DYNAMIC RESPONSE OF A BEAM WITH A CRACK SUBJECT TO A MOVING MASS

An iterative modal analysis approach is developed to determine the effect of transverse cracks on the dynamic behavior of simply supported undamped Bernoulli-Euler beams subject to a moving mass. The presence of crack results in higher deflections and alters the beam response patterns. In particular, the largest deflection in the beam for a given speed takes longer to build up, and a discontinuity appears in the slope of the beam deflected shape at the crack location. Crack effects become more noticeable as crack depth increases. The effect of the inertia force due to the moving mass is, in general, qualitatively similar and additive to the effect of the crack. The exact effect of crack and mass depends on the speed, time, crack size, crack location, and the moving mass level. Other approximate methods, namely a stationary mass model and a single iteration technique, are also evaluated. The stationary mass approach is useful for light moving masses (<20% of beam mass) and cracks at mid-span. For other cases, the errors can be unacceptably large. The results of the single-iteration approximation are quite close to the iterative modal analysis approach, which indicates that this approximate solution is an excellent tool for the analysis of the moving mass problem.     

CRACK IDENTIFICATION IN VIBRATING BEAMS USING THE ENERGY METHOD

An energy-based numerical model is developed to investigate the influence of cracks on structural dynamic characteristics during the vibration of a beam with open crack(s). Upon the determination of strain energy in the cracked beam, the equivalent bending stiffness over the beam length is computed. The cracked beam is then taken as a continuous system with varying moment of intertia, and equations of transverse vibration are obtained for a rectangular beam containing one or two cracks. Galerkin's method is applied to solve for the frequencies and vibration modes. To identify the crack, the frequency contours with respect to crack depth and location are defined and plotted. The intersection of contours from different modes could be used to identify the crack location and depth.     

VIBRATION-BASED DIAGNOSTICS OF FATIGUE DAMAGE OF BEAM-LIKE STRUCTURES

The analytical investigation of vibration of damaged structures is a complicated problem. This problem may be simplified if a structure can be represented in the form of a beam with corresponding boundary and loading conditions. In this connection, free vibrations of an elastic cantilever Bernoulli-Euler beam with a closing edge transverse crack is considered in the present work as a model of a structure with a fatigue crack. The modelling of bending vibrations of a beam with a closing crack is realized based on the solutions for an intact beam and for a beam with an open crack. The algorithm of consecutive (cycle-by-cycle) calculation of beam mode shapes amplitudes is presented. It is shown that at the instant of crack opening and closing, the growth of the so-called concomitant mode shapes which differ from the initially given mode shape takes place. Moreover, each of the half-cycles is characterized by a non-recurrent set of amplitudes of concomitant modes of vibration and these amplitudes are heavily dependent on the crack depth.The vibration characteristics of damage based on the estimation of non-linear distortions of the displacement, acceleration and strain waves of a cracked beam are investigated, and the comparative evaluation of their sensitivity is carried out.     

The use of roving discs and orthogonal natural frequencies for crack identification and location in rotors

A variety of approaches that have been developed for the identification and localisation of cracks in a rotor system, which exploit natural frequencies, require a finite element model to obtain the natural frequencies of the intact rotor as baseline data. In fact, such approaches can give erroneous results about the location and depth of a crack if an inaccurate finite element model is used to represent an uncracked model. A new approach for the identification and localisation of cracks in rotor systems, which does not require the use of the natural frequencies of an intact rotor as a baseline data, is presented in this paper. The approach, named orthogonal natural frequencies (ONFs), is based only on the natural frequencies of the non-rotating cracked rotor in the two lateral bending vibration x–z and y–z planes. The approach uses the cracked natural frequencies in the horizontal x–z plane as the reference data instead of the intact natural frequencies. Also, a roving disc is traversed along the rotor in order to enhance the dynamics of the rotor at the cracked locations. At each spatial location of the roving disc, the two ONFs of the rotor–disc system are determined from which the corresponding ONF ratio is computed. The ONF ratios are normalised by the maximum ONF ratio to obtain normalised orthogonal natural frequency curves (NONFCs). The non-rotating cracked rotor is simulated by the finite element method using the Bernoulli–Euler beam theory. The unique characteristics of the proposed approach are the sharp, notched peaks at the crack locations but rounded peaks at non-cracked locations. These features facilitate the unambiguous identification and locations of cracks in rotors. The effects of crack depth, crack location, and mass of a roving disc are investigated. The results show that the proposed method has a great potential in the identification and localisation of cracks in a non-rotating cracked rotor.     

