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Vibration of Cracked Structures: A State of the Art Review
Abstract
The presence of a crack in a structural member introduces a local flexibility that affects its vibration response. Moreover, the crack will open and close in time depending on the rotation and vibration amplitude. In this case the system is nonlinear. Furthermore, if general motion is considered, the local stiffness matrix description of the cracked section of the shaft leads to a coupled system, while for an uncracked shaft the system is decoupled. This means that the crack introduces new harmonics in the spectrum. In fact, in addition to the second harmonic of rotation and the subharmonic of the critical speed, two more families of harmonics are observed: 1.(1) higher harmonics of the rotating speed due to the nonlinearity of the closing crack, and2.(2) longitudinal and torsional harmonics are present in the start-up lateral vibration spectrum due to the coupling.Over 500 papers on the subject were published in the past 10 yrs. A wealth of analytical, numerical and experimental investigations now exists. However, a consistent cracked bar vibration theory is yet to be developed. There are still many unanswered questions, especially in the area of closing cracks in rotating shafts.
... It is assumed herein also that the membrane force vanishes. Substituting now (15) in (10) leads to the fourth-order equation ...
... The jump of the slope of the deflection of the nanoarch is coupled with the generalized stresses at the cross-section with defects. Following Dimarogonas [15], Anifantis and Dimarogonas [3] the quantities θ j will be treated as generalized displacements. In the linear elastic fracture mechanics, the generalized stresses P i and generalized displacements u i (i = 1, ..., 6) are coupled as ...
... The boundary conditions corresponding to the free edge are presented by (11) and (12). Making use of (15) and (35) one can check that the requirements at the free edge are satisfied if X n (β) + X n (β)(1 + A n ) = 0 and X n (β) + X n (β)(1 + A n ) = 0. ...
The free vibrations of elastic circular arches made of a nano-material are considered. A method of determination of eigenfrequencies of nanoarches weakened with stable cracks is developed making use of the concept of the massless spring and Eringen's nonlocal theory of elasticity. The aim of the paper is to evaluate the sensitivity of eigenfrequencies on the geometrical and physical parameters of the nanoarch.
... The effect of local cracks on the vibration characteristics is first studied by Kirmsher [4] in 1944. The theoretical background of the vibration characteristics of a cracked shaft is then developed by Dimarogonas [5] . Pafelias [6] with the knowledge of vibrational property of a cracked shaft and extensive experimental analysis, proposed the concept of harmonic generation in presence of a crack. ...
... Many reviews have been performed in the past on nonlinear system identification methods, analysis of vibration characteristics of different members [5] , application of nonlinear dynamics on SHM [39] , modeling of nonlinear crack-wave interaction [40] , etc. However, to the best of author's knowledge a complete review dedicated to the nonlinear breathing crack from modeling to different damage detection techniques is not taken up before. ...
... If an impulse force of amplitude is applied for a short time duration, the response of the linear system is expressed as Eq. (5) . ...
Swift development of technology for monitoring complex structures demands major attention on the precision of damage detection methods. The early detection of any type of deterioration or degradation of structures is of paramount