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.     

Structural damage localization and evaluation based on modal data via a new evolutionary algorithm

This paper is intended to present a method for the localization and evaluation of damage in plates based on the changes in natural frequencies and mode shapes of the damaged plate using an optimization approach. The colonial competitive algorithm is employed to detect damage (or damages) in plates by optimizing a damage function. The performance of the proposed method is demonstrated by implementing the technique to two examples; a shear wall and a four-fixed supported plate with and without modal data noise including one or a large number of damages. The results confirm the applicability and efficiency of the presented method in detecting damage localization and quantification in the shear walls. Furthermore, the proposed method is implemented to the four-fixed supported plate aimed at demonstrating the high sensitivity of the proposed method in quantitative estimation of damaged plate structures. Finally, the reliability of the presented method is explored through the comparison of the obtained results and those of the other methods. It is concluded that the proposed method can be viewed as a powerful and robust method for structural damage detection in plate structures.     

Damage detection and denoising in two-dimensional structures using curvelet transform by wrapping method

The main objectives of this study are to present a vibration-based damage identification method and also a denoising mode shape approach applicable to two-dimensional structures using curvelet transform. For this purpose, the curvelet transform via wrapping method is employed. The reliability of the proposed technique is demonstrated through a verification study by comparing the results of numerical and those of the experimental data in plate structures. Two case studies, one-story and three-story shear walls assuming damages at arbitrary locations, are examined in which different noise levels are included. Good agreement between the simulated and assumed damage in both example is demonstrated. The results confirm the robustness and high performance of the proposed method in detecting the damage in plate structure and eliminating the noises.     

THREE-DIMENSIONAL MODELING FOR THIN PLATE-LIKE STRUCTURES INCLUDING SURFACE EFFECTS BY USING STATE SPACE METHOD

<正>A three-dimensional(3-D)approach based on the state space method is proposed to study size-dependent mechanical properties of ultra-thin plate-like elastic structures considering surface effects.The structure is modeled as a laminate composed of a bulk bounded with upper and bottom surface layers,which are allowed to have different material properties from the bulk layer.State equations,including the surface properties of the structure,can be established on the basis of 3-D fundamental elasticity to analyze the size-dependent static characteristics of the thin plate-like structure.Compared with two-dimensional plate theories based size-dependent models for thin film structures in literature,the present 3-D approach is exact,which can provide benchmark results to assess the accuracy of 2-D plate theories and various numerical approaches. To show the feasibility of the proposed approach,a 3-D analytical solution for a simply supported plate-like thin structure including surface layers is derived.An algorithm is proposed for the calculation of the state equations obtained to ensure that the numerical results can reveal the surface effects clearly even for extremely thin surface layers.Numerical examples are carried out to exhibit the surface effects and some discussions are provided based on the results obtained.     

Waveform fractal dimension for mode shape-based damage identification of beam-type structures

Mode shape-based structural damage identification has been a research focus during the last couple of decades. Most of the existing methods need a numerical or measured baseline mode shape serving as a reference to identify damage, and this requirement extremely limits the practicability of the methods. Recently, waveform fractal dimension such as Katz’s waveform fractal dimension (KWD) has been explored and applied to mode shape for crack identification without a baseline requirement. In this study, different from the popular KWD, an approximate waveform capacity dimension (AWCD) is formulated first, from which an AWCD-based modal abnormality algorithm (AWCD-MAA) is systematically established. Then, the basic characteristics of AWCD-MAA on abnormality detection of mode shapes, e.g., crack localization, crack quantification, noise immunity, etc., are investigated based on an analytical crack model of cantilever beams using linear elastic fracture mechanics. In particular, from the perspective of isomorphism, a mathematical solution on the use of applying waveform fractal dimension to higher mode shapes for crack identification is originally proposed, from which the inherent deficiency of waveform fractal dimension to identify crack when implemented to higher mode shapes is overcome. The applicability and effectiveness of the AWCD-MAA is validated by an experimental program on damage identification of a cracked composite cantilever beam using smart piezoelectric sensors/actuators (i.e., Piezoelectric lead–zirconate–titanate (PZT) and polyvinylidene fluoride (PVDF)). The proposed AWCD-MAA provides a novel, viable method for crack identification of beam-type structures without baseline requirement, and it largely expands the scope of fractal in structural health monitoring applications.     

