基于裂缝梁动力特性和自振频率的参数敏感性

基于Bernoulli-Euler理论,将开口裂缝梁视为变截面梁,利用模态摄动方法建立了一种求解带任意数量开口裂缝简支梁和连续梁动力特性的半解析分析方法。在等截面无损梁的模态子空间内将裂缝梁的变系数微分方程的求解转化为非线性代数方程组的求解;利用无损梁的自振频率和振型函数摄动求解裂缝梁的模态参数;通过矩形开口裂缝简支梁和两跨连续梁的动力试验验证了笔者方法的准确性;最后,利用开口裂缝梁动力特性的半解析解研究了简支梁和两跨连续梁的自振频率对裂缝尺寸和位置的敏感性。     

Free vibration analysis of Euler-Bernoulli beam with double cracks

In this paper, the influence of two open cracks on the dynamic behavior of a double cracked simply supported beam is investigated both analytically and experimentally. The equation of motion is derived by using the Hamilton’s principle and analyzed by numerical method. The simply supported beam is modeled by the Euler-Bemoulli beam theory. The crack sections are represented by a local flexibility matrix connecting three undamaged beam segments. The influences of the crack depth and the position of each crack on the vibration mode and the natural frequencies of a simply supported beam are analytically clarified for the single and double cracked simply supported beam. The theoretical results are also validated by a comparison with experimental measurements.     

Crack detection in a beam with an arbitrary number of transverse cracks using genetic algorithms

In this paper, a crack detection approach is presented for detecting depth and location of cracks in beam-like structures. For this purpose, a new beam element with an arbitrary number of embedded transverse edge cracks, in arbitrary positions of beam element with any depth, is derived. The components of the stiffness matrix for the cracked element are computed using the conjugate beam concept and Betti’s theorem, and finally represented in closed-form expressions. The proposed beam element is efficiently employed for solving forward problem (i.e., to gain precise natural frequencies and mode shapes of the beam knowing the cracks’ characteristics). To validate the proposed element, results obtained by new element are compared with two-dimensional (2D) finite element results and available experimental measurements. Moreover, by knowing the natural frequencies and mode shapes, an inverse problem is established in which the location and depth of cracks are determined. In the inverse approach, an optimization problem based on the new finite element and genetic algorithms (GAs) is solved to search the solution. It is shown that the present algorithm is able to identify various crack configurations in a cracked beam. The proposed approach is verified through a cracked beam containing various cracks with different depths.     

Crack detection in beams using experimental modal data and finite element model

In this paper, an analytical, as well as experimental approach to the crack detection in cantilever beams by vibration analysis is established. An experimental setup is designed in which a cracked cantilever beam is excited by a hammer and the response is obtained using an accelerometer attached to the beam. To avoid non-linearity, it is assumed that the crack is always open. To identify the crack, contours of the normalized frequency in terms of the normalized crack depth and location are plotted. The intersection of contours with the constant modal natural frequency planes is used to relate the crack location and depth. A minimization approach is employed for identifying the cracked element within the cantilever beam. The proposed method is based on measured frequencies and mode shapes of the beam.     

Structural damage identification of a cantilever beam using excitation force level control

Most studies in damage identification so far have concentrated on comparing modal parameters of a structure with an open crack with those of an intact structure. In this study, a new damage identification method for beam-like structures with a fatigue crack is proposed, which does not require comparative measurement on an intact structure but several measurements at different level of excitation forces on the cracked structure. The idea comes from the fact that dynamic behavior of a structure with a fatigue crack changes with the level of the excitation force. In other words, a beam with a real fatigue crack would behave as an intact beam at low excitation forces due to the crack closure. The 2nd spatial derivatives of frequency response functions along the longitudinal direction of a beam are used as the sensitive indicator of crack existence. Then, weighting function is employed in the averaging process in frequency domain to account for the modal participation of the differences between the dynamic behavior of beam with a fatigue crack at the low excitation and one at the high excitation. Subsequently, a damage index is defined such that the location and level of the crack may be identified. Finally, it is shown that damage identification method using the proposed damage index is very successful through experiment and finite element analysis.     

