A 2-D continuous wavelet transform of mode shape data for damage detection of plate structures

A two-dimensional (2-D) continuous wavelet transform (CWT)-based damage detection algorithm using “Dergauss2d” wavelet for plate-type structures is presented. The 2-D CWT considered in this study is based on the formulation by Antoine et al. (2004). A concept of isosurface of 2-D wavelet coefficients is proposed, and it is generated to indicate the location and approximate shape or area of the damage. The proposed algorithm is a response-based damage detection technique which only requires the mode shapes of the damaged plates. This algorithm is applied to the numerical vibration mode shapes of a cantilever plate with different types of damage to illustrate its effectiveness and viability. A comparative study with other two 2-D damage detection algorithms, i.e., 2-D gapped smoothing method (GSM) and 2-D strain energy method (SEM), is performed, and it demonstrates that the proposed 2-D CWT-based algorithm is superior in noise immunity and robust with limited sensor data. The algorithm is further implemented in an experimental modal test to detect impact damage in an FRP composite plate using smart piezoelectric actuators and sensors, demonstrating its applicability to the experimental mode shapes. The present 2-D CWT-based algorithm is among a few limited studies in the literature to explore the application of 2-D wavelets in damage detection, and as demonstrated in this study, it can be used as a viable and effective technique for damage identification of plate- or shell-type structures.     

On irregularity-based damage detection method for cracked beams

A new damage detection technique using irregularity profile of a structural mode shape is proposed in this paper. The mode-shape of a cracked beam is first obtained analytically by using a general function. Its irregularity profile is then extracted from the mode shape by a numerical filter. The location and size of the crack in the beam can be determined by the peak value appearing on the irregularity profile. Two types of numerical filters, i.e., triangular and Gaussian, are examined. It has been found that the former filter is more effective in damage detection than the latter one. Numerical simulations suggest that the irregularity-based method requires a relatively low measurement resolution. Noise stress tests are carried out to demonstrate the effectiveness and robustness of this method under the influence of noise. As a validation, the proposed method is applied to detect crack damage in an E-glass/epoxy laminated composite beam. The successful detection of the crack in the composite beam demonstrates that the irregularity-based method is capable of assessing both the location and size of the crack and can be used efficiently and effectively in damage identification and health monitoring of beam-type structures.     

小波分析在悬臂梁裂纹识别中的应用

基于空间信号的小波分析理论,将含裂纹悬臂梁前四阶振型信息直接用于小波变换,小波系数在空间域上的突变反映了裂纹的存在并指出了裂纹的位置.本文分析了前四阶振型对小波识别结果的敏感性,利用小波系数模极大值在尺度上的表现与Lipschitz指数之闻的关系建立了集中因子和裂纹深度之间的关系,以此来估计裂纹深度.鉴于实测信号往往是含噪声信号,分析了噪声对识别结果的影响规律.数值算例表明利用sym4小波对含裂纹梁的四阶振型信息进行小波分析可以准确地识别出裂纹的位置和深度;高阶振型对结构损伤较为敏感,高阶振型更适合于微裂纹和含噪声信息的处理,但高阶振型的非线性也会给裂纹识别带来一定的困难.使用本文方法进行结构裂纹参数识别,噪声对裂纹位置的影响只是指示清晰度的影响,基本不会产生错误的识别,而对裂纹深度的影响远比对位置的影响复杂,由于小波系数混入了噪声成分,从而增加了集中因子的取值,致使识别结果总是比真实结果偏大.     

基于裂纹附加模态Timoshenko梁的裂纹损伤识别

将梁中横向裂纹等效为无质量扭转弹簧，并忽略其对梁剪切变形的影响，得到的具有任意裂纹数目Timoshenko 梁自振模态的统一显示解析表达式．将裂纹梁的自振模态分为基本模态和裂纹附加模态，利用最小二乘拟合，建立了利用裂纹附加模态函数的梁裂纹损伤识别方法．通过数值模拟开展了简支单裂纹梁以及悬臂和固支双裂纹梁等的裂纹损伤识别，考察了测量误差对损伤识别的影响，数值结果表明本文所提出的裂纹损伤识别方法对裂纹位置的识别精度高于对裂纹损伤程度的识别精度；随着测量误差的增加，裂纹位置及裂纹损伤程度的识别误差增加，但仍在可接受的范围内，故该裂纹损伤识别方法在实际工程中具有一定的应用价值．     