Perturbation methods for the analysis of the dynamic behavior of damaged plates

This paper presents an analytical method for the analysis of the dynamic behavior of damaged plates. The proposed approach allows the derivation of mode shapes and corresponding curvature modes for plates with various kinds of defects. Damage is modeled as a localized reduction in the plate thickness. Both point and line defects are considered to model notches or line cracks and delaminations in the plate. Small thickness reductions are considered so that the dynamic behavior of the damage plate can be analyzed through perturbations with respect to the undamaged modes. Results are presented to demonstrate the sensitivity of the curvature modes with respect to the considered low damage levels. Also, the curvature modes are used for the estimation of the strain energy of the plate and for the formulation of a damage index which can be used to provide damage location and extent information.     

DAMAGE DETECTION IN BUILDINGS USING A TWO-STAGE SENSITIVITY-BASED METHOD FROM MODAL TEST DATA

Many multi-story or highrise buildings consisting of a number of identical stories are usually considered as periodic spring-mass systems. The general expressions of natural frequencies, mode shapes, slopes and curvatures of mode shapes of the periodic spring-mass system by utilizing the periodic structure theory are derived in this paper. The sensitivities of these mode parameters with respect to structural damages, which do not depend on the physical parameters of the original structures, are obtained. Based on the sensitivity analysis of these mode parameters, a two-stage method is proposed to localize and quantify damages of multi-story or highrise buildings. The slopes and curvatures of mode shapes, which are highly sensitive to local damages,are used to localize the damages. Subsequently, the limited measured natural frequencies, which have a better accuracy than the other mode parameters, are used to quantify the extent of damages within the potential damaged locations. The experimental results of a 3-story experimental building demonstrate that the single or multiple damages of buildings, either slight or severe,can be correctly localized by using only the slope or curvature of mode shape in one of the lower modes, in which the change of natural frequency is the largest, and can be accurately quantified by the limited measured natural frequencies with noise pollution.     

Nondestructive damage detection in structures using changes in compliance

A methodology to identify damage in a structure is presented in this paper. The method utilizes a new form of damage index based on the changes in the distribution of the compliance of the structure due to damage. The changes in the compliance distribution are obtained using the mode shapes of the pre-damaged and the post-damaged state of the structure. The validity of the method is demonstrated using numerically generated data from beam structures and experimental data from a free–free beam structure with inflicted damage. In the numerical and experimental examples, the damage identification performance of the proposed method is compared with that of the existing strain-energy-based method. The results of the numerical and experimental studies indicate that the proposed compliance-based damage index method can be used in damage identification of the structure.     

Nondestructive testing method based on lamb waves for localization and extent of damage

Based on Lamb wave analysis of propagation in plate-like structures, a damage detection method is proposed that not only locates the position of the damage accurately but also estimates its size. Similar damage detection methods focus only on localization giving no quantitative estimation of extent. To improve detection, we propose two predictive circle methods for size estimation. Numerical simulations and experiments were performed for an aluminum plate with a hole. Two PZT configurations of different sizes were designed to excite and detect Lamb waves. From cross-correlation analysis, the damage location and extent can be determined. Results show that the proposed method enables a better quantitative resolution in detection, the size of the inspection area influences the accuracy of damage identification, and the closer is the inspected area to the damage, the more accurate are the results. The method proposed can be developed into a multiple-step detection method for multi-scale analysis with prospective accuracy.     

