Interval analysis of mode shapes to identify damage in beam structures

Various damage detection methods have been proposed by several researchers in the past few decades. Amongst them, the efficiency of mode shapes in detecting damage has been demonstrated by many researchers when further processed. In most cases, the processing involves expansion or reduction of the mode shape data. However, vital information that are damage-prints are often lost during processing of the mode shape data. In addition, most of these processes involve long and complex computation, thus, leading to inaccurate damage identification. In this study, a simple and fast damage identification technique is proposed to identify damage in beam structures. Interval analysis is applied to the mode shapes of a beam structure in the damaged and undamaged states. The interval situations of each of the beam's segment via mode shape are derived to obtain the upper and lower bounds and the derived bounds are compared. To establish a relationship for identify the damaged point, a possibility of damage existence is defined for each segment of the beam structure. The mode shape increment is defined as the increase in the mode shape value. Furthermore, a damage measure index that provide enhance damage information is obtained as the product of the possibility of damage existence and mode shape increment. A numerical model of a simply supported steel beam is applied to demonstrate this method by imposing damage through thickness reduction of elements in segments. In addition, a parametric analysis is carried out to evaluate noise effect by considering varying damage severities and different noise levels. The results showed that this method is simple and provides considerable accurate results.     

Evaluation of crack locations in beam using artificial neural network-based modified curvature damage index

Although the frequency response-curvature methodology is commonly used to detect irregularities in mechanical and civil structures, the artificial neural network-based frequency response-curvature damage index method may have good efficacy in the detection and localization of structural damages. By utilizing experimental data sets, a novel method is proposed to pinpoint a saw-cut damage location and the degree of damage in beam models. Using a dynamic data logger, the frequency response function of a beam model is obtained for altering damage levels at different positions. As frequency response data contains environmental and operational noise, the accuracy of obtained results may get reduced. To improve the accuracy by reducing the noise effect, the experimentally obtained frequency response data is trained through an artificial neural network. Using central difference approximation, the sets of trained modal data are utilized to determine the improved mode shape curvature. The curvature damage index is then obtained by using the improved mode shape curvature for different damaged scenarios to ultimately identify structural damages.     

Flexibility-based structural damage identification using Gauss–Newton method

Structural damage will change the dynamic characteristics, including natural frequencies, modal shapes, damping ratios and modal flexibility matrix of the structure. Modal flexibility matrix is a function of natural frequencies and mode shapes and can be used for structural damage detection and health monitoring. In this paper, experimental modal flexibility matrix is obtained from the first few lower measured natural frequencies and incomplete modal shapes. The optimization problem is then constructed by minimizing Frobenius norm of the change of flexibility matrix. Gauss–Newton method is used to solve the optimization problem, where the sensitivity of flexibility matrix with respect to structural parameters is calculated iteratively by only using the first few lower modes. The optimal solution corresponds to structural parameters which can be used to identify damage sites and extent. Numerical results show that flexibility-based method can be successfully applied to identify the damage elements and is robust to measurement noise.     

Identification of Added Mass in the Composite Plate Structure Based on Wavelet Packet Transform

A new concept has been introduced that the combination of rotational mode shape with two‐dimensional wavelet packet transform to detect the added mass (damage) in a glass fibre reinforced polymer composite plate structure. Wavelet packet transform is an advanced signal processing tool that can magnify the abnormality features in the signal. Rotational mode shapes are sensitive to damage in beam and plate structures. The proposed method employs an added mass, which slides to different locations to alter the local and global dynamic characteristics of the structure. Finite element analysis is carried out to obtain the first three rotational bending mode shapes, from the damaged plate structure, then used as input to two‐dimensional wavelet packet transform. The numerical results of normalised diagonal detail wavelet packet coefficients show a peak at single or multiple added mass (damage) locations of a plate structure for two different boundary conditions. This method seems to be sensitive to relatively small amount of damage to the plate structure. A simple parametric study is carried out for the damage extent quantification. In addition, investigation with noise‐contaminated signals shows its feasibility in the real applications.     