考虑轴力二阶效应悬臂梁的支承及裂纹参数识别

建立了轴向压力作用下悬臂裂纹梁边界支承和裂纹损伤程度识别方法．首先,将悬臂梁边界非完整支承等效为竖向和扭转弹簧、梁中开裂纹等效为内部扭转弹簧,利用Laplace变换,得到了边界弹性支承、考虑轴向压力二阶效应、具有任意裂纹数目Euler-Bernoulli悬臂梁弯曲挠度的解析解．其次,提出了边界弹性支承弹簧柔度和裂纹等效扭转弹簧柔度的识别方法．最后,通过数值试验,考察了轴向压力,裂纹深度以及测量误差等对识别结果的影响,说明了本文考虑轴向压力二阶效应的悬臂梁边界支承弹簧柔度及裂纹等效扭转弹簧柔度识别方法的适用性和可靠性,结果表明：相比于应变测量误差,挠度测量误差对裂纹损伤程度识别结果影响更加敏感,且轴向压力对裂纹损伤程度识别影响较小,因此,应严格控制挠度的测量误差．同时,边界支承扭转弹簧柔度的识别误差大于其竖向弹簧柔度识别误差．这些结果为实际工程中边界非完整支承悬臂裂纹梁的参数识别提供了指导．     

悬臂梁中开闭裂纹的损伤识别

利用裂纹诱导弦挠度函数,建立了悬臂Euler-Bernoulli中开闭裂纹位置、深度、初始张开角等损伤参数的识别方法。为此,首先将梁中开闭裂纹等效为单向扭转弹簧,给出了考虑裂纹缝隙效应的裂纹梁等效抗弯刚度,并得到悬臂Euler-Bernoulli开闭裂纹梁弯曲挠度的显式闭合解及裂纹诱导弦挠度函数,证明了裂纹诱导弦挠度的分段线性函数。其次,基于单向扭转弹簧的性质,建立了通过多步加载进行梁中开闭裂纹参数及其上下侧属性的识别方法。最后,通过数值算例验证了本文所建立的开闭裂纹损伤识别方法的适用性和可靠性,考察了裂纹分布位置、深度和初始张开角以及裂纹识别区间和挠度测量误差等参数对识别结果的影响,结果表明:当裂纹处于张开状态时,裂纹处裂纹诱导弦挠度斜率改变量随着施加荷载的增加而增加;当裂纹闭合时,其裂纹诱导弦挠度斜率改变量将保持为常量;裂纹损伤参数的识别误差随测量误差的增加而增加,但整体识别结果具有较高的精度,较好的鲁棒性。     

基于有限测点信息的结构损伤识别柔度法

利用有限测点获得结构的模态参数,提出了基于有限测点的结构损伤识别的柔度法。该方法是通过仅考虑结构柔度的灵敏度分析,以结构各自由度的损伤信息为条件,选择对结构柔度变化敏感的自由度为测点,并利用有限测点的信息提出了结构完备模态振型的重建技术。在此基础上,对柔度矩阵做关于结构物理参数变化量的一阶泰勒展开,来确定结构单元的损伤因子对结构进行损伤识别。从而实现利用结构有限测点的模态信息来识别结构的损伤,解决了测试结构的模态振型的不完整给结构的损伤诊断带来的困难。通过数值算例说明了该方法的有效性。     

铝合金蚀坑损伤参量评述

将蚀坑损伤参量分成4 类:尺寸、形状、统计和试验参数,回顾了蚀坑深度、投影面积、长宽比、矩形比、表面腐蚀率、分形维数、预腐蚀系数等影响剩余寿命的主要损伤参量,总结了蚀坑等效成初始裂纹的方法.     

Tikhonov's regularization approach in mode shape curvature analysis applied to damage detection

In this paper an on-going research effort aimed at detecting and localizing damage in plate-like structures by using mode shape curvature based damage detection algorithm is described. The proposed damage index uses exclusively mode shape curvature data from the damaged structure. This method was originally developed for beam-like structures. In this paper, the method is generalized to plate-like structures which are characterized by two-dimensional mode shape curvature. To calculate mode shape curvatures from the measured mode shapes, three approaches are proposed: the first one is the well-known central difference approximation, the other two are classical approaches based on Tikhonov's regularization technique with smoothing functional. The applicability and effectiveness of the proposed damage detection algorithms are demonstrated experimentally on an aluminium plate containing mill-cut damage. The validity of the method is assessed by comparing the identification results of the experimental test case to the results obtained from the simulated test case. The modal frequencies and the corresponding mode shapes of the aluminium plate are obtained via finite element models for numerical simulations and by using a scanning laser vibrometer (SLV) for the experimental study.     