DAMAGE DETECTION STRATEGY FOR RETICULATED STRUCTURES BASED ON INCOMPLETE STRAIN MODE

Damage detection based on strain responses of vibration is highly attractive for monitoring long-span reticulated structures.However,there are a lot of structure members in reticulated structures and it is impossible to install strain sensors in each member.Therefore,how to locate and quantify damages with the incomplete mode shapes obtained from few strain sensors is a challenge topic.A new strategy,named incomplete strain mode damage detection(ISMDD) strategy,is proposed in this paper.In the strategy,the distribution of the strain sensors in the reticulated structures can be optimized through sensitive analysis on strain mode perturbation matrix,which can be obtained by perturbation theory.Mode assurance criterion(MAC) value is applied in damage location,and the members with relative large MAC values are defined as damage members.In addition,damage index obtained by solving the perturbation equation is used for damage quantification.Numerical analysis on a long-span reticulated structure,including damage location and quantification for single-and multi-member damages,detection for different damage quantity,the effect analysis of sensor quantity,are performed to verify the effectiveness of the proposed ISMDD strategy.It can be shown from the analysis that the ISMDD strategy is effective in damage location and quantification for both single-and multi-member damages.And the quantity of strain sensors has effect on damage location,but has no obvious influence on damage quantification.Additionally,the anti-noise pollution ability analysis of the ISMDD strategy is carried out,which shows that the ISMDD strategy has excellent anti-noise pollution ability for both single-and multi-damaged members.     

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.     

Numerical modeling of PZT-induced Lamb wave-based crack detection in plate-like structures

This paper presents the numerical modeling and simulations of PZT-induced Lamb wave propagation in plate-like structures by using the spectral finite element method. A novel spectral plate finite element, which can efficiently model the three-dimensional (3D) behavior of Lamb waves, is proposed. In the formulation, linear displacement distributions in the thickness direction are assumed for both the PZT layer and the base plate. A way to avoid the thickness locking is proposed and used in the formulations. Two examples, one for the validation of the proposed two-dimensional (2D) spectral finite element and the other for the demonstration of crack detection in plates, are presented and discussed. The contact between the two faces of crack is considered. Numerical results show that (1) only the anti-symmetric mode is prone to thickness locking thus remedy should be made only on this part, (2) the proposed 2D spectral finite element can adequately model the Lamb wave propagation in plate-like structures and the complex scattering for the crack, and (3) crack location can be well determined by a PZT-induced Lamb wave-based diagnosis algorithm.     

Simultaneous Measurement of Plate Natural Frequencies and Vibration Mode Shapes Using ESPI

The natural frequencies and vibration mode shapes of flat plates are simultaneously measured using ESPI. The method involves measuring the surface shape of a vibrating plate at high frame rate using a modified Michelson interferometer and high-speed camera. The vibration is excited here by impact; white (random) noise could alternatively be used. Fourier analysis of the acquired data gives the natural frequencies and associated mode shapes. The analytical procedure used has the advantage that it simultaneously identifies full-field quantitative images of all vibration modes with frequencies up to half the sampling frequency. In comparison, the ESPI time-averaging and the traditional Chladni methods both require that the plate be excited at each natural frequency to allow separate qualitative measurements of the associated mode shapes. The Instrumented Hammer method and Laser Doppler Vibrometry give quantitative measurements but require sequential sampling of individual points on the test surface to provide full-field results. Example ESPI measurements are presented to illustrate the use and capabilities of the proposed plate natural frequency and mode shape measurement method.     

Displacement and strain of vibrating structures using time-average holography

A technique for computing the bending strain resulting from the resonant modal deformation of vibrating plate-like structures is described. Interferometric fringes obtained by time-average holography are used as the basis for generating a mathematically continuous series approximation of a plate-like structure's normal displacement. The terms of the series consist of the clamped-free or free-free eigenfunctions of a simple beam. The bending strain is then obtained by computing the second derivative of the displacement series. The coefficients of the terms of the displacement series are computed for a given segment of a cantilevered plate-like structure based upon the holographic-fringe values lying along the same plate segment. A linear least-squares-solution routine is used to solve for the series coefficients, called modal weighting coefficients, in terms of the normal displacement values obtained from the holographic-fringe value. A ‘best fit’ solution is thus obtained for the plate displacement. This least-squares approach in conjunction with the fact the beam-series functions exactly satisfy the plate's geometric boundary conditions and approximately satisfy the plate's natural boundary conditions, results in a displacement series that yields quite accurate displacement and bending strain values. The technique described above has been programmed for use on a Tektronix 4010 interactive computer terminal. The accuracy and effectiveness of the computer algorithm, called HOLOCURVE, is checked by determining the displacement and bending strain at selected segment locations for three different modes of vibration of a cantilevered plate. The results are compared to those of an eigenvalue analysis carried out on a finite-element model of the plate using the finite-element computer program, NASTRAN. The HOLOCURVE results are in excellent agreement with the finite-element analysis except for cases where the bending strain is essentially zero as in the case of chordwise segment for a torsional-mode shape.     

