Transverse Open Crack Research Articles

Natural frequencies of a Timoshenko beam with cracks

A problem of calculating of natural frequencies of a Timoshenko beam weakened by a finite number of transverse open cracks is considered. The problem is solved for both known crack models. In one model every crack is simulated by a single, massless rotational spring. In the other model every crack is simulated by two massless springs (one extensional and another one rotational). An effective method for calculation of the natural frequencies of a vibrating beam with cracks, which was previously successfully applied to Euler-Bernoulli beams, is extended to the case of Timoshenko beam. The developed method makes it possible to significantly reduce the order of the determinant, the zeros of which are natural frequencies. Numerical examples are considered. The results are compared with the known where it is possible.

Investigating the modal behaviors of a deep beam with a transverse open crack

This paper investigates the modal behaviors of a deep beam with a transverse open crack. Unlike cracked shallow beams with aspect ratio (L/h) > 10, cracked deep beams with aspect ratio < 5 simultaneously exhibit near the crack both axial-bending coupling mode caused by mode I crack and shear (or sliding) mode caused by mode II crack, referred to as axial-bending-shear modal coupling. Therefore, for accurate modal behaviors prediction, appropriate compatibility equations accounting for both modes I and II crack should be employed for deriving a frequency equation. Because the derived frequency equation is nonlinear with respect to modal frequency, the modal frequencies are calculated through the Newton–Raphson method. The corresponding mode shapes are determined by substituting the calculated modal frequencies into the spectral solutions for either side of the crack. The validity of the proposed method was demonstrated through comparison to the finite element analysis and experimental results.

A concerted experimental and numerical approach for frequency based non-destructive analysis of Bi-directional cracked laminated composite beams

ABSTRACT The effects of single and multiple transverse open cracks on the dynamic properties of woven fibre laminated composite beam (LCB) under free vibration by experimental and numerical approaches are explored in this research work. The experimental modal testing is performed n laminated glass/epoxy composite beams using vibration Fast Fourier Transformation (FFT) analyser system. The finite element simulation software ABAQUS is used as the platform for numerical modal analysis to arrive at the natural frequencies. Within the test domain of modal analysis, the simulated results are finely agreed with the empirical test results. The effects of crack location, relative crack depth, boundary conditions and fibre orientations on the modal frequency of the cracked LCB are explored under free vibration. The results of the study conclude that the maximum decrease in frequencies is observed for 0° ply oriented single and multi-cracked LCB. Furthermore, the results elicit that the presence of crack positioned nearer to the restrained end exhibits more reduction in vibration frequencies for any particular crack depth. Within the frequency domain, the present research approach can be employed as an effective tool for the detection and analysis of crack or crack-like defects in LCB structures in any applied field of engineering.

DAMAGE DETECTION IN A STEEL BEAM USING VIBRATION RESPONSE

Changes in vibration signals, natural frequency and modal parameters were used for damage detection in a steel beam. Several tests were performed on simply supported steel beams having open transverse cracks with varying depths and locations. Frequencies for the first three vibration modes were generated from the analytical, numerical, and experimental models. Analytical frequencies were calculated assuming Euler-Bernoulli beam and solving partial differential equations. Vibration signals were collected using a portable digital vibrometer and analyzed through Fast Fourier Transform. The finite element beam models were calibrated using experimental results. The graphical plots of the normalized frequency from the experimental study and FE models showed similar pattern for the first mode. The nodal points were identified as the locations where the damaged frequency equals the frequency of an intact beam. The existence of damage in the beam specimen was confirmed by comparing the natural frequency of a beam at its intact and damaged states. Equations relating normalized frequency with varying crack depth and location were proposed.

Determining the Severity of Open and Closed Cracks Using the Strain Energy Loss and the Hill-Climbing Method

Evaluating the integrity of structures is an important issue in engineering applications. The use of vibration-based techniques has become a common approach to assessing cracks, which are the most frequently occurring damage in structures. When involving an inverse method, it is necessary to know the influence of the position and the geometry of the crack on the modal parameter changes. The geometry of the crack, both in size and shape, defines the damage severity (DS). In this study, we present a method (DS-SHC) used for estimating the DS for closed and open transverse cracks in beam-like structures using the intact and damaged beam deflections under its weight and a Stochastic Hill Climbing (SHC) algorithm. After describing the procedure of applying DS-SHC, we calculate for a prismatic cantilever beam the severities for different crack types and depths. The results are tested by comparing the DS obtained with DS-SHC with those acquired from dynamic tests made using professional simulation software. We obtained a good fit between the severities determined in these two ways. Subsequently, we performed laboratory experiments and found that the severities obtained with the DS-SHC method can accurately predict the frequency changes due to the crack. Hence, these severities are a valuable tool for damage detection.

