Localization of a delamination and estimation of its length in a composite laminate beam by the VSHM and pattern recognition methods

The purpose of this study was to investigate the delamination damage in laminate composite beams in order to adapt the vibration-based structural health monitoring (VSHM) method for laminated structures. The analysis was concentrated on the vibration characteristics of laminated specimens, in particular, on the first several natural frequencies of a composite laminate beam with a delamination damage. The core of this work is an experimental investigation into the vibration response of a composite laminate beam and its changes caused by delaminations of different sizes and different location in the beam. The aim was to determine how the first six harmonic frequencies are changed by a delamination, and the results show that they can be successfully used to clarify the presence, location, and dimensions of delaminations in a composite beam. A pattern recognition analysis was used to locate the damage, while its detection and evaluation were performed by using changes in the harmonic frequencies. A finite-element analysis was carried out, and the variations in the natural frequencies due to delamination are found to be in good agreement with experimental results.     

Vibration monitoring of cylindrical shells using piezoelectric sensors

From linear vibration theory for beams and plates, one can express the response as a linear combination of its natural modes. For beams, these eigenfunctions can be shown to be mutually orthogonal for any boundary conditions. For plates, orthogonality of the modes is not guaranteed, but can be proven for various boundary conditions. Modal analysis for beams and plates allows the system response to be broken down into simpler vibration models, due to the orthogonality of the modes. Here the modal analysis approach is extended to the vibration of thin cylindrical shells. The longitudinal, radial, and circumferential displacements are coupled with each other, due to Poisson's ratio and the curvature of the shell. As will be shown, the mode shapes can be solved analytically with numerically determined coefficients. The immediate application of this work will be for modal sensing of cylindrical shell vibrations using thin piezoelectric films.     

Finite volume analysis of adaptive beams with piezoelectric sensors and actuators

This paper presents a finite volume (FV) formulation for the free vibration analysis and active vibration control of the smart beams with piezoelectric sensors and actuators. The governing equations based on Timoshenko beam theory are discretized using the finite volume method. For the purpose of forced vibration control of beam structures, the negative velocity feedback controller is designed for the single-input, single-output system. To achieve the best effect, the piezoelectric sensors and actuators are coupled with the host structure in different positions and then the performance of the designed control system is evaluated for each position. In the test examples, first the shear locking free feature of the present formulation is demonstrated. This has been performed by doing static and natural frequency analysis of some reference models. Then, the capability of the proposed method for the prediction of uncontrolled forced vibration response and active vibration control of a beam structure is studied.     

An analytical solution for dynamic behavior of a beam–column frame with a tip body

This paper analyzes the vibration characteristics of a beam-column frame, typical examples of which are often found in optical pickup actuators of optical disc drives (ODDs) and many architectural structures. The dynamic behaviour of this beam structure is predicted by solving mathematically its vibration characteristics governed by beam configurations. For practical applications and simplicity in the analysis, the vibration analysis for the structure is limited to lateral and longitudinal directions of the beams. As a result, mode and modal frequencies are obtained from mathematical expressions. The accuracy of vibration characteristics, which is mathematically induced, is demonstrated by a finite element (FE) analysis. Finally, it is shown that mode shapes are modified by using design values with the mathematical expressions.     

An electromechanically coupled intrinsic,mixed variational formulation for geometrically nonlinear smart composite beam

The work presented in this article is the outcome of a combined strategy of a mathematical tool for 2D cross-sectional analysis, i.e., Variational Asymptotic Method (VAM) as well as the 1D exact beam analyzer, i.e., the intrinsic mixed variational formulation for modeling and analysis of Piezoelectric-laminated composite beams. This work talks about a novel approach of mixed variational formulation to analyze a two-way electromechanically coupled piezoelectric composite beam. In a classical intrinsic mixed variational approach for a passive structure, the 1D exact beam model deals only with mechanical degrees of freedom. In the present case, an extra 1D electrical degree of freedom has been incorporated. A computational code is developed based on the present theory to solve the two-way coupled electromechanical beam problem. In the present case, we have validated the static results for sensor application. Both linear and nonlinear results have been discussed. Results obtained are very promising and are helpful in building a platform where design, optimization and nonlinear analysis of composite ‘smart’ beams in a multibody framework can be done faster while maintaining acceptable accuracy.     

