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。     

Crack detection in simply supported beams without baseline modal parameters by stationary wavelet transform

There are significant changes in the vibration responses of cracked structures when the crack depth is significant in comparison to the depth of the structure. This fact enables the identification of cracks in structures from their vibration response data. However when the crack is relatively small, it is difficult to identify the presence of the crack by a mere observation of the vibration response data. A new approach for crack detection in beam-like structures is presented and applied to cracked simply supported beams in this paper. The approach is based on finding the difference between two sets of detail coefficients obtained by the use of the stationary wavelet transform (SWT) of two sets of mode shape data of the beam-like structure. These two sets of mode shape data, which constitute two new signal series, are obtained and reconstructed from the modal displacement data of a cracked simply supported beam. They represent the left half and the modified right half of the modal data of the simply supported beam. SWT is a redundant transform that doubles the number of input samples at each iteration. It provides a more accurate estimate of the variances at each scale and facilitates the identification of salient features in a signal, especially for recognising noise or signal rupture. It is well known that the mode shape of a beam containing a small crack is apparently a single smooth curve like that of an uncracked beam. However, the mode shape of the cracked beam actually exhibits a local peak or discontinuity in the region of damage. Therefore, the mode shape ‘signal’ of a cracked beam can be approximately considered as that of the uncracked beam contaminated by ‘noise’, which consists of response noise and the additional response due to the crack. Thus, the modal data can be decomposed by SWT into a smooth curve, called the approximation coefficient, and a detail coefficient. The difference of the detail coefficients of the two new signal series includes crack information that is useful for damage detection. The modal responses of the damaged simply supported beams used are computed using the finite element method. For real cases, mode shape data are affected by experimental noise. Therefore, mode shape data with a normally distributed random noise are also studied. The results show that the proposed method has great potential in crack detection of beam-like structures as it does not require the modal parameters of an uncracked beam as a baseline for crack detection. The effects of crack size, depth and location, and the effects of sampling interval are examined.     

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.     

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。     

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.     

Repair of a cracked Timoshenko beam subjected to a moving mass using piezoelectric patches

This paper presents an analytical method for the application of piezoelectric patches for the repair of cracked beams subjected to a moving mass. The beam equations of motion are obtained based on the Timoshenko beam theory by including the dynamic effect of a moving mass traveling along a vibrating path. The criterion used for the repair is altering the first natural frequency of the cracked beam towards that of the healthy beam using a piezoelectric patch. Conceptually, an external voltage is applied to actuate a piezoelectric patch bonded on the beam. This affects the closure of the crack so that the singularity induced by the crack tip will be decreased. The equations of motion are discretized by using the assumed modes method. The cracked beam is modeled as number of segments connected by two massless springs at the crack locations (one, extensional and the other, rotational). The relationships between any two spans can be obtained by considering the compatibility requirements on the crack section and on the ends of the piezoelectric patch. Using the analytical transfer matrix method, eigensolutions of the system can be calculated explicitly. Finally, numerical simulations are performed with respect to different conditions such as the moving load velocity. It is seen that when the piezoelectric patch is used, the maximum deflection of the cracked beam approaches maximum deflection of the healthy beam.     

基于有限元位移模式的含裂纹梁结构动力学模型

针对裂纹的存在将降低梁的刚度的实际情形,首先根据断裂力学理论,引入裂纹梁因裂纹扩展而释放的应变能表达式,然后根据金属材料的特点,运用有限元位移法建立裂纹梁单元的动力学模型,再在梁单元模型的基础上应用有限元位移法建立裂纹梁结构的动力学方程。研究表明：基于有限元位移模式所建立的动力学方程较好地体现了裂纹梁动态性能与其结构参数和裂纹参数之间的内在关系,反映了裂纹的位置及长度对含裂纹梁结构动态性能的影响,为建立含裂纹梁结构动力学模型提供了一种新的有效方法。最后通过实例对理论分析结果进行了验证。     

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.     

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.     

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.     

Simulation on the motion of crankshaft with a slant crack in crankpin

A new model for vibration analysis of a crankshaft with a slant crack in crankpin is proposed, and the influence of crack depth on the transient response of a cracked crankshaft is investigated. A slant cracked shaft element is developed by deducing the local flexibility due to a slant crack. The frequently occurred slant crack in crankpin is studied, and a new finite element model of crankshaft including the slant crack in crankpin, which combines the slant cracked shaft element and Timoshenko beam elements, is derived. The support of engine block and the switching behaviour of the crack are considered, and the non-linear equation of motion for cracked crankshaft-bearing system is set up in a rotating coordinate system. The motion of a crankshaft of a four in-line cylinder engine with and without an initial crack is simulated. The influence of the crack depth on the transient response is investigated. The numerical simulation demonstrates that the current model is valid for simulating the motion of cracked crankshaft system. The results show that a useful foundation is laid for crack detection of crankshaft.     