Frequency Shift Curve Based Damage Detection Method for Beam Structures

Vibration based damage detection methods play an important role in the maintenance of beam structures such as bridges. However, most of them require the accurate measurement of structural mode shapes, which may not be easily satisfied in practice. Since the measurement of frequencies is more accurate than that of mode shapes, this paper proposes a frequency shift curve (FSC) method, based on the equivalence between the FSC due to auxiliary mass and the mode shape square, which has been demonstrated to be effective in structural damage detection. Two damage indices based on the FSC are developed, which are called the local outlier detection index and the global outlier detection index, respectively. The efficiency and reliability of the proposed method are demonstrated by numerical simulations and experimental results. Compared with traditional methods, this method can provide reliable results without the requirement of fixing many sensors on the structure.     

Modal parameter identification and vibration based damage detection of a multiple cracked cantilever beam

This paper presents a detailed investigation on the modal parameter identification and vibration based damage detection of a multiple cracked cantilever beam with hollow circular cross-section. To consider multiple crack effects, a cantilever beam including cracks is considered for six damage scenarios. Finite element models are constituted in ANSYS software for numerical solutions. The results are validated by experimental measurements. Ambient vibration tests are performed to extract the dynamic characteristics using Enhanced Frequency Domain Decomposition (EFDD) and Stochastic Subspace Identification (SSI) methods. Calculated and measured natural frequencies and mode shapes for undamaged and damaged beams are compared with each other. Automated model updating is carried out using the modal sensitivity method based on Bayesian parameter estimation to minimize the differences for damage detection. In addition, modal assurance criterion (MAC) and coordinated modal assurance criterion (COMAC) factors are obtained from the mode shapes and two set of measurements to establish the correlation between the measured and calculated values for damage location identification.     

结构破损定位的单元模态应变能变化率法

提出了结构单元模态应变能的概念，并导出了基于单元模态应变能变化率的结构破损位置的诊断方法。该方法仅以结构破损前后的模态振型和单元刚度矩阵为输入信息。算例的数值结果表明，该方法简便、有效。     

Damage detection in beam through change in measured frequency and undamaged curvature mode shape

This paper develops and presents a technique on the identification of the damage location and depth in a beam using a frequency contour method. Unlike the frequency contour methods presented in the literature, the one presented in this work does not require neither the solving of the governing differential equation analytically to obtain the contour lines, nor the calculation of the frequencies by varying all damage severities and locations in order to use an inverse problem to plot the contours. In other words, the detection method does not require the modelling of the damage. In this work, a contour line is plotted using only the value of the changes in the measured natural frequencies and the vectors of the curvature mode shapes of the intact structure which can be calculated numerically or can be derived from the measured mode shapes. In order to verify the practical applicability of the developed theoretical method, a numerical simulation and an experimental framework on cantilever beam have been designed and the results confirm the applicability of the developed method in the real applications.     

Local Damage Detection with the Global Fitting Method Using Mode Shape Data in Notched Beams

Damage in a structure alters its dynamic characteristics such as frequency response functions and modal parameters. The present study extends the ‘gapped smoothing method’ for identifying the location of structural damage in a beam by introducing the ‘global fitting method.’ The procedure uses only the mode shape data obtained from a damaged structure with an assumption that the undamaged structure is homogeneous and uniform and the damage size is small. The sensitivity of damage detection algorithm is evaluated using a finite element analysis (FEA) of a few beams having a notch. Structural irregularity index (SSI) was used to identify the locations and size of damage. The ability to detect damage was enhanced by averaging SSI over a few modes. A statistical procedure was applied to identify damage with respect to background noise. A methodology and quantitative criteria was developed to select the optimum excitation grid spacing. Numerical results showed that the present method can detect both narrow (13 mm width) and wide damage (126 mm width) associated with less than 3% local thickness reductions. Experimental results validated the numerical results and detected the depth to thickness ratio about 41% and 35% for the wide and narrow notch beams, respectively. The present method showed improved resolution on detecting the location and size of damage in a beam over the previous methods using mode shape data reported in literature.     