Characterisation and Modelling of Piezoceramic Actuator Hystereses

Piezoelectric ceramics are often used as actuators in smart structures technology. In the vast majority of papers dealing with this topic only linear constitutive relations are used. However, the electric field-strain relations of such actuators show hysteretic behaviour, which means that the piezoelectric coupling coefficient is not constant. In this study the hysteresis of a mechanically unconstrained actuator is obtained using the Michelson interferometry. The hysteretic behaviour is modelled by a Preisach model. Using these experimental data, for the modelling of an active structure with embedded piezoelectric actuators the actual coupling coefficient can then be determined with the help of the Preisach model. With this procedure the actuation strain of an embedded actuator, including the physical nonlinearities, can be calculated using the material characteristics obtained for an unconstrained actuator. For an experimental validation of the method outlined above, a Lead Zirconate Titanate (PZT) actuator is characterised experimentally and then glued to a cantilever beam. Then, the tip displacement of the actuated beam is determined experimentally and simulated numerically using the above method. The experimental and numerical results agree reasonably well if the shear lag due to the bonding layer between the actuator and the structure is taken into consideration. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatial Random Material Property Model for Vibration of Composites

Investigation of vibration and buckling of thin walled composite structures is very sensitive to parameters like uncertain material properties and thickness imperfections. Because of the manufacturing process and others, thin walled composite and other structures show uncertainties in material properties, and other parameters which cannot be reduced by refined discretization. These parameters are mostly spatial distributed in nature. Here I introduce a semivariogram type material property model to predict the spatial distributed material property (like young's modulus) over the structure. The computation of semivariogram parameters needs the local material properties over a prespecified gird. The material properties at each grid have been obtained by considering a statistically homogeneous representative volume element (RVE) at each gird. According to random nature of the spatial arrangement of fibers, the statistically homogeneous RVE is obtained using image processing. The effective material properties of the RVE have been obtained numerically with the help of periodic boundary condition. The methodology is applied to a composite panel model and modal analysis has been carried. The results of the modal analysis (eigen values and mode shapes) are compared with experimental modal analysis results which are in good agreement. Using the presented material property model we can better predict the vibration characteristics of the thin walled composite structures with the inherent uncertainties. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

轴向运动梁的横向振动分析

论述了轴向运动梁横向振动问题以及研究轴向运动梁横向振动问题的方法,指出对轴向运动梁横向振动问题研究中存在的一些错误并进行了更正.针对一端可看作固定边界条件的轴向运动悬臂梁,基于连续体的模态叠加法,推导出含自重效应的轴向运动梁动力响应的计算公式,进行实例计算,并对计算结果进行了详细的讨论,得出影响轴向运动梁振动响应的因素主要有速度和运动方向.     

Nonlinear analysis of buckling,free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment

Employing Euler–Bernoulli beam theory and the physical neutral surface concept, the nonlinear governing equation for the functionally graded material beam with two clamped ends and surface-bonded piezoelectric actuators is derived by the Hamilton’s principle. The thermo-piezoelectric buckling, nonlinear free vibration and dynamic stability for the piezoelectric functionally graded beams, subjected to one-dimensional steady heat conduction in the thickness direction, are studied. The critical buckling loads for the beam are obtained by the existing methods in the analysis of thermo-piezoelectric buckling. The Galerkin’s procedure and elliptic function are adopted to obtain the analytical solution of the nonlinear free vibration, and the incremental harmonic balance method is applied to obtain the principle unstable regions of the piezoelectric functionally graded beam. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and accuracy of the present analysis and solving method. Simultaneously, validation of the results achieved by rule of mixture against those obtained via the Mori–Tanaka scheme is carried out, and excellent agreements are reported. The effects of the thermal load, electric load, and thermal properties of the constituent materials on the thermo-piezoelectric buckling, nonlinear free vibration, and dynamic stability of the piezoelectric functionally graded beam are discussed, and some meaningful conclusions have been drawn.     