An electro-mechanical impedance model of a cracked composite beam with adhesively bonded piezoelectric patches

A model of a laminated composite beam including multiple non-propagating part-through surface cracks as well as installed PZT transducers is presented based on the method of reverberation-ray matrix (MRRM) in this paper. Toward determining the local flexibility characteristics induced by the individual cracks, the concept of the massless rotational spring is applied. A Timoshenko beam theory is then used to simulate the behavior of the composite beam with open cracks. As a result, transverse shear and rotatory inertia effects are included in the model. Only one-dimensional axial vibration of the PZT wafer is considered and the imperfect interfacial bonding between PZT patches and the host beam is further investigated based on a Kelvin-type viscoelastic model. Then, an accurate electro-mechanical impedance (EMI) model can be established for crack detection in laminated beams. In this model, the effects of various parameters such as the ply-angle, fibre volume fraction, crack depth and position on the EMI signatures are highlighted. Furthermore, comparison with existent numerical results is presented to validate the present analysis.     

Toughening through multilayering in TiN–AlTiN films

The damage response of columnar multilayers of TiN and AlTiN to Vickers indentation is studied through focused ion beam machining and elastic modelling. Multilayers display an enhanced resistance, which increases with layer refinement, to the multiple fracture modes that appear at high loads in these materials, including edge (nested) cracks and inclined shear cracks. Measurements of layer thickness reveal that multilayers display additional modes of plastic deformation that lead to permanent compression and bending of the film. An elastic model of contact deformation in a bilayer where plasticity is mimicked by greatly enhanced elastic compliance of the film is used to rationalize the trends in crack resistance. It is shown that the enhanced toughness is not due to any increase in the strain capacity (hardness/modulus) of the film material, brought about by multilayering.     

Flexural vibration of non-uniform beams having double-edge breathing cracks

Flexural vibration of non-uniform Rayleigh beams having single-edge and double-edge cracks is presented in this paper. Asymmetric double-edge cracks are formed as thin transverse slots with different depths at the same location of opposite surfaces. The cracks are modelled as breathing since the bending of the beam makes the cracks open and close in accordance with the direction of external moments. The presented crack model is used for single-edge cracks and double-edge cracks having different depth combinations. The energy method is used in the vibration analysis of the cracked beams. The consumed energy caused by the cracks opening and closing is obtained along the beam's length together with the contribution of tensile and compressive stress fields that come into existence during the bending. The total energy is evaluated for the Rayleigh-Ritz approximation method in analysing the vibration of the beam. Examples are presented on simply supported beams having uniform width and cantilever beams which are tapered. Good agreements are obtained when the results from the present method are compared with the results of Chondros et al. and the results of the commercial finite element program, Ansys^©. The effects of breathing in addition to crack depth's asymmetry and crack positions on the natural frequency ratios are presented in graphics.     

THE DYNAMIC STIFFNESS MATRIX METHOD IN FORCED VIBRATION ANALYSIS OF MULTIPLE-CRACKED BEAM

The dynamic behaviour of a beam with numerous transverse cracks is studied. Based on the equivalent rotational spring model of crack and the transfer matrix for beam, the dynamic stiffness matrix method has been developed for spectral analysis of forced vibration of a multiple cracked beam. As a particular case, when the excitation frequency is close to zero, the solution for static response of beam with an arbitrary number of cracks has been obtained exactly in an analytical form. In general case, the effect of crack number and depth on the dynamic response of beam was analyzed numerically.     

A method for determining the location and depth of cracks in double-cracked beams

The influence of two transverse open cracks on the antiresonances of a double cracked cantilever beam is investigated both analytically and experimentally. It is shown that there is a shift in the antiresonances of the cracked beam depending on the location and size of the cracks. These antiresonance changes, complementary with natural frequency changes, can be used as additional information carrier for crack identification in double cracked beams. Experimental results from tests on plexiglas beams damaged at different locations and different magnitudes are found to be in good agreement with theoretical predictions. Based on the results of the present work, an efficient prediction scheme for crack localization and characterization in double cracked beams is proposed.     

Porcelain Insulator Crack Location and Surface States Pattern Recognition Based on Hyperspectral Technology

A porcelain insulator is an important part to ensure that the insulation requirements of power equipment can be met. Under the influence of their structure, porcelain insulators are prone to mechanical damage and cracks, which will reduce their insulation performance. After a long-term operation, crack expansion will eventually lead to breakdown and safety hazards. Therefore, it is of great significance to detect insulator cracks to ensure the safe and reliable operation of a power grid. However, most traditional methods of insulator crack detection involve offline detection or contact measurement, which is not conducive to the online monitoring of equipment. Hyperspectral imaging technology is a noncontact detection technology containing three-dimensional (3D) spatial spectral information, whereby the data provide more information and the measuring method has a higher safety than electric detection methods. Therefore, a model of positioning and state classification of porcelain insulators based on hyperspectral technology is proposed. In this model, image data were used to extract edges to locate cracks, and spectral information was used to classify the surface states of porcelain insulators with EfficientNet. Lastly, crack extraction was realized, and the recognition accuracy of cracks and normal states was 96.9%. Through an analysis of the results, it is proven that the crack detection method of a porcelain insulator based on hyperspectral technology is an effective non-contact online monitoring approach, which has broad application prospects in the era of the Internet of Things with the rapid development of electric power.     