Improvement of damage detection methods based on experimental modal parameters

The objective of this paper is to introduce a new method for structural damage detection based on experimentally obtained modal parameters. The new method is suitable for detection of fatigue damage occurring in an aluminium cantilever beam. The damage has been practically realised as saw cuts of different sizes and at different locations. The first step of analysis included an attempt of damage identification with the most often used damage indicators based on measured modal parameters. For that purpose special signal processing technique has been proposed improving the effectiveness of indicators tested. However the results obtained have not been satisfactory. That was the motivation for defining new damage indicators (frequency change based damage indicator, Hybrid Damage Detection method), utilising the change of natural frequencies and any mode shape (measured or modelled) as the measurement of frequencies is much less time consuming in comparison to total mode shape measurement. It has been shown that the proposed technique is suitable for damage localisation in beam-like structures.     

Matrix of the covariance of covariance of acceleration responses for damage detection from ambient vibration measurements

A new matrix on the covariance of covariance is formed from the auto/cross-correlation function of acceleration responses of a structure under white noise ambient excitation. The components of the covariance matrix are proved to be function of the modal parameters (modal frequency, mode shape, and damping parameter) of the structure. Information from all the vibration modes of the structure limited by the sampling frequency contributes to these components. The formulated covariance matrix contains more information on the vibration modes of the structure that cannot be obtained by the general methods for extracting modal parameters. When the component of the covariance matrix is used for damage detection, it is found more sensitive to local stiffness reduction than the first few modal frequencies and mode shapes obtained from ambient excitation. A simply supported 31 bar plane truss structure is studied numerically where a multiple damage scenario with different noise levels is identified with satisfactory results.     

Examining the trend in loss of flexural stiffness of simply supported RC beams with various crack severity using model updating

The flexural stiffness of simply supported cracked reinforced concrete beams was determined by model updating. The beams were 150 mm wide, 250 mm deep and 2200 mm long. Different FE models were created which include a datum and models with a single crack at three different locations along the length of the beam. The mode shape equation was obtained by using non-linear regression. The equation used in the regression was the generalized solution of transverse vibration of a prismatic beam. Local flexural stiffness, EI, at each coordinate point was derived by substituting the regressed data by using the centered-finite-divided-difference formula. Experimental modal analysis was performed on a control beam and beams with load-induced cracks at predetermined loading. Results from FE analyses showed the trend in the loss of stiffness was similar to the results obtained on the experimental beams. The more severe the damage, the higher the loss of stiffness and the loss patterns are similar for damage at different locations along the beam. The updating technique is able to indicate the trend in the loss of stiffness as a result of cracks of varying severity in the RC beams showing good agreement with experimental results.     

Four-beam model for vibration analysis of a cantilever beam with an embedded horizontal crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton's principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.     

Improved independent component analysis based modal identification of higher damping structures

Structural modal parameter identification under ambient excitation has strong engineering value and theoretical significance. As the most popular tool for solving Blind Source Separation (BSS) problems, Independent Component Analysis (ICA) is able to directly extract the time-domain modal parameters, including frequencies, damping ratios and modal shapes. ICA, however, has a fatal flaw of failing to identify structures with higher damping. To overcome the flaw above, the paper proposes a new method named “ICA + IDT”. Firstly, free vibration response of a structure is obtained from structural outputs under ambient excitation. Inverse damping transfer (IDT) is employed to turn a highly damped signal into a low damping response signal without changing of frequencies and mode shapes. Then, structural modal parameters are extracted from the low damping response signal by ICA. Finally, the identified damping ratios are adjusted to eliminate the impact of IDT. To verify the effectiveness and applicability of IDT + ICA proposed herein, two numerical simulations—mass-spring model and simply supported concrete beam—and an experiment model of three-story steel frame are built, and the analysis results reveal that presented method can identify structures with higher damping effectively.     

