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.     

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.     

利用小波变换的双裂纹悬臂梁参数识别实验研究

将小波变换直接用于含裂纹悬臂梁的不同模态,利用小波系数在空间域上的突变指示裂纹位置,通过定义集中因子标定裂纹深度。并分析和讨论了悬臂梁前四阶模态对损伤的敏感性。通过本文的实验研究,检验了sym4小波在工程应用中的适用性,并指出二阶模态是较为敏感的损伤信息,有助于提高缺陷识别的精度和可靠性。     

利用传递矩阵法对阶梯悬臂梁的裂纹参数识别

基于Bernoulli-Euler梁振动理论,以等效弹簧模拟裂纹引起的局部软化效应,利用传递矩阵法推导阶梯悬臂梁振动频率的特征方程,对于含多个裂纹以及复杂边界条件的阶梯梁,仅需求解4×4的行列式即可获得相应的频率特征方程。直接利用该特征方程,提出两种有效估计裂纹参数的方法———等值线法和目标函数最小化法,并应用两段阶梯悬臂梁的数值算例说明方法的有效性。算例结果表明,只需结构前三阶频率即可识别裂纹位置和深度。应用“零设置”可减小计算频率与理论频率不相等对识别结果的影响。等值线法可以直观给出裂纹位置和裂纹深度参数,目标函数最小化法可给出最优的裂纹参数结果,并且该方法可推广应用到含多个裂纹复杂梁(如非完全固支、弹性支撑等)结构的裂纹参数识别中。     

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.     

悬臂梁裂纹参数的识别方法

以梁振动理论作为基础 ,将含裂纹梁的振动问题转化为由弹性铰联接两个弹性梁系统的振动问题 ,得到理论计算含裂纹梁振动频率的特征方程。由此特征方程计算得到裂纹深度参数和位置参数变化时悬臂梁振动固有频率的变化规律。利用计算裂纹悬臂梁振动固有频率的特征方程 ,提出一种辩识裂纹深度和位置参数的数值计算方法。并通过对模拟悬臂梁裂纹的分析说明文中方法的有效性。     

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     

基于固有频率的悬臂梁裂缝识别

结构中裂缝的存在使其模态参数发生改变 ,如局部刚度减小、阻尼增大、固有频率降低。把裂缝梁模拟成由扭曲弹簧连接 ,并对其前三阶固有频率的变化与裂缝位置和深度之间的关系进行计算和分析 ;利用特征方程以及前三阶固有频率 ,通过作图法对裂缝参数进行识别。识别结果证明 ,这种方法精度较高、简单可行 ,可用于机械工程实时监测。     

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bernoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method (DTM) has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.     

Crack identification in a cantilever beam under uncertain end conditions

Crack identification in structures by changes of their dynamic behavior has been studied in the past, and various methods were developed enabling the calculation of crack location along a beam, by using the variations in the natural frequencies between the initial undamaged state and a later, cracked beam. Application of this procedure to cantilever beams may result in unacceptably large errors, due to changes in clamp rigidity between measurements in the two states. The present research studies the problem of crack identification in a cantilever when clamp rigidity is unknown, and may change with time. An identification method is developed, which requires monitoring of three natural bending frequencies. Crack location may then be found by using a universal curve, i.e. independent of any beam property (geometry or material). The proposed method was verified by numerical simulation and experiment.     

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.     

小波分析识别裂纹参数中的噪声影响

鉴于信号和噪声进行多尺度小波变换时,其小波系数模极大值在多尺度上表现出截然不同的性质,直接将悬臂梁含噪声的前四阶模态进行小波分析,从小波系数模极大值在多尺度上的特性可以定位损伤,讨论高阶模态在损伤识别中的优缺点.定义一个和小波系数直接相关的集中因子,建立其与裂纹深度之间的关系,并分析模态中的噪声对裂纹深度识别的影响规律.研究可对实测环境中小波分析识别损伤提供一定的参考.     

Frequency error based identification of cracks in beam-like structures

A crack identification method of a single edge cracked beam-like structure by the use of a frequency error function is presented in this paper. First, the dynamic theory of Euler-Bernoulli beams was employed to derive the equation of the natural frequency for a single edge cracked cantilever beam-like structure. Subsequently, the cracked section of the beam was simulated by a torsional spring. The flexibility model of the torsional spring due to the crack was estimated by fracture mechanics and energy theory. Thereafter, a function model was proposed for crack identification by using the error between the measured natural frequencies and the predicted natural frequencies. In this manner, the crack depth and crack position can be determined when the total error reaches a minimum value. Finally, the accuracy of the natural frequency equation and the viabilty of the crack identification method were verified in the case studies by the measured natural frequencies from the literature. Results indicate that the first two predicted natural frequencies are in good agreement with the measured ones. However, the third predicted natural frequency is smaller than the measured natural frequency. In the case of small measured frequency errors, the predicted crack parameters are in good agreement with the measured crack parameters. However, in the case of large measured frequency errors, the predicted crack parameters only give roughly estimated results.     

