多加筋圆柱壳体振动特性的导纳法研究

弹性多加筋圆柱壳体由于其结构的特点，广泛应用于工程实际，因此对其进行振动研究具有重要的工程价值和应用意义。文章利用薄板和梁的导纳公式推导了多加筋圆柱壳体在任意位置简谐力激励下的弯曲振动响应，并对特定参数的多加筋圆柱壳体的振动能量分布进行了数值仿真，结果表明振动能量在加筋处呈现急剧衰减。还对不加筋圆柱壳以及加不同数目筋的圆柱壳的振动响应进行了对比，得到了加筋和加不同叛目的筋对结构振动能量分布和传播的影响。     

Stress intensity factors for cracked T-sections and dynamic behaviour of T-beams

This paper presents an extension of a simple and convenient method proposed by Kienzler and Herrmann [An elementary theory of defective beams. Acta Mech 1986;62:37-46] to estimate the stress intensity factors of cracked beams and bars. This method is based on an elementary beam theory estimation of the strain energy release as the crack is widened into a fracture band. As an extension, the power of the simple beam theory analysis is demonstrated by application to cracked T-beams subjected to a bending moment, shear forces and a torsion. Moreover, the present work addresses the coupled bending-torsional vibration of cracked T-beams within the context of the dynamic stiffness matrix method of analysing structures.     

Free vibration analysis of uniform and stepped cracked beams with circular cross sections

This paper presents a novel numerical technique applicable to analyse the free vibration analysis of uniform and stepped cracked beams with circular cross section. In this approach in which the finite element and component mode synthesis methods are used together, the beam is detached into parts from the crack section. These substructures are joined by using the flexibility matrices taking into account the interaction forces derived by virtue of fracture mechanics theory as the inverse of the compliance matrix found with the appropriate stress intensity factors and strain energy release rate expressions. To reveal the accuracy and effectiveness of the offered method, a number of numerical examples are given for free vibration analysis of beams with transverse non-propagating open cracks. Numerical results showing good agreement with the results of other available studies, address the effects of the location and depth of the cracks on the natural frequencies and mode shapes of the cracked beams. Modal characteristics of a cracked beam can be employed in the crack recognition process. The outcomes of the study verified that presented method is appropriate for the vibration analysis of uniform and stepped cracked beams with circular cross section.     

非对称Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀Bernoulli-Euler薄壁梁单元自由振动的控制运动微分方程,推导了其精确的动态传递矩阵.采用Bernoulli-Euler弯扭耦合梁理论,假定梁横截面没有任何对称性,考虑了薄壁梁在两个方向的弯曲振动及翘曲刚度的影响.动态传递矩阵可以用于计算非对称薄壁梁及其集合体的精确固有频率和模态形状.针对具体的算例,给出了各种边界条件下固有频率的数值结果并与文献中已有的结果进行了比较,还讨论了翘曲刚度对固有频率和模态形状的影响,结果表明如果忽略翘曲刚度的影响,可能得到毫无意义的结果.     

基于能量有限元法的功能梯度梁振动分析

功能梯度材料(Functionally Graded Material,FGM)由于其优良的结构性能和重要的应用价值,近些年来得到了广泛的研究和关注。采用能量有限元法对功能梯度梁和耦合梁的弯曲振动特性进行研究,推导了功能梯度材料梁的能量密度控制方程、能量有限元矩阵方程以及耦合梁的能量有限元方程,从而得到梁中的能量密度和能量流。以一简支功能梯度梁为例,分别采用该方法和传统有限元法计算了梁弯曲振动时的能量密度,通过对比验证了能量有限元法求解的准确性。在此基础上进一步对耦合功能梯度梁结构的能量密度和能量流进行了求解,得到其能量分布特征。该研究为基于能量有限元法分析复杂功能梯度材料结构的振动特性提供了理论基础。     