Experimental and computational study on dynamic analysis of cracked simply supported structures under moving mass

In this study, the influences of crack parameters like crack location from the left end, crack height, number of cracks and the magnitude of mass and the velocity of the transit load on the vertical displacements of the cracked simply supported beams subjected to transit mass are investigated. The transverse open cracks with numerous damage scenerios are considered for the mathematical modelling of the system. The governing equations of motion for the system have been obtained and the equations have been solved by the help of Duhamel integral technique. The theoretical formulation has been exemplified with numerical studies. By utilizing ANSYS Workbench 2020, transient structural analysis has been carried out. The mode shapes and the frequency ratios of damaged simply supported beam have also been determined. To validate the numerical and FEM models, the experiments with damaged beams have been carried out in the laboratory. It has been proven that the results of the theoretical and FEM models are well convergent with the experimental data. The results gathered from the numerical analysis, FEA as well as experimental study have been presented with comparative graphs and tables. The outcomes of the examinations have been interpreted in the conclusions part. It has been observed that subject parameters are of considerable significance on the time dependent response of the cracked beams.

Dynamic Modeling and Nonlinear Characteristics of a Cracked Bolt-Disk Combined Rotor System

Abstract Different from the crack on the rotating shaft, the crack on the bolt, which is a connecting part of the bolt-disk combined rotor, is a kind of local defect. The local crack on the bolt under high pretension is always in open state, and it increases the overall vibration of the combined rotor significantly in practice. This paper studies the modeling of the crack on the bolt and nonlinear dynamic behaviors of the cracked bolt-disk rotor system. The circumferential bolts with a transverse open crack are treated as several bar elements under the assumption that each bolt has the same original tensile extension length. The cracked correction coefficient is introduced to describe the decreasing amount of bolt's tension due to crack. After this coefficient is obtained according to finite element method, the stiffness matrix of circumferential bolts with crack is built based on total potential energy. The dynamic model consists of a time-independent stiffness matrix for perfect bolts, a time-variant reductive stiffness, and an additional moment. As a result, the crack in bolt reduces rotor's nonlinear stability and leads to greater vibration and fluctuation. In addition, crack depth has much larger influence than crack location on the dynamic behaviors.

Crack detection in a rotary shaft analytical and experimental analyses: A review

Detection of depth of cracks and its positions is the broad region of investigator over the universe. The article discussed a critical review of recent research done on crack identification and localization in rotary machine shaft using experimental modal analysis. Crack detection and localization in rotor shaft is very essential in various engineering applications. An enormous vibration in a rotary shaft causes premature failures of machine components. It is required to recognize the damage in time to avoid catastrophic failure of rotary machine shaft. Experimental investigation is used to observe their vibration characteristics like natural frequency of vibration, damping and mode shapes of structures. Natural frequency of rotary shaft can be obtained using the modal analysis technique. Also trained machine learning (ML) based artificial neural networks (ANN) can be applied to detect crack depth and its location. From the review by different research papers, it is observed that a lot of the research has been done on rotary machine shafts with open transverse cracks. Excessive speed and frequency for identified cracks position as well as cracks depth can be used to train an ANN. By using ML based artificial neural networks, location of cracks, and its depth identified precisely.

Crack Identification and Localization In Structural Beams Using Numerical and Experimental Modal Analysis- A Review

This article presents a critical review of recent research done on crack identification and localization in structural beams using numerical and experimental modal analysis. Crack identification and localization in beams are very crucial in various engineering applications such as ship propeller shafts, aircraft wings, gantry cranes, and Turbo machinery blades. It is necessary to identify the damage in time; otherwise, there may be serious consequences like a catastrophic failure of the engineering structures. Experimental modal analysis is used to study the vibration characteristics of structures like natural frequency, damping and mode shapes. The modal parameters like natural frequency and mode shapes of undamaged and damaged beams are different. Based on this reason, structural damage can be detected, especially in beams. From the review of various research papers, it is identified that a lot of the research done on beams with open transverse crack. Crack location is identified by tracking variation in natural frequencies of a healthy and cracked beam

A closed-form frequency equation of an arbitrarily supported beam with a transverse open crack considering axial–bending modal coupling

This paper presents a closed-form frequency equation that predicts the modal coupling between the adjacent axial and bending modes of a beam with a transverse open crack. The final form of the proposed frequency equation is simply expressed in terms of the uncoupled frequency equations of intact and cracked beams for any type of support condition of the beam by introducing an axial–bending modal coupling factor. The validity of the proposed frequency equation is demonstrated through a comparison with the numerical results from a finite element analysis and with experimental data. For relatively deep cracks, the overall effect of axial–bending modal coupling on the axial modal frequencies is more significant than that on the bending modal frequencies.