Non-linear dynamics of a cracked cantilever beam under harmonic excitation

The presence of cracks in a structure is usually detected by adopting a linear approach through the monitoring of changes in its dynamic response features, such as natural frequencies and mode shapes. But these linear vibration procedures do not always come up to practical results because of their inherently low sensitivity to defects. Since a crack introduces non-linearities in the system, their use in damage detection merits to be investigated. With this aim the present paper is devoted to analysing the peculiar features of the non-linear response of a cracked beam.The problem of a cantilever beam with an asymmetric edge crack subjected to a harmonic forcing at the tip is considered as a plane problem and is solved by using two-dimensional finite elements; the behaviour of the breathing crack is simulated as a frictionless contact problem. The modification of the response with respect to the linear one is outlined: in particular, excitation of sub- and super-harmonics, period doubling, and quasi-impulsive behaviour at crack interfaces are the main achievements. These response characteristics, strictly due to the presence of a crack, can be used in non-linear techniques of crack identification.     

Three-dimensional structure optimal design for extending fatigue life by using biological algorithm

This paper presents some applications of a new structural shape optimization procedure for maximizing fatigue life or inspection intervals for damage tolerant structures. In this approach, a new and simple method, which we termed FAST (Failure Analysis of Structures), for estimating the stress intensity factor for cracks at a notch, as well as an extension of the biological algorithm was employed to study the problem of optimization with fatigue life as the design objective. Research by the authors has demonstrated that the optimum shape for minimizing stress is not necessarily the optimum shape for static strength or fatigue life of a damage tolerant structure. The examples are presented that highlight this difference. The optimal shapes for stress are compared with optimized shapes found for static strength with different crack lengths. These are also compared with optimized shapes found for maximum fatigue life. The choice of initial crack size was found to have a significant effect on the optimal shapes for the structures presented.     

Experimental and numerical analysis of vibrating cracked plates at resonant frequencies

Owing to the advantages of noncontact and fullfield measurement, an optical system called the amplitude fluctuation electronic speckle pattern interferometry (AFESPI) method with an out-of-plane setup is employed to investigate the vibration of a cantilever square plate with a crack emanating from one edge. Based on the fact that clear fringe patterns will be shown by the AFESPI method only at resonant frequencies, both the resonant frequencies and the vibration mode shapes can be obtained experimentally at the same time. Three different crack locations will be discussed in detail in this study. One is parallel to the clamped edge, and the other two are perpendicular to the clamped edge. The numerical finite element calculations are compared with the experimental results, and good agreement is obtained for resonant frequencies and mode shapes. The influences of crack locations and lengths on the vibration behavior of the clamped cantilever plate are studied in terms of the dimensionless frequency parameter (λ ²) versus crack length ratio (a/L). The authors find that if the crack face displacements are out of phase, a large value of stress intensity factor may be induced, and the cracked plate will be dangerous from the fracture mechanics point of view. However, there are some resonant frequencies for which the crack face displacements are completely in phase, causing a zero stress intensity factor, and the cracked plate will be safe.     

Free vibration analysis of a von Koch beam

In this paper, the free undamped motion of a cantilever von Koch beam is investigated. The reduction of the stiffness and mass matrices leads to simple analytical recursive relationships depending on the fractal dimension of the structure. Results are then extended to perform a detailed modal analysis, which suggests peculiar scaling laws for the natural frequencies and modal shapes of the structure. Energy considerations are also provided. Finally, the potentiality of the von Koch beam as a fractal antenna is examined in terms of resonant frequencies.     

梁结构中裂纹参数识别方法研究

以等效弹簧模型来模拟裂纹引起的局部软化效应,将该模型同Bernoulli-Euler梁理论、模态分析方法以及断裂力学原理等结合起来,利用传递矩阵法导出含裂纹梁振动的各种边界条件下的特征方程通解。借助于特征方程,提出两种识别裂纹深度和位置参数的数值方法,最后,通过对含裂纹悬臂梁的分析说明文中方法的有效性。     

含裂纹一维欧拉梁的裂纹无效位置分析

基于集中柔度模型,建立了含裂纹一维欧拉梁的频率方程,以此为基础探讨了裂纹无效位置的求解方法。数值计算结果显示,裂纹无效位置和位移振型节点并不一致。进一步的理论推导证明裂纹无效位置就是曲率模态振型的零点位置,从曲率和力学性能基本参数的关系分析,这一结论是合理的。本文结论对于实验、测试方案设计有指导意义。     