Sub-Structural Parameter Identification Including Cracks of Beam Structure Using PZT Patch

This paper presents the numerical simulation of damage identification of beam structures using novel One Dimensional (1-D) PZT patch model with sub-structuring applied to the PZT Patch boundaries. The 1-D PZT patch model is simple and requires less computational effort than 2-D or 3-D models. A hybrid element constituted of 1-D beam element and a PZT sensor is used with reduced material properties which is very convenient for beams and is a novel application. The accuracy of 1-D hybrid element with cracked structure is first verified experimentally. The stiffness, damping and crack parameters are identified by minimizing the deviation between the predicted and measured voltage from the PZT patch using an optimization algorithm. The numerical signals are polluted with 5% Gaussian noise to simulate experimental noise. Numerical studies are performed on a beam and a nine member frame structure for sub-structural parameter and multiple crack identification. The PZT patches are attached at either ends of beam member to be identified. Sub-structuring is done to isolate the beam with patch, thus reducing the size of model to be identified. The numerical results show significant improvement in identification accuracy compared to other methods.     

Fourier Analysis of Mode Shapes of Damaged Beams

This paper investigates the effect of damage on beams with fixed boundary conditions using Fourier analysis of the mode shapes in spatial domain. A finite element model is used to obtain the mode shapes of a damaged fixed-fixed beam. Then the damaged beams are studied using a spatial Fourier analysis. This approach contrasts with the typical time domain application of Fourier analysis for vibration problems. It is found that damage causes considerable change in the Fourier coefficients of the mode shapes. The Fourier coefficients, especially the higher harmonics, are found to be sensitive to both damage size and location and amplify the changes in the mode shape due to the damage. Therefore, we formulate a damage index in the form of a vector of Fourier coefficients which is robust and unique for a given damage size and damage location. The effect of noise in the mode shape data is considered and it is found that Fourier coefficients provide a useful indication of damage even in the presence of noise. Various damage levels are considered and it is found that higher modes are needed to detect small amount of damage.     

Exact reconstruction of multiple concentrated damages on beams

In this paper, an exact procedure for the reconstruction of multiple concentrated damages on a straight beam is proposed. The concentrated damages are modelled as Dirac’s delta distributions capturing the effect of concentrated stiffness reduction. The presented procedure requires the knowledge of vibration mode shape displacements together with the relevant natural frequency, for the reconstruction of each damage position and intensity. The exact solution of the inverse problem at hand has been pursued by exploiting the analytical structure of the explicit closed form expressions provided for the vibration mode shapes of beams in the presence of an arbitrary number of cracks. The proposed procedure is first presented under the hypothesis that the displacements of a vibration mode shape are known at the cracked cross-sections. In this case, explicit closed form expressions of the crack severities are formulated. A further simple reconstruction approach allows the evaluation of the exact positions and intensity of the concentrated damages, if displacements of two vibration mode shapes are known at a single cross-section between two consecutive cracks. The proposed reconstruction procedure is applied for the identification of multiple cracks on a free–free beam where measurements have been simulated by means of a finite element analysis.     

结构固有频率的灵敏度分析及修改技术

本文给出了在结构上进行点加质量和两点间加杆的固有频率灵敏度分析和结构修改的方法。对一空间框架结构进行的有限元分析和实验结果表明，该方法给出的灵敏位置确切，误差小，使用方便，实用性好     

A new data reduction scheme for mode I wood fracture characterization using the double cantilever beam test

This paper describes experimental and numerical studies on double cantilever beam test applied to fracture characterization of wood in mode I. A new data reduction scheme based on the beam theory and specimen compliance is proposed in order to overcome the difficulties inherent to crack monitoring during propagation. A cohesive damage model adapted to wood is used to simulate the test. The cohesive properties are evaluated using an inverse method based on a developed Genetic Algorithm through an optimisation strategy. The results demonstrate the effectiveness of the proposed methodology as a suitable data reduction scheme for the double cantilever beam test.     

