV-holography and finite element method

An experimental and numerical investigation into the structural behaviour of symmetrically laminated carbon fibre-epoxy composite rectangular plates subjected to vibration. The laminated composite plates are composed on layers of Grafil XAS carbon fibres preimpregnated in 914C Fibredux epoxy resin and each plate was vibrated by a piezoelectric transducer (PZT) attached onto its surface. The specimens tested were of two different length to width ratios and of symmetric stacking sequence. In this study the short edges of the plate were of various combinations of clamp and free support conditions, and the long edges of the plate were of various combinations of free and simple support conditions. Experimental and finite element studies were carried out in parallel. The experimental vibrational response of the test plates were obtained using a TV-holography technique. The comparison between experimental results and finite element results are reasonably good in all the cases studied.     

Vibration analysis of angle-ply laminated composite plates with an embedded piezoceramic layer

An optical full-field technique, called amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), is used in this study to investigate the force-induced transverse vibration of an angle-ply laminated composite embedded with a piezoceramic layer (piezolaminated plates). The piezolaminated plates are excited by applying time-harmonic voltages to the embedded piezoceramic layer. Because clear fringe patterns will appear only at resonant frequencies, both the resonant frequencies and mode shapes of the vibrating piezolaminated plates with five different fiber orientation angles are obtained by the proposed AF-ESPI method. A laser Doppler vibrometer (LDV) system that has the advantage of high resolution and broad dynamic range also is applied to measure the frequency response of piezolaminated plates. In addition to the two proposed optical techniques, numerical computations based on a commercial finite element package are presented for comparison with the experimental results. Three different numerical formulations are used to evaluate the vibration characteristics of piezolaminated plates. Good agreements of the measured data by the optical method and the numerical results predicted by the finite element method (FEM) demonstrate that the proposed methodology in this study is a powerful tool for the vibration analysis of piezolaminated plates.     

旋转振荡板的尾流旋涡脱落模式研究

目的:寻找不同振频和振幅条件下旋转振荡板尾流所有旋涡脱落模式,画出各模式的区域分布图,总结旋涡脱落规律。方法:通过风洞实验与数值模拟确定雷诺数Re=2 300时,尾流旋涡脱落模式为2S,2P,P+S的典型工况,验证实验结果与仿真结果是否具有一致性。板的旋转振幅变化范围为:5°～50°,无量纲振频f_eH/V_∞的变化范围为0.01～0.129,通过改变数值模拟的参数找出所有旋涡脱落模式。结果:确定了四种旋涡脱落模式的典型工况,每种模式下实验与仿真结果对比良好,在研究范围内共发现了十一种旋涡脱落模式。结论:旋转振荡板尾流的旋涡脱落模式随着振幅、振频的增加不断改变,最终全部成为2S模式。其中振频对旋涡脱落模式变化的影响较大,总体趋势是从复杂的涡结构逐渐转变成单一结构。     

Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations

This paper deals with free vibration analysis of radially functionally graded circular and annular sectorial thin plates of variable thickness, resting on the Pasternak elastic foundation. Differential quadrature method (DQM) is used to yield natural frequencies of the circular/annular sectorial plates under simply-supported and clamped boundary conditions on the basis of the classical plate theory (CPT). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is assumed to remain constant. The validity of the present solution is first examined by studying the convergence of the frequency parameters. Then, a comparison of results with those available in literature confirms the excellent accuracy of the present approach. Afterwards, the frequency parameters of the circular/annular sectorial thin plates with uniform, linear, and quadratic variations in thickness are computed for different boundary conditions and various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

Vibration analysis for piezoceramic rectangular plates using Ritz's method with equivalent constants

The transverse vibration of piezoceramic rectangular thin plates is investigated theoretically and experimentally using the Ritz's method incorporated with the defined equivalent constants. The equivalent constants are derived by comparing the characteristic equations of transverse resonant frequencies between isotropic and piezoceramic disks. By replacing the Poisson's ratio and flexural rigidity with the equivalent constants, the well-known Ritz's method can be used to investigate the transverse vibration of piezoceramic rectangular plates. Two different types of boundary conditions - clamped-free-free-free (CFFF) and clamped-free-clamped-free (CFCF) - are analyzed in this paper. For the experimental measurement, two optical techniques - amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV) - are used to validate the analytical results. Both the transverse vibration modes and resonant frequencies of piezoceramic rectangular plates are obtained by the AF-ESPI method. Numerical calculations using the finite-element method (FEM) are performed, and the results are compared with the theoretical analysis and experimental measurements. Excellent agreements are obtained for results of both resonant frequencies and mode shapes. According to the theoretical calculations with different equivalent Poisson's ratios, resonant frequency variations versus aspect ratios ranging from 0.1 to 10 also are discussed for the first several modes in the work.