Free vibration analysis of cracked beams by using rigid segment method

This paper deals with the analysis of influence of crack parameters to the modal characteristics of beams at various boundary conditions by using rigid segment method. The beam was discretized by a number of rigid segments which were connected by elastic joints with three degrees of freedom, while the crack was described by cracked element based on fracture mechanics. This model allows detection of coupling between the axial and transverse vibrations under the special boundary conditions. The proposed approach covers both the Euler–Bernoulli and Timoshenko beam model. The efficiency of the method was shown through the few numerical examples.     

Genetic algorithm based wireless vibration control of multiple modal for a beam by using photostrictive actuators

The lanthanum-modified lead zirconate titanate (PLZT) actuator, which are capable of converting photonic energy to mechanical motion, have great potential in applications of remote structural vibration control of smart structures and machines. In this paper, a novel genetic algorithm based controlling algorithm for multi-modal vibration control of beam structures via photostrictive actuators is proposed. Two pairs of photostrictive actuators are laminated with the beams and the alternation of light irradiation is in accordance with the changing of the corresponding modal velocity direction. The modal force indexes for beams with different boundary conditions are derived and a binary-coded GA is used to optimize the locations and sizes of photostrictive actuators to maximize the modal force index and guarantee the overall modal force index induced by two pairs of photostrictive actuators is positive. The control effect of multiple vibration modes of beams under irradiation of set/variable light intensity is analyzed. Numerical results demonstrate that the method is robust and efficient, and the use of strategically positioned actuator patches can effectively control the first two bending modes that dominate the structural vibration.     

Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load

This work deals with a study of the vibrational properties of functionally graded nanocomposite beams reinforced by randomly oriented straight single-walled carbon nanotubes (SWCNTs) under the actions of moving load. Timoshenko and Euler-Bernoulli beam theories are used to evaluate dynamic characteristics of the beam. The Eshelby-Mori-Tanaka approach based on an equivalent fiber is used to investigate the material properties of the beam. An embedded carbon nanotube in a polymer matrix and its surrounding inter-phase is replaced with an equivalent fiber for predicting the mechanical properties of the carbon nanotube/polymer composite. The primary contribution of the present work deals with the global elastic properties of nano-structured composite beams. The system of equations of motion is derived by using Hamilton’s principle under the assumptions of the Timoshenko beam theory. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. In order to evaluate time response of the system, Newmark method is also used. Numerical results are presented in both tabular and graphical forms to figure out the effects of various material distributions, carbon nanotube orientations, velocity of the moving load, shear deformation, slenderness ratios and boundary conditions on the dynamic characteristics of the beam. The results show that the above mentioned effects play very important role on the dynamic behavior of the beam and it is believed that new results are presented for dynamics of FG nano-structure beams under moving loads which are of interest to the scientific and engineering community in the area of FGM nano-structures.     

Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories

This paper investigates the linear free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). Two types of CNT reinforced beams, namely uniformly distributed CNT reinforced (UD-CNT) beams and functionally graded CNT reinforced (FG-CNT) beams, are considered. It is assumed that the SWCNTs are aligned along the beam axial direction and the distribution of the SWCNTs may vary through the thickness of the beam. The virtual strain and kinetic energies of the FG-CNT composite beam are obtained using the classic variational method of Hamilton’s principle and then solved by the p-Ritz method. Vibration frequency parameters for the FG-CNT beams based on the first order and third order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam span to depth ratio and end support conditions on the free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first order and the third order beam theories are also performed and the differences in vibration frequencies between these two theories are highlighted.     

Elactic buckling of thin-walled beams with sandwich and double bends flanges

This of the paper are two thin–walled beams with sandwich and double bends flanges. Cross section of these beams is of C type. The beams are simply supported and subjected to a couple of moment – the pure bending. Geometric propeties (warping functions and inertia moments) of two sections with sandwich and double bendsflanges are separately described by dimensionless parameters. Values of critical loads for family of thin–walled beams are numerically determined on the ground of analytical solution. A comparative analysis for selected beams with the use of FEM is performed. Morover the values of critical loads for a family of thin–walled beams are experimentally researched in the Material Strength Laboratoey of the Poznan University of Technology. Finally, the results of the investigation for thin–walled beams are compared in paper. Results of the calculation are presented in tables and figures. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