Detection of multiple cracks using frequency measurements

A method for detection of multiple open cracks in a slender Euler-Bernoulli beams is presented based on frequency measurements. The method is based on the approach given by Hu and Liang [J. Franklin Inst. 330 (5) (1993) 841], transverse vibration modelling through transfer matrix method and representation of a crack by rotational spring. The beam is virtually divided into a number of segments, which can be decided by the analyst, and each of them is considered to be associated with a damage parameter. The procedure gives a linear relationship explicitly between the changes in natural frequencies of the beam and the damage parameters. These parameters are determined from the knowledge of changes in the natural frequencies. After obtaining them, each is treated in turn to exactly pinpoint the crack location in the segment and determine its size. The forward, or natural frequency determination, problems are examined in the passing. The method is approximate, but it can handle segmented beams, any boundary conditions, intermediate spring or rigid supports, etc. It eliminates the need for any symbolic computation which is envisaged by Hu and Liang [J. Franklin Inst. 330 (5) (1993) 841] to obtain mode shapes of the corresponding uncracked beams. The proposed method gives a clear insight into the whole analysis. Case studies (numerical) are presented to demonstrate the method effectiveness for two simultaneous cracks of size 10% and more of section depth. The differences between the actual and predicted crack locations and sizes are less than 10% and 15% respectively. The numbers of segments into which the beam is virtually divided limits the maximum number of cracks that can be handled. The difference in the forward problem is less than 5%.     

Dynamics of anisotropic composite cantilevers weakened by multiple transverse open cracks

In this paper an exact solution methodology, based on Laplace transform technique enabling one to analyze the bending free vibration of cantilevered laminated composite beams weakened by multiple non-propagating part-through surface cracks is presented. Toward determining the local flexibility characteristics induced by the individual cracks, the concept of the massless rotational spring is applied. The governing equations of the composite beam with open cracks as used in this paper have been derived via Hamilton's variational principle in conjunction with Timoshenko's beam model. As a result, transverse shear and rotatory inertia effects are included in the model. The effects of various parameters such as the ply-angle, fiber volume fraction, crack number, position and depth on the beam free vibration are highlighted. The extensive numerical results show that the existence of multiple cracks in anisotropic composite beams affects the free vibration response in a more complex fashion than in the case of beam counterparts weakened by a single crack. It should be mentioned that to the best of the authors' knowledge, with the exception of the present study, the problem of free vibration of composite beams weakened by multiple open cracks was not yet investigated.     

Structural damage identification using transfer matrix with lumped crack properties

A novel damage detection scheme is developed for detecting multiple cracks in beams, based on a transfer matrix (TM) approach. Lumped Crack TM of a beam element with multiple cracks is derived based on lumped crack properties. A cracked beam element is assumed as two intact beam elements connected with a hinge or torsional spring. The crack is modelled as an element of zero length and mass, but with elastic properties. Lumped crack approach is simpler for multiple cracks than the possible alternative methods. The state vector at a node includes displacements, forces and moments at that node; when it is multiplied with TM the state vector at the adjacent node can be obtained. The crack identification strategy used here, involves measuring the initial state vector at a node, in the zone of interest. The displacements at the adjacent nodes are measured and these are predicted using TM. Using an optimization algorithm the unknown crack parameters in the TM are solved by minimizing the deviation between measured and predicted displacements. The method is verified using several numerical models as well as experiments with cracked beams. The TM is shown to be suitable for local identification and also fast and accurate compare to other methods.     

The continuous crack flexibility model for crack identification

The presence of a crack in a structural member introduces a local flexibility that affects its dynamic response. Moreover, the crack will open and close in time depending on the loading conditions and vibration amplitude. The changes in dynamic characteristics can be measured and lead to an identification of the structural changes which eventually might lead to the detection of a structural flaw. The results of various independent evaluations of changes in the natural frequency of vibrations of cracked structural elements are reported. A crack model of a continuous flexibility, found with fracture mechanics methods using the displacement field in the vicinity of the crack developed recently is used here. The analytical results for the cracked elements behaviour based on the continuous crack flexibility vibration theory were correlated with numerical solutions, the lumped-crack beam vibration analysis and experimental results obtained on aluminium and steel beams with open cracks.     