Three-steps-meshing based multiple crack identification for structures and its experimental studies

Multiple crack identification plays an important role in vibration-based crack identification of structures. Traditional crack detection method of single crack is difficult to be used in multiple crack diagnosis. A three-step-meshing method for the multiple cracks identification in structures is presented. Firstly, the changes in natural frequency of a structure with various crack locations and depth are accurately obtained by means of wavelet finite element method, and then the damage coefficient method is used to determine the number and the region of cracks. Secondly, different regions in the cracked structure are divided into meshes with different scales, and then the small unit containing cracks in the damaged area is gradually located by iterative computation. Lastly, by finding the points of intersection of three frequency contour lines in the small unit, the crack location and depth are identified. In order to verify the effectiveness of the presented method, a multiple cracks identification experiment is carried out. The diagnostic tests on a cantilever beam under two working conditions show the accuracy of the proposed method: with a maximum error of crack location identification 2.7% and of depth identification 5.2%. The method is able to detect multiple crack of beam with less subdivision and higher precision, and can be developed as a multiple crack detection approach for complicated structures.     

裂纹悬臂梁的扭转弹簧模型及其实验验证

将含裂纹悬臂梁转化为由扭转弹簧联接两段弹性梁构成的连接体,得到理论计算含裂纹梁振动频率的特征方程。确立了求解裂纹梁固有频率的数值计算流程．计算得到了裂纹深度和位置变化时裂纹悬臂梁振动固有频率的变化规律。进行了裂纹悬臂梁的弯曲振动台架实验,验证了本文提出的扭转弹簧模型及固有频率数值计算方法的有效性。     

基于小波分析的悬臂梁裂纹参数识别方法研究

通过对含裂纹悬臂梁的应变能信号进行小波分析,悬臂梁的裂纹位置可由小波系数的局部极大值给出,并通过小波系数局部极大值定义集中因子和裂纹深度之间的关系,以此估计裂纹深度.数值算例表明, 利用sym4小波对含裂纹梁的应变能信号进行小波分析,可以准确识别出裂纹的位置和深度,这一方法很容易推广应用到结构的在线监测中.     

基于区间B样条小波有限元的转子裂纹定量识别

研究一种基于区间B样条小波有限元的转子横向裂纹定量识别方法.构造包含转动惯量影响的区间B样条小波Rayleigh梁单元,高精度求解裂纹转子前三阶固有频率,获得裂纹相对位置和相对深度作为变量的固有频率解曲面.然后将实测的裂纹转子前三阶固有频率作为裂纹识别问题的输入,利用三条等高线的交点定量识别出裂纹存在的相对位置和相对深度.数值仿真和试验研究结果表明,该方法鲁棒性强,单元数量少,辨识精度和效率高,为转子系统裂纹定量识别提供了新方法.     

基于压电增益特性进行梁中缺陷的识别

驱动元件PZT片和传感元件PVDF膜粘贴于自由梁表面,通过测试压电增益,试验获取梁中不同缺陷尺寸下的固有频率,根据固有频率的变化,实现缺陷的识别,梁中的缺陷采用等效线性弹簧模拟,描绘出不同模态下刚度与缺陷可能位置曲线,根据曲线的交点,得出缺陷位置与尺寸,相比于实际的缺陷位置与尺寸,自由梁弯曲激振下识别的结果满足一定的精度。     

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.     

Nonlinear vibrational response of a single edge cracked beam

The nonlinear vibrational response of a breathing cracked beam was investigated. The study was done by using a new crack stiffness model to examine some of the nonlinear behaviors of a cantilever beam with a breathing crack. The quadratic polynomial stiffness equation of the cracked beam was derived based on the hypothesis that the breathing process of a crack depends on the vibration magnitude. The Galerkin method combined with the stiffness equation was used to simplify the cracked beam into a Single-degree-of-freedom (SDOF) lumped system with nonlinear terms. The multi scale method was adopted to analyze the nonlinear amplitude frequency response of the beam. The applicability of the stiffness model was discussed and parameter sensitivity studies on the dynamic response were carried out by the SDOF model for a cantilever beam. Results indicate that the new stiffness model provides an efficient tool to study the vibrational nonlinearities introuduced by the breathing crack. Therefore, it might be used to develop a nonlinear identification method of a crack in a beam.