Crack detection in a beam using wavelet transform and photographic measurements

An optimistic attempt for the detection of a crack in a beam using discrete wavelet transform is proposed. Finite element model of the simply supported beam with a transverse open crack is considered. Crack in a beam causes a slope discontinuity in the elastic line. The deflection of the beam is used to locate the slope discontinuity using wavelet transform. The effect of measurement noise on the wavelet decomposition level is studied. An inexpensive digital image based technique is employed to obtain the deflection of the cracked beam for experimental investigations. White pixels due to dust particles are removed using connected component algorithm. A suitable edge detection technique is selected for finding the edged image on the basis of peak signal to noise ratio values. The deflection of the non-cracked edge of the beam is used as an input signal for wavelet transform to detect the crack location. Detection of a crack in the beam with different crack depths is investigated experimentally. The proposed method works well with numerically simulated as well as experimentally measured cracked beam deflections.

Stability behavior of two-crack functionally graded shaft in a rotor-disc system: finite element approach

Stability behavior of a rotor-disc system having a functionally graded (FG) shaft with fully open transverse cracks are studied through a finite element (FE) approach, considering the rigid end bearing conditions, various crack parameters, shaft internal damping and geometrical parameters. FG shaft is modeled with considering the effect of internal damping using Timoshenko beam theory (TBT)using two nodded with four degrees of freedom (DOFs) per node are taken into the formulation. Zirconia (ZrO2) and stainless steel (SS) are considered as FG material where material properties are considered temperature dependent and radially graded in the cracked FG shaft. Local flexibility coefficients (LFCs) are derived analytically with depth of crack, gradient index and temperature using Castigliano's theorem and Paris’s equation to compute the modified stiffness matrix at each instant of shaft rotation. Carrying out several numerical examples, the present formulation is validated with availableresults. Influences of shaft internal damping coefficients, gradient index, temperature gradient, shaft’s slenderness and relative locations of crack on the dynamic responses of the FG cracked shaft system are computed and discussed in detail.

Free vibrations of laminated composite beams with multiple edge cracks: Numerical model and experimental validation

Free vibration analysis of laminated composite beams including open transverse cracks is presented by using a shear-deformable thirteen degrees-of-freedom finite element model, which considers extension-twist, bending-twist and bending-extension couplings, and the Poisson's effect. Lagrange's equation is employed to derive the equations of motion. Trial functions are selected as cubic Lagrange polynomials for deflections while quadratic for the others in derivation of element mass and stiffness matrices. Damage is introduced into the beam by reducing stiffness of elements within damaged zone. Comparisons are made with the available literature to show the present element's accuracy. Experimental validation is also performed with through ambient vibration tests on a cantilever laminated composite beam under six damage scenarios. According to the results, the methodology presented in this study provides a simple and quick way in a sufficient accuracy to determine the natural frequencies and mode shapes of laminated composite beams with open transverse non-propagating edge cracks.

Study of the Effect of an Open Transverse Crack on the Vibratory Behavior of Rotors Using the h-p Version of the Finite Element Method

In this paper, we use the hybrid h-p version of the finite element method to study the effect of an open transverse crack on the vibratory behavior of rotors, the one-dimensional finite element Euler-Bernoulli beam is used for modeling the rotor, the shape functions used are the Hermite cubic functions coupled to the special Legendre polynomials of Rodrigues. The global matrices of the equation of motion of the cracked rotor are derived by the application of the Lagrange equation taking into account the local variation in the shaft’s stiffness due to the presence of the crack, and the stiffness of the cracked element of the shaft are determined using the time-varying stiffness method. Numerical results generated by a program developed in MATLAB show the rapidity of the convergence of the h-p version of FEM compared to the classical version, after the validation of our results with theoretical and experimental results and other obtained with the simulator ANSYS Workbench, a parametric study was provided to show the influence of the depth and position of the crack on the vibratory behavior of a symmetrical and asymmetrical rotor.