Spatial Wavelet Approach to Local Matrix Crack Detection in Composite Beams with Ply Level Material Uncertainty

Wavelet coefficients based on spatial wavelets are used as damage indicators to identify the damage location as well as the size of the damage in a laminated composite beam with localized matrix cracks. A finite element model of the composite beam is used in conjunction with a matrix crack based damage model to simulate the damaged composite beam structure. The modes of vibration of the beam are analyzed using the wavelet transform in order to identify the location and the extent of the damage by sensing the local perturbations at the damage locations. The location of the damage is identified by a sudden change in spatial distribution of wavelet coefficients. Monte Carlo Simulations (MCS) are used to investigate the effect of ply level uncertainty in composite material properties such as ply longitudinal stiffness, transverse stiffness, shear modulus and Poisson’s ratio on damage detection parameter, wavelet coefficient. In this study, numerical simulations are done for single and multiple damage cases. It is observed that spatial wavelets can be used as a reliable damage detection tool for composite beams with localized matrix cracks which can result from low velocity impact damage.     

Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 3. Analytical and numerical determination of natural frequencies of longitudinal and bending vibrations of beams with transverse cracks. Solution

Based on the concepts of linear fracture mechanics, we derive analytical expressions for the determinational of natural frequencies of longitudinal and bending vibrations of beams which are rectangular in cross section, are fixed in different ways, have variable ratios of the section height to the beam length, and have transverse cracks of various types. The results of the analytical solution are compared with those obtained by the finite-element method as well as with experimental data obtained by the authors and other scientists. The analytical solution under consideration is shown to be quite simple and provide a fairly good accuracy of the results obtained. Using a cantilever beam with one or two symmetrical edge cracks or a central through crack as an example, we consider the possible dependence of the relative change in natural frequencies of vibration on the relative crack length, crack location, and the vibration mode of a beam. We discuss the possible methods of evaluating the crack size and location from the results of experimental determination of the change in natural vibration frequencies of a cracked beam. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 19–31, July–August, 1999.     

含多裂纹损伤圆弧曲梁自由振动扰动的有限元网格自适应分析

该文建立圆弧形曲梁裂纹的截面损伤缺陷比拟方案,实施微裂纹损伤诱发截面弱化,实现多裂纹深度、位置、数目的模拟。引入变截面Timoshenko梁的h型有限元网格自适应分析方法,求解含裂纹损伤圆弧曲梁自由振动问题,得到优化的网格和满足预设误差限的高精度自振频率和振型解答,研究多裂纹损伤对圆弧曲梁振型的扰动行为。数值算例表明,该算法中网格非均匀加密可适应裂纹损伤引起的振型变化,应用于各类曲梁夹角和裂纹损伤分布工况下的自由振动研究,定量分析了多裂纹损伤深度、数目、分布对圆弧曲梁自振频率和振型的扰动影响,检验了该文算法的精确性和实用性。     

周期结构空腹梁的动态特性研究

设计了一种内部周期性挖空的梁，它在中高频具有良好的带通和带阻特性，带阻频率范围内的弹性波不能传播。空腹梁由周期单元串联而成，把周期单元分解成薄梁和刚性联接杆等子结构，推导了Timoshenko梁纵向、弯曲振动导纳，给出了刚性联接杆振动导纳，利用传递矩阵法计算得到了周期结构空腹梁的力传递率和带隙位置。数值模拟计算表明，当激励频率在周期结构的阻带之内时，周期结构空腹梁上的位移响应和传递到基础的力响应将大大衰减。     

Crack identification in rotating shafts by coupled response measurements

In this paper a new method is applied in rotating cracked shafts to identify the depth and the location of a transverse surface crack. A local compliance matrix of different degrees of freedom is used to model the transverse crack in a shaft of circular cross section, based on available expressions of the stress intensity factors and the associated expressions for the strain energy release rates. It is known that when a crack exists in a structure, such as a beam, then the excitation in one-direction causes coupled vibrations in other directions. This property is used here to identify the crack. The shaft is modeled as a rotating Timoshenko beam including the gyroscopic effect and the axial vibration due to coupling. The method used here is based on the measurements of the axial vibration response due to different excitation frequencies and shaft revolutions. The figures presented are used to identify the crack.     

On determination of the natural frequency of transverse and longitudinal vibrations of a cracked beam. Part 2. Experimental and calculation results

The results of experimental investigation of the relationship between the natural frequency of the first mode bending and longitudinal vibrations of titanium alloy and alloyed steel cantilever beams and the crack parameters (crack depth and location) are presented together with the data on calculation of the relative change of the vibration frequency of cracked beams, which are obtained by the proposed analytical approaches. Based on a comparative analysis, we choose fairly simple and sufficiently accurate formulas for calculating the frequencies of bending and longitudinal vibrations of a cantilever beam with an open or closing crack. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 45–53, May–June, 1999.     