脆性材料裂纹扩展的分形运动学

大量实验结果表明脆性材料裂纹扩展路径是弯弯曲曲的，不规则的，断裂表面粗糙不平，表现出分形特征。本文应用分形几何推导出脆性材料裂纹扩展的分形运动学公式，得到了动、静态断裂韧性与分形裂纹扩展速度、裂纹长度、微结构特征量δ＿ｃ以及分形维数之间的关系；推导出了裂纹扩展速度的计算公式，得出宏观量测的裂纹扩展速度Ｖ＿０应小于分形裂纹扩展速度Ｖ；并推导出直接根据材料动、静态断裂韧性和裂纹扩展速度Ｖ＿０估算材料裂纹扩展路径的分维计算公式．本文理论分析结果与实验值在定性定量上达到了较好的一致。     

Construction of quasi-brittle and quasi-ductile fracture diagrams based on necessary and sufficient criteria

Materials with a regular structure characterized by quasi-brittle and quasi-ductile fractures are considered in the case where the characteristic linear dimension of the structural element is known. Necessary and sufficient fracture criteria are constructed using the Neuber-Novozhilov approach. A modified Leonov-Panasyuk-Dugdale model for an opening mode crack is proposed where the width of the prefracture zone coincides with the width of the plasticity zone. For the critical parameters of quasi-brittle fracture (tensile stress, length of prefracture zones, stress intensity factors), relations are obtained that allow material fracture to be considered in the case where the crack length is negligible compared to the characteristic linear dimension of the structural element. A fracture diagram obtained using the critical stresses calculated from the necessary and sufficient criteria is considered in a wide range of crack lengths. The elastoplastic problem of extension of a plate with a central crack is solved using the finite-element method. The dimensions and shape of the plastic zone near the crack tip are determined for different levels of loads corresponding to quasi-brittle and quasi-ductile fracture. The obtained results are analyzed to estimate the width of the prefracture zone and the critical crack opening.     

DAMAGE DETECTION BASED ON OPTIMIZED INCOMPLETE MODE SHAPE AND FREQUENCY

For the purpose of structural health monitoring, a damage detection method combined with optimum sensor placement is proposed in this paper. The back sequential sensor placement(BSSP) algorithm is introduced to optimize the sensor locations with the aim of maximizing the 2-norm of information matrix, since the EI method is not suitable for optimum sensor placement based on eigenvector sensitivity analysis. Structural damage detection is carried out based on the respective advantages of mode shape and frequency. The optimized incomplete mode shapes yielded from the optimal sensor locations are used to localize structural damage. After the potential damage elements have been preliminarily identified, an iteration scheme is adopted to estimate the damage extent of the potential damage elements based on the changes in the frequency. The effectiveness of this method is demonstrated using a numerical example of a 31-bar truss structure.     

含双裂纹圆截面悬臂梁的损伤识别研究

裂纹的萌生和扩展直接影响构件的振动响应,对构件的安全可靠性具有重要影响．本文以圆截面悬臂梁为对象,结合转角模态振型和模态频率等高线,研究了一种双裂纹识别技术．首先,基于应力强度因子和卡氏定理推导了无裂纹梁单元和含裂纹梁单元的刚度矩阵;在此基础上,建立了含裂纹圆截面悬臂梁的有限元动力学方程;然后,结合裂纹对梁转角模态振型和模态频率的影响,提出了双裂纹识别策略．最后,通过算例讨论了双裂纹识别策略的可行性．结果表明,圆截面悬臂梁的模态转角在裂纹位置出现突变,裂纹深度越大转角突变值越大;将识别出的裂纹位置作为已知参数,通过模态频率等高线法,可以准确地识别出双裂纹的深度．     

Crack identification in beams using wavelet analysis

In this paper a simple method for crack identification in beam structures based on wavelet analysis is presented. The fundamental vibration mode of a cracked cantilever beam is analyzed using continuous wavelet transform and both the location and size of the crack are estimated. The position of the crack is located by the sudden change in the spatial variation of the transformed response. To estimate the size of the crack, an intensity factor is defined which relates the size of the crack to the coefficients of the wavelet transform. An intensity factor law is established which allows accurate prediction of crack size. The viability of the proposed method is investigated both analytically and experimentally in case of a cantilever beam containing a transverse surface crack. In the light of the results obtained, the advantages and limitations of the proposed method as well as suggestions for future work are presented and discussed.