基于波传播和阻抗特性的裂纹梁损伤识别

基于结构振动波传播理论,讨论了在简谐力作用下,裂纹简支梁的弯曲波动解。为了描述由裂纹引起的梁中波传播的不连续特性,引入弯曲弹簧模型来模拟裂纹,并在此基础上提出了利用梁结构驱动点阻抗特性的裂纹损伤识别方法。以一裂纹简支梁为例进行了数值分析,得到了裂纹简支梁的驱动点阻抗特性曲线。从该曲线可以发现,梁的第一阶谐振频率和反谐振频率都随裂纹的出现而减小,并且频率减少量随裂纹尺寸的增大而增加。结合裂纹梁第一阶谐振频率与驱动点位置关系曲线,利用曲线上出现的突变点,准确地识别了梁的损伤状态和裂纹损伤位置。最后,利用已识别的裂纹位置和第一阶固有频率定量地识别了裂纹尺寸。     

Second Generation Wavelet Finite Element and Rotor Cracks Quantitative Identification Method

The presence of cracks in the rotor is one of the most dangerous and critical defects for rotating machinery. Defect of fatigue cracks may lead to long out-of-service periods, heavy damages of machines and severe economic consequences. With the method of finite element, vibration behavior of cracked rotors and crack detection was received considerable attention in the academic and engineering field. Various researchers studied the response of a cracked rotor and most of them are focused on the crack detection based on vibration behavior of cracked rotors. But it is often difficult to identify the crack parameters quantitatively. Second generation wavelets (SGW) finite element has good ability in modal analysis for singularity problems like a cracked rotor. Based on the fact that the feature of SGW could be designed depending on applications, a multiresolution finite element method is presented. The new model of SGW beam element is constructed. The first three natural frequencies of the rotor with different crack location and size were solved with SGW beam elements, and the database for crack diagnosis is obtained. The first three metrical natural frequencies are employed as inputs of the database and the intersection of the three frequencies contour lines predicted the normalized crack location and size. With the Bently RK4 rotor test rig, rotors with different crack location and size are tested and diagnosed. The experimental results denote the cracks quantitative identification method has higher identification precision. With SGW finite element method, a novel method is presented that has higher precision and faster computing speed to identify the crack location and size.     

连续小波变换在梁结构损伤诊断中的应用研究

为了检测出梁中的裂缝或因刚度降低引起的损伤,对有损伤简支梁的振型曲线进行连续小波变换．从小波系数出现模极大值有效地识别损伤的存在以及裂缝位置和刚度下降段的位置。基本振型是用小波变换识别裂缝的最佳振型．用损伤位置处振幅较大的振型曲线来识别最清楚,对有噪声影响的振型曲线同样可以用本文方法进行识别。通过分析和计算获得满意结果．在梁结构损伤诊断中具有较高的应用价值。     

基于小波分析的裂纹梁的损伤识别

阐述了一种基于小波变换的含裂纹梁的损伤识别方法,利用含裂纹梁的一阶模态阵型作为小波分析的力学特征信号,识别损伤的位置和大小.利用小波分析系数的模极大值随分析尺度的传播定位损伤的位置,计算针对于损伤频率信号的能量判断损伤的大小.与以前的小波分析方法相比,此方法确定损伤位置的可靠性高,能识别微小的损伤.利用能量守恒定理和小波分析频段细化的能力,裂纹的定量分辨率高.     

Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions

An analytical approach for crack identification procedure in uniform beams with an open edge crack, based on bending vibration measurements, is developed in this research. The cracked beam is modeled as two segments connected by a rotational mass-less linear elastic spring with sectional flexibility, and each segment of the continuous beam is assumed to obey Timoshenko beam theory. The method is based on the assumption that the equivalent spring stiffness does not depend on the frequency of vibration, and may be obtained from fracture mechanics. Six various boundary conditions (i.e., simply supported, simple–clamped, clamped–clamped, simple–free shear, clamped–free shear, and cantilever beam) are considered in this research. Considering appropriate compatibility requirements at the cracked section and the corresponding boundary conditions, closed-form expressions for the characteristic equation of each of the six cracked beams are reached. The results provide simple expressions for the characteristic equations, which are functions of circular natural frequencies, crack location, and crack depth. Methods for solving forward solutions (i.e., determination of natural frequencies of beams knowing the crack parameters) are discussed and verified through a large number of finite-element analyses. By knowing the natural frequencies in bending vibrations, it is possible to study the inverse problem in which the crack location and the sectional flexibility may be determined using the characteristic equation. The crack depth is then computed using the relationship between the sectional flexibility and the crack depth. The proposed analytical method is also validated using numerical studies on cracked beam examples with different boundary conditions. There is quite encouraging agreement between the results of the present study and those numerically obtained by the finite-element method.     