Energy release rate and stress field calculation for debonding crack extension at the fibre-matrix interface during single-fibre pull-out

For the micromechanical modelling of the macroscopic failure of fibre-reinforced composites the formulation of a critical parameter for initiation and extension of debonding cracks at the fibre-matrix interface is essential. This point is discussed for the 'fibre pull-out' specimen, a test commonly used to measure the adhesion quality of fibre-matrix systems. Some of the simplifying assumptions fundamental to shear lag theory-based models of the fibre pull-out test are compared with results from a detailed finite element (FE) model to examine their validity. The FE model strongly contradicts assumptions made with the shear lag theory that the axial stress gradient in the matrix can be neglected from the equilibrium equation. A critical interface shear strength is commonly used as a measure of adhesion quality. But for elastic materials the nature of the stress concentrations at the fibre end and interface crack-tip are singular. Therefore a fracture mechanic approach is better suited for a debonding criterion than a simple finite shear strength. The energy release rate shows a minimum for short crack lengths and may stabilize the moving crack.     

CRACK DETECTION IN BEAM-TYPE STRUCTURES USING FREQUENCY DATA

A practical method to non-destructively locate and estimate size of a crack by using changes in natural frequencies of a structure is presented. First, a crack detection algorithm to locate and size cracks in beam-type structures using a few natural frequencies is outlined. A crack location model and a crack size model are formulated by relating fractional changes in modal energy to changes in natural frequencies due to damage such as cracks or other geometrical changes. Next, the feasibility and practicality of the crack detection scheme are evaluated for several damage scenarios by locating and sizing cracks in test beams for which a few natural frequencies are available. By applying the approach to the test beams, it is observed that crack can be confidently located with a relatively small localization error. It is also observed that crack size can be estimated with a relatively small size error.     

Meshless formulation using NURBS basis functions for eigenfrequency changes of beam having multiple open-cracks

This paper presents a meshless formulation using non-uniform rational B-spline (NURBS) basis functions, and its applications to evaluate natural frequencies of a beam having multiple open-cracks. Node-based NURBS basis functions are used to construct the approximation function. The characteristic differentiability of the NURBS basis functions allows it to represent a function having specific degrees of smoothness and/or discontinuity. The discontinuity can be incorporated simply by assigning multiple knots at those locations. Hence, it can yield exact solutions having interior discontinuous derivatives. These advantages of NURBS are well known, and have been used extensively in graphical approximation of geometrical surfaces. However, it is seldom used in other engineering applications. To model the multiple open-cracks in a beam, quartic NURBS basis functions are employed and quadruplicate knots are assigned at the crack locations. Hence, it is capable to model the abrupt changes of slope (the first derivative of displacement) across a crack. In the present applications, additional equivalent massless rotational springs are inserted at the crack locations to represent the local flexibility caused by the cracks. As such, the cracked beam can be treated in the usual manner as a continuous beam. By adopting the meshless Petrov–Galerkin formulation, a generalized stiffness matrix for the cracked beam can be derived. Compared to the conventional finite element method, the present method does not require a finite element mesh for the purposes of interpolation and numerical integration. The advantages and effectiveness of the present method is illustrated in solving the eigenfrequencies of a beam having multiple open-cracks of different depths.     

NON-DESTRUCTIVE EVALUATION OF HORIZONTAL CRACK DETECTION IN BEAMS USING TRANSVERSE IMPACT

Numerical and experimental studies for crack detection in beam employing transverse impact are presented. In the numerical study, a beam model of wave propagation is developed to calculate the time history of beam response before, over and after the crack region. It is expected that the resulting wave in the beam will be scattered by the crack and will carry information on the location and geometry of the crack. Experiments using a scanning laser vibrometer on specimens containing simulated crack are then conducted to verify the numerical results. Comparison study between the numerical results and experimental observations are conducted; good correlation between theory and experiment is observed. The beam model of wave propagation and adaptive multilayer perceptron networks (MLP) are then used for inverse identification of crack parameters (i.e., crack location, depth and length) in the beams. Time-domain displacement responses calculated using the present beam model containing predetermined crack parameters are used as training data for the MLP. Once the MLP is trained, the MLP networks are then employed for inverse determination of an unknown crack in a beam using experimental displacement responses measured with a scanning laser vibrometer. Examples show that the procedure performs well for the determination of a wide range of values for the location, depth and length of the crack.     