Identification of crack size via an energy approach

An energy method for identifying the size of a crack at given location in structures using one measured eigencouple of the cracked structures is presented. The method utilizes the maximum strain energies of the structures both with and without a crack and the additional strain energy induced by the crack to construct the energy balance equation from which the size of the crack is evaluated through an iteration procedure. A pair of measured vibration frequency and mode shape of the cracked structure is used in a free vibration analysis to derive for its maximum strain energy. The maximum inertia force of the cracked structure is then applied to the uncracked structure with known stiffness and the resulting strain energy of the uncracked structure is obtained in a finite element analysis. Fracture mechanics is used to derive for the additional strain energy induced by the crack. Experimental investigations of several cracked free-free beams are performed to validate the proposed method. Examples of the identification of crack sizes for a number of damaged beam structures are given to further illustrate the feasibility and effectiveness of the present method. Overall the results are encouraging showing that the present method has the prospect of becoming an alternative approach for crack size detection.     

压电功能梯度层合梁的力-电-热耦合梁单元及最优振动控制

给出了一个压电功能梯度层合梁振动分析的两节点力-电-热耦合梁单元,并将其用于功能梯度层合梁的振动最优控制。在这个多场耦合梁单元中,功能梯度材料的等效力学性能用Voigt或Mori-Tanaka模型表征;梁的位移场用Shi改进的三阶剪切变形板理论描述;压电层的电势场用Layer-wise理论分层表征,且呈高阶非线性电势场的压电层可离散成数个子层。用Hamilton原理推导了压电功能梯度梁的力-电-热耦合单元列式,用拟协调元法给出了多场耦合梁单元的高计算效率的显式单元刚度矩阵,以及采用线性二次型(LQR)最优控制算法进行压电功能梯度层合梁的最优振动控制。使用所得力-电-热耦合梁单元进行了压电功能梯度层合梁的静力和动力分析。数值算例表明,所得力-电-热耦合梁单元可靠、准确和高效,LQR最优控制算法得到最优控制电压可有效抑制功能梯度梁的振动且实现控制系统能量的优化。     

Augmented ABCD Matrix for Non-specular Diffraction of Gaussian Beams

Abstract

Optical beams reflected or transmitted by a dielectric interface undergo several non-specular deformations: changes in the beam amplitude, lateral, focal and angular shifts of the beam axis and a magnification of the beam width. In this communication the non-specular interaction of the beam with, in general, multilayered, planar dielectric structure is described in terms of the ray transfer matrix formalism. A relationship between the beam deformations and a 3 × 3 ray transfer matrix of the structure is built. Transfer matrix elements are shown in a simple form dependent on the real deformations.     

Vibration Evaluation of Wrinkled Membrane Inflated Beam

Inflated booms are under increasing application in various membrane spacecraft structures. A specific emphasis is often placed on determining static characteristics of inflated beams in bending. There is a lack of knowledge in dynamic conditions, especially for simple and accurate solutions for wrinkled inflated beams. This article deals with the theoretical evaluation on the natural frequencies of wrinkled inflated beams. Three factors, including the inflated pressure, the wrinkles, and the aspect ratio, are introduced into the vibration equation of the beam to predict the first-order natural frequency of the wrinkled inflated beams. The predictions on the first-order natural frequency agree well with the vibration simulations. The parametric studies on the vibration characteristics of the wrinkled inflated beam are performed in the end. Results reveal that the wrinkles decrease the first-order natural frequency of the inflated beam greatly. The first-order frequency of the wrinkled inflated beam is sensitive to the aspect ratio. The results and conclusions are good references to the design and the wrinkling control of the inflated booms.     

简支梁裂纹位置识别的一种简单方法

由等效线弹簧来模拟裂纹引起的软化效应,基于铁摩辛柯梁理论得到含裂纹简支梁横向振动的频率计算式,由此获得识别裂纹位置的一种近似方法。文中利用梁的二维有限元模态分析数据进行裂纹位置的识别,结果表明该法在较宽的高跨比范围内,有好的效果;裂纹的深度对识别精度影响不大。