I-beam Crack Identification Based on Study of Local Flexibility due to Crack

Local flexibility of crack plays an important role in crack identification of structures.Analytical methods on local flexibility in a cracked beam with simple geometric crossing sections,such as rectangle,circle,have been made,but there are some difficulties in calculating local flexibility in a cracked beam with complex crossing section,such as pipe and I-beam.In this paper,an analytical method to calculate the local flexibility and rotational spring stiffness due to crack in I-beam is proposed.The local flexibility with respect to various crack depths can be calculated by dividing a cracked I-beam into a series of thin rectangles.The forward and inverse problems in crack detection of I-beam are studied.The forward problem comprises the construction of crack model exclusively for crack section and the construction of a numerically I-beam model to gain crack detection database.The inverse problem consists of the measurement of modal parameters and the detection of crack parameters.Two experiments including measurement of rotational spring stiffness and prediction of cracks in I-beam are conducted.Experimental results based on the current methods indicate that relative error of crack location is less than 3%,while the error of crack depth identification is less than 6%.Crack identification of I-beam is expected to contribute to the development of automated crack detection techniques for railway lines and building skeletons.     

Discussion on calculation of the local flexibility due to the crack in a pipe

The stress intensity factor plays an important role in the calculation of the local flexibility due to the crack in a cracked structure. Many researches on the stress intensity factors in a cracked beam or rotor have been made, but there are some difficulties in calculating the stress intensity factors in a cracked pipe. Maiti et al. obtained the local flexibility due to the crack in a pipe experimentally by deflection and natural frequency methods without calculating the stress intensity factor. In this paper, the stress intensity factor in a cracked pipe can be calculated by dividing a cracked pipe into a series of thin annuli, and a method to calculate the local flexibility due to the crack in a pipe is presented. The experiment results are given to verify the effectiveness of such a method.     

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     

Repair of notched beam under dynamic load using piezoelectric patch

In this paper, the repair of a cracked beam under an external dynamic load employing the electro-mechanical characteristic of piezoelectric material to induce a local moment is presented. Conceptually, an external voltage is applied to actuate a piezoelectric patch bonded on the beam to effect closure of a crack so that the singularity at the crack tip under dynamic load may be decreased. Globally, this has the effect of altering the resonant frequency of the cracked beam towards that of the healthy beam, which is the criterion used for the repair. To demonstrate the repair methodology, a cantilever beam is used as an illustration, where the repair moment coefficient and the voltage required are mathematically derived. The relationship between repair moment coefficient, crack parameters and length of piezoelectric patch is investigated. The difference between the proposed repair criterion and an earlier published criterion for cracked beam under static load is also shown. A numerical example is used to study the effectiveness of the proposed repair methodology and its results are compared with those from 3-D finite element analyses using ABAQUS 6.4 as one means of verification.     

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.     

Free vibration analysis of beams with non-ideal clamped boundary conditions

A non-ideal boundary condition is modeled as a linear combination of the ideal simply supported and the ideal clamped boundary conditions with the weighting factors k and 1-k, respectively. The proposed non-ideal boundary model is applied to the free vibration analyses of Euler-Bernoulli beam and Timoshenko beam. The free vibration analysis of the Euler-Bernoulli beam is carried out analytically, and the pseudospectral method is employed to accommodate the non-ideal boundary conditions in the analysis of the free vibration of Timoshenko beam. For the free vibration with the non-ideal boundary condition at one end and the free boundary condition at the other end, the natural frequencies of the beam decrease as k increases. The free vibration where both the ends of a beam are restrained by the non-ideal boundary conditions is also considered. It is found that when the non-ideal boundary conditions are close to the ideal clamped boundary conditions the natural frequencies are reduced noticeably as k increases. When the non-ideal boundary conditions are close to the ideal simply supported boundary conditions, however, the natural frequencies hardly change as k varies, which indicate that the proposed boundary condition model is more suitable to the non-ideal boundary condition close to the ideal clamped boundary condition.     

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.