线性主动结构及模态(Ⅱ)——离散系统及梁

在“线性主动结构及模态(Ⅰ)”中给出了主动结构的基本概念及模态的若干属性的基础上,对主动离散系统及主动梁的属性做了进一步的讨论,包括稳定性和正交性,并用具体实例对模态做了解释．将伴随结构概念在梁结构中推广,具体讨论了两种配置的主动梁,它们分别代表离散传感和作动及分布传感和作动的配置,并给出了用主动梁振型和伴随主动梁振型表示的正交性条件．用实例给出了同位和非同位主动刚度梁的特征值随反馈大小的变化．     

Parametric excitation of a piezoelectrically actuated system near Hopf bifurcation

This paper deals with investigation into the stability analysis for transverse motions of a cantilever micro-beam, which is axially loaded due to a voltage applied to the piezoelectric layers located on the lower and upper surfaces of the micro-beam. The piezoelectric layers are pinned to the open end of the micro-beam and not bonded to it through its length. Application of the DC and AC piezoelectric actuations creates steady and time varying axial forces. The equation of the motion is derived using variational principal, and discretized using modal expansion theorem. The differential equations of the discretized model are a set of Mathieu type ODEs, whose stability analysis is performed using Floquet theory for multiple degree of freedom systems. Considering first two eigen-functions in the modal expansion theorem leads in the prediction of flutter type of instability as a consequence of Hopf bifurcation, which is not seen in the reduced single degree of freedom system. The object of the present study is to passively control the flutter instability in the proposed model by applying AC voltage with suitable amplitude and frequency to the piezoelectric layers. The effect of various parameters on the stability of the structure, including damping coefficient, amplitude of the DC and AC voltages, and the frequency of the applied AC voltage is studied.     

压电超材料梁中弹性波带隙特性与界面传输的主动控制

该文采用周期压电负电容电路,研究了弹性波超材料梁中带隙特性的主动控制问题.该系统利用外部电路改变所连接压电材料的材料参数,从而改变结构的等效参数,实现对带隙特性的调控.通过对单胞进行控制,可观察到主动控制系统作用时带隙的产生与消失.构造了含有交界面的弹性波超材料梁结构,分析了主动控制系统对波动界面传输特性的影响.     

Piezoelectric energy harvesting from vibrations of a beam subjected to multi-moving loads

This study is intended to investigate piezoelectric energy harvesting from vibrations of a beam induced by multi-moving loads. Various multi-moving loads are analyzed by considering various parameters. The system of equations for electro-mechanical materials is derived by using the generalized Hamilton's principle under the assumptions of the Euler–Bernoulli beam theory. The electromechanical behavior of piezoelectric harvesters in a unimorph configuration is analyzed using finite element method. The Newmark's explicit integration technique is adopted for the transient analysis. The predictions of the results of the finite element models are verified by that of the available solutions. The effects of piezoelectric bonding location, velocity and number of moving loads as well as time lags between moving loads on the produced power are investigated. The numerical results show that the investigated parameters have significant effects on the energy harvesting from a vibration of beams under the action of multi-moving loads.     

Active suppression of nonlinear composite beam vibrations by selected control algorithms

This paper is focused on application of different control algorithms for a flexible, geometrically nonlinear beam-like structure with Macro Fiber Composite (MFC) actuator. Based on the mathematical model of a geometrically nonlinear beam, analytical solutions for Nonlinear Saturation Controller (NSC) are obtained using Multiple Scale Method. Effectiveness of different control strategies is evaluated by numerical simulations in Matlab–Simulink software. Then, the Digital Signal Processing (DSP) controller and selected control algorithms are implemented to the physical system to compare numerical and experimental results. Detailed analysis for the NSC system is carried out, especially for high level of amplitude and wide range of frequencies of excitation. Finally, the efficiency of the considered controllers is tested experimentally for a more complex autoparametric “L-shape” beam system.     

Forced vibration of curved beams on two-parameter elastic foundation

Forced vibration analysis of curved beams on two-parameter elastic foundation subjected to impulsive loads are investigated. The Timoshenko beam theory is adopted in the derivation of the governing equation. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method. The solutions obtained are transformed to the real space using the Durbin’s numerical inverse Laplace transform method. The static and forced vibration analysis of circular beams on elastic foundation are analyzed through various examples.