空间碎片超高速撞击充气压力容器前壁准静态破坏分析

针对空间碎片超高速撞击充气压力容器前壁发生准静态破坏问题,将其简化为受双向拉应力的圆孔边双裂纹的线弹性断裂问题进行处理;并在数值模拟及理论分析的基础上建立了充气压力容器前壁发生准静态破坏的预报模型,得到了当球形弹丸撞击速度为7.0 km/s时、壁厚为1.0 mm的Al5754圆柱形压力容器前壁发生准静态破坏的临界应力曲...     

Three-dimensional characterisation and modelling of small fatigue corner cracks in high strength Al-alloys

The growth of fatigue cracks at small length scales is known to be influenced by a variety of factors, including local microstructure, varying stress states and crack shape. High resolution computed tomography allows for sub-micron resolution imaging of failure processes in small test coupons undergoing in situ cyclic loading, providing detailed three-dimensional (3D) assessment of propagation processes across the entire crack front (surface and depth). In this work fatigue crack growth has been examined in an advanced Direct Chill (DC) cast aluminium alloy, along with a fine grained powder-metallurgy alloy. The latter is identified as a model material, offering considerably simpler microscopic crack paths than the DC cast alloy, and hence a means of separating bulk mechanical effects (such as stress state variations across a crack front and plasticity induced closure) from microstructural effects (such as crystallographic deflection and roughness induced crack closure). Crack growth has been studied in both materials under both constant amplitude (CA) and single peak overload (OL) conditions. Experimental results are presented in the present paper, particularly in relation to micromechanical understanding of failure. A modelling approach based on those results, and some typical results, is also presented.     

Identification of multiple cracks in a beam using natural frequencies

A simple method to identify multiple cracks in a beam is presented. The cracks are modeled as rotational springs and the forward problem is solved using the finite element method. The inverse problem is solved iteratively for the locations and sizes of the cracks using the Newton-Raphson method. Numerical examples are provided for the identification of triple cracks in a cantilever beam as well as double cracks. The detected crack locations and sizes are in excellent agreement with the actual ones.     

Multi-cracks detection of a beam-like structure based on the on-vehicle vibration signal and wavelet analysis

In this paper, a new method for detecting a multi-cracked beam-like structure subjected to a moving vehicle is presented. The crack model is adopted from fracture mechanics. The dynamic response of the bridge-vehicle system is measured directly from the moving vehicle. When moving along the structure, the moving vehicle causes small distortions in the dynamic response of the bridge-vehicle system at the crack locations. In general, these small distortions are difficult to detect visually. However, wavelet transform has recently emerged to be an effective method of detecting such small distortions. Large values (peaks) in the wavelet transform indicate the existence of the cracks. The locations of the cracks are pinpointed by positions of peaks of the wavelet transform and the velocity of the moving vehicle. Numerical results show that the method can detect cracks as small as 10% of the beam height. The proposed method is applicable for low velocity-movements while high velocity-movements are not recommended. The method presents an idea for measuring the vibration directly from the vehicle for crack detection problem in practice.     

Identification of embedded horizontal cracks in beams using measured mode shapes

Mode shapes (MSs) have been extensively used to detect structural damage. This paper presents two new non-model-based methods that use measured MSs to identify embedded horizontal cracks in beams. The proposed methods do not require any a priori information of associated undamaged beams, if the beams are geometrically smooth and made of materials that have no stiffness discontinuities. Curvatures and continuous wavelet transforms (CWTs) of differences between a measured MS of a damaged beam and that from a polynomial that fits the MS of the damaged beam are processed to yield a curvature damage index (CDI) and a CWT damage index (CWTDI), respectively, at each measurement point. It is shown that the MS from the polynomial fit can well approximate the measured MS and associated curvature MS of the undamaged beam, provided that the measured MS of the damaged beam is extended beyond boundaries of the beam and the order of the polynomial is properly chosen. The proposed CDIs of a measured MS are presented with multiple resolutions to alleviate adverse effects caused by measurement noise, and cracks can be identified by locating their tips near regions with high values of the CDIs. It is shown that the CWT of a measured MS with the n-th-order Gaussian wavelet function has a shape resembling that of the n-th-order derivative of the MS. The crack tips can also be located using the CWTs of the aforementioned MS differences with second- and third-order Gaussian wavelet functions near peaks and valleys of the resulting CWTDIs, respectively, which are presented with multiple scales. A uniform acrylonitrile butadiene styrene (ABS) cantilever beam with an embedded horizontal crack was constructed to experimentally validate the proposed methods. The elastic modulus of the ABS was determined using experimental modal analysis and model updating. Non-contact operational modal analysis using acoustic excitations and measurements by two laser vibrometers was performed to measure the natural frequencies and MSs of the ABS cantilever beam, and the results compare well with those from the finite element method. Numerical and experimental crack identification can successfully identify the crack by locating its tips.