Analytical Buckling Loads of Columns Weakened Simultaneously with Transverse Cracks and Partial Delamination

This paper focuses on development of a new mathematical model and its analytical solution for buckling analysis of elastic columns weakened simultaneously with transverse open cracks and partial longitudinal delamination. Consequently, the analytical solution for buckling loads is derived for the first time. The critical buckling loads are calculated using the proposed analytical model. A parametric study is performed to investigate the effects of transverse crack location and magnitude, length and degree of partial longitudinal delamination, and different boundary conditions on critical buckling loads of weakened columns. It is shown that the critical buckling loads of weakened columns can be greatly affected by all the analyzed parameters. Finally, the presented results can be used as a benchmark solution.

Stability and bifurcation of a flexible rod-fastening rotor bearing system with a transverse open crack

The stability and bifurcation of a flexible rod-fastening rotor bearing system with a transverse open crack in a fastening rod are investigated. The nonlinear dynamic model of a cracked rod-fastening rotor system is established based on the finite element method. A methodology is introduced where shooting method, path-following technique, and Floquet theory are combined for determining the periodic solutions and stability margins of the system. The effects of crack depth and mass eccentricities on the system are studied by numerical simulations. Results show the system stability will reduce due to the presence of crack, two saddlebacks occur on the periodic-doubling borderline whose bottom location corresponds to the two resonant peak of bearing node, and effects of the crack and mass eccentricity play a dominant position in different conditions respectively. Comparisons between the cracked rotor system and the intact ones referred in the literature indicate that some special characteristics of cracked rod-fastening rotor system in motion orbits and frequency components can be used to detect the presence of crack and its depth.

Finite Element Analysis and Experimental Validation of Transfer Function of Rotating Shaft System With Both an Open Crack and Anisotropic Support Stiffness

In this paper, transfer function of rotating shaft system for detecting transverse open crack is developed. Rotating shaft system is modeled using one-dimensional finite element method (1D-FEM), and quantitative analysis is performed. Open crack is modeled as weak asymmetry rotating with shaft's rotation. It is known that, when both open crack and support stiffness anisotropy coexist, various frequency components of shaft's vibration are generated through their successive interaction. This paper evaluates the order of these components, and concludes that first five main components are enough to investigate interaction of open crack and support stiffness anisotropy. Then, five sets of transfer functions for these components are derived. The validity of this set of transfer functions is confirmed by numerical simulation. Moreover, excitation experiment utilizing active magnetic bearing (AMB) is performed, and the validity of derived transfer function was verified experimentally.

Experimental modal analysis of curved composite beam with transverse open crack

Curved composite beams are the most widely preferred structures due to their high strength, low weight, corrosion and heat resistance. The aim of this study is to examine the modalities and natural frequencies of curved cracked composite (CCC) beams under different boundary conditions. Numerical and experimental free vibration analyses of curved composite beams with transverse cracks of various depths and locations are presented. Fixed-Fixed and Fixed-Free (cantilever) boundary conditions are used for both numerical and experimental studies. It is observed that the natural frequency values of the modes decrease as the crack depth increase depend on the crack position in the material.

A New Form of Frequency Equation for Functionally Graded Timoshenko Beams with Arbitrary Number of Open Transverse Cracks

The present paper addresses the problem of free vibration in multiple cracked beams made of functionally graded material (FGM). Governing equations of the beam vibration are established using the Timoshenko beam theory, power law of material grading along the beam thickness and rotational spring model of crack. The actual position of neutral plane differed from the mid-plane due to the material properties variation is also taken into account. The frequency equation obtained in a simple form provides an efficient tool for natural frequency analysis and may also be used for solving the inverse problems such as identification of material properties or cracks in the beams from natural frequencies. Numerical examples have been carried out to validate the proposed theory and to investigate the effect of cracks, material and geometry parameters on natural frequencies of the beams.

Study of Effect of Crack on Natural Frequencies of Euler Bernoulli Cantilever Beam

In this paper study of the effect of transverse open crack on natural frequencies of Euler Bernoulli cantilever beam has been done. Mathematical modeling has been done using flexibility approach. Flexibility at crack location is calculated using fracture mechanics approach. Natural frequencies for first six modes are obtained for different crack locations and crack sizes. It is observed that when crack size increases for a given crack location, the natural frequencies for the beam decrease. For a crack size with variable crack location, natural frequency has less effect near the node.