基于摄动法的多条裂纹欧拉梁特征模态分析

基于摄动理论推导了带多条开口裂纹的欧拉梁的特征模态参数的理论计算公式。采用最直接的方式将梁开口裂纹模拟成梁微段内的横截面折减并用δ函数表达了带开口裂纹的梁沿轴线的截面惯矩和线质量等物理参数。基于此,建立了裂纹梁动力微分方程,并采用一阶摄动理论推导得到了梁的模态频率和振型计算公式。简支梁及悬臂梁算例研究表明,该方法具有很好的精度,与有限元模拟结果及实验结果都能很好地吻合。并采用此方法分析了裂纹深度和位置对带多条开口裂纹梁的特征模态参数的影响。结果表明,裂纹对各阶模态频率虽然影响有限,但其引起的各阶频率变化有着明显的模式,可用于结构损伤定位;裂纹对模态振型影响不明显,但对模态曲率影响比较大,可用于结构损伤位置和程度的诊断。     

Crack detection by modal analysis in 3D beams based on FEM

During the past decade, the development of Structural Health Monitoring (SHM) systems has gained considerable interest in several engineering fields. The possibility of monitoring structure status under various operational conditions has resulted in a drastic change in the maintenance schemes. The introduction of SHM systems would allow a transition of the maintenance strategy from scheduled basis (i.e. Time Based) to a condition basis. The development of SHM systems relies on the availability of reliable techniques to acquire the characteristic features of damage through experimental measurements. In this paper, the flexural vibration of a cantilever beam, fixed-roller beam, and portal frame with a transverse surface crack is considered. The natural frequencies and mode shapes are computed by the Finite Element Method (FEM) using SAP2000 Software for various crack locations and depths. The computed mode shapes are animated to give the designers a good picture on the failure modes and shapes. The modal frequencies are experimentally measured and compared with the computed values and showed good compatibility. The sensitivity of the modal frequencies to a crack increases when the crack is near the fixed end, and decreases as the crack moves towards the free end of the beam or frame. The present investigations have revealed the following generalized results. The presence of crack in structure leads to an appreciable difference in the dynamic response. The findings based on condition monitoring technique can be utilized in various industrial applications, particularly for fault detection on structures. The proposed technique represents actually a non-destructive procedure having great benefits for health monitoring of structures.     

白噪声激励下的复合材料层合结构损伤检测的内积向量法

利用白噪声激励下结构各测点的动力学响应之间的互相关函数,引入了内积向量（Inner Product Vector,IPV）的概念,提出了应用内积向量进行结构损伤检测的方法,并分析了测试噪声对基于内积向量的损伤检测方法的影响。通过对复合材料层合梁的分层损伤检测仿真算例,验证了该方法检测损伤的有效性以及较高的抗噪能力。     

通过PVDF阵列测量阶梯梁体积位移的实验

以阶梯梁为例,通过一组阵列式压电薄膜(Polyvinylidene fluoride,PVDF)测量阶梯梁的体积位移。在梁表面均匀布置一组相同形状的矩形PVDF阵列,通过测量PVDF与激励力之间的频率响应函数,然后利用伪逆方法结合截断奇异值法(The truncated singular value decomposition,TSVD)设计这组PVDF阵列的加权系数,从而得到阶梯梁体积位移。实验结果表明,该方法能够准确测量到阶梯梁的体积位移,具有不需计算结构模态、附加质量小、与激励力位置无关等优点。     

应振型损伤特征量在混凝土结构损伤诊断中的应用

为检测混凝土构件损伤的出现、损伤位置及损伤程度，本利用位移振型特征量推导了相应的应变振型特征量，并利用Matlab软件编制了混凝土悬臂梁的有限元振动计算程序，对混凝土悬臂梁在发生损伤前后的位移和应变振型特征量进行了仿真计算。结果表明，位移和应变振型特征量均可确定混凝土悬臂梁上存在的损伤，并可确定损伤位置。应变振型特征量较位移振型特征量对损伤更敏感，可用于混凝土构件的损伤检测。     

Mechanical fatigue and spectrum analysis of small-satellite structure

Mechanical fatigue due to environmental loads and spectrum analysis due to launch loads of the primary structure of a low cost, low-earth orbit small satellite intended for earth observation missions are presented. The payload of the satellite under consideration is a precise optical unit to image the earth’s surface having a mass of 45 kg. 3-D Finite Element Model for the satellite structure is generated by applying substructure method. Modal analysis is required to determine natural frequencies of the satellite and define its mode shape. Then, ranking of mode shapes according to specific constraint is performed. Harmonic analysis at resonance frequencies with the highest ranking is done and cumulative fatigue damage analysis is performed. Spectrum analysis is performed for Small Sat structure to verify the satellite structure reliability under all dynamic random vibration loads applied during transportation and launch cases.