基于改进残余力向量法的桁架结构损伤诊断

提出一种基于改进残余力向量法的桁架结构损伤诊断方法. 先由灵敏度分析, 求出结构刚度联系矩阵,再由刚度联系矩阵将损伤后的刚度摄动矩阵展开成对角矩阵,代入 残余力向量方程,得到由刚度联系矩阵表示的新的残余力向量方程,此方程可以直接求解, 即可诊断出桁架结构的损伤杆件及其损伤程度. 对于实测中难以获得完备振型的情况,采用 模态扩阶的方法来获得完备的测试振型. 最后以一桁架结构进行数值仿真分析,证实了该方 法的有效性.     

Damage detection method in complicated beams with varying flexural stiffness

A damage detection method for complicated beam-like structures is proposed based on the subsection strain energy method (SSEM), and its applicability condition is introduced. For a beam with the continuously varying flexural stiffness and an edge crack, the SSEM is used to detect the crack location effectively by numerical modal shapes. As a complicated beam, the glass fiber-reinforced composite model of a wind turbine blade is studied based on an experimental modal analysis. The SSEM is used to calculate the damage index from the measured modal parameters and locate the damage position in the blade model successfully. The results indicate that the SSEM based on the modal shapes can be used to detect the damages in complicated beams or beam-like structures for engineering applications.     

基于卡车加载和差分曲率的简支梁桥预警新指标

差分曲率指标在实际用于损伤识别时受到加载方式的限制.为此,首先采用了一种卡车加载方式,通过两次卡车加载可实现集中载荷的工况,并对简支梁桥建立起了相应的损伤识别方法,提出了新的"卡车载荷差分曲率指标(truck load difference curvature indicator,TLDCI)".基于实测的竖向位移数据得到的TLDCI图形可识别损伤的出现和位置信息,从而实现简支梁桥的状态预警.TLDCI方法的计算简单而且不需要与结构完好状态下的数据进行对比.所提出方法的有效性通过算例得到了证明,并最后讨论了提高其稳定性的措施.     

Wavelet packet based damage identification of beam structures

Most of vibration-based damage detection methods require the modal properties that are obtained from measured signals through the system identification techniques. However, the modal properties such as natural frequencies and mode shapes are not such a good sensitive indication of structural damage. The wavelet packet transform (WPT) is a mathematical tool that has a special advantage over the traditional Fourier transform in analyzing non-stationary signals. It adopts redundant basis functions and hence can provide an arbitrary time-frequency resolution. In this study, a damage detection index called wavelet packet energy rate index (WPERI), is proposed for the damage detection of beam structures. The measured dynamic signals are decomposed into the wavelet packet components and the wavelet energy rate index is computed to indicate the structural damage. The proposed damage identification method is firstly illustrated with a simulated simply supported beam and the identified damage is satisfactory with assumed damage. Afterward, the method is applied to the tested steel beams with three damage scenarios in the laboratory. Despite the noise is present for real measurement data, the identified damage pattern is comparable with the tests. Both simulated and experimental studies demonstrated that the WPT-based energy rate index is a good candidate index that is sensitive to structural local damage.     

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.     

基于位移响应协方差参数的数值仿真和实测损伤识别研究

本文使用结构位移响应协方差参数进行结构损伤识别,首先推导和建立位移响应协方差参数的解析公式,它是结构频率、振型和阻尼等模态参数的函数,结构物理参数的改变会导致该协方差参数的改变;对一个七层框架结构进行数值模拟分析来演示该方法的有效性,通过比较结构不同状态下各单元位移响应协方差参数CoD的分布曲线,研究各单元CoD与损伤程度的关系曲线,发现损伤位置处的CoD改变最大,其次是对称和附近单元,通过单损伤和多损伤工况研究分析,表明基于结构损伤前后CoD的改变,能成功判定损伤发生和识别出损伤位置,最后把该方法应用于一个实验室简支钢梁的损伤识别,通过对锤击振动下的加速度响应进行二次积分得到位移响应,并比较钢梁损伤前后的CoD,得到损伤概率向量,成功识别出损伤位置。该方法具有较好的噪声鲁棒性,无需结构分析模型,计算简便,具有较好的工程应用性。