RAT矩函数在模态振形数据去噪中的应用

提出将RAT矩函数应用于三维结构模态振型数据去噪的方法,使用RAT矩函数对结构的振型数据进行矩变换,从振型数据中提取出包含所有振型信息的特征矩集,将幅值较小的矩值作为噪声从特征矩集中剔除,再利用特征矩集重构振型图像,即能实现对振型数据去噪。采用一个仿真的机匣结构,以正态分布的随机噪声模拟真实情况下的噪声,在不同噪声强度下利用RAT矩对振型数据去噪,来验证RAT矩函数的去噪能力。结果表明:RAT矩函数具有良好的抗噪性,能大幅度减小噪声对模态振型的影响。     

受弯梁中开裂纹的位置识别与分析

利用有限元计算判定受弯梁中开裂纹的位置 ,从中得出 :同正常梁相比 ,裂纹梁的固有频率与振型的变化不但与裂纹深度而且与裂纹位置有关 ,因而 ,通过裂纹梁低阶固有频率及振型的变化情况可以判定裂纹的位置。对于裂纹较浅的情况 ,直接利用振型与固有频率的变化很难判定裂纹的位置 ,必须借用一些特征参数来提高识别的敏感性 ,这样 ,裂纹梁中早期裂纹的识别也是可行的     

含多条裂纹梁的模态与振动疲劳寿命分析

基于Paris公式,提出了一种含多条裂纹梁疲劳寿命预估的方法。在模态分析中,基于传递矩阵方法,利用无质量的弯曲弹簧等效裂纹,提出一种求解含有多条裂纹梁固有振型的方法,分析裂纹数目、裂纹位置、裂纹深度对裂纹梁固有频率的影响。在振动疲劳分析中,研究了在简谐激励作用下裂纹数目对裂纹尖端应力强度因子的影响。通过Paris疲劳裂纹扩展方程和同步分析法,考虑裂纹梁振动与裂纹扩展的相互作用,分析了裂纹数目和裂纹位置对裂纹梁疲劳寿命的影响。结果表明,裂纹数量、裂纹位置和深度对梁的模态参数和疲劳寿命有重要影响。     

Forced responses of cracked cantilever beams subjected to a concentrated moving load

An analytical method is developed to present the dynamic response of a cracked cantilever beam subject to a concentrated moving load. The cracked beam system is modeled as a two-span beam and each span of the continuous beam is assumed to obey Euler–Bernoulli beam theory. The crack is modeled as a rotational spring with sectional flexibility. Considering the compatibility requirements on the crack, the relationships between these two spans can be obtained. By using the analytical transfer matrix method, eigensolutions of this cracked system are obtained explicitly. The forced responses can be obtained by the modal expansion theory using the determined eigenfunctions. Some numerical results are shown to present the crack effects (crack extent, location of the crack) and are studied for different speeds of the moving load.     

An experimental study on damage detection of structures using a timber beam

Using vibration methods for the damage detection and structural health monitoring in bridge structures is rapidly developing. However, very little work has so far been reported on timber bridges. This paper intends to address such shortcomings by experimental investigation on a timber beam using a vibration based method to detect damage. A promising damage detection algorithm based on modal strain energy was adopted and modified to locate/evaluate damage. A laboratory investigation was conducted on a timber beam inflicted with various damage scenarios using modal tests. The modal parameters obtained from the undamaged and damaged state of the test beam were used in the computation of damage index, were then applied using a damage detection algorithm utilising modal strain energy and a statistical approach to detect location of damage. A mode shape reconstruction technique was used to enhance the capability of the damage detection algorithm with limited number of sensors. The test results and analysis show that location of damage can be accurately identified with limited sensors. The modified method is less dependent on the number of modes selected and can detect damage with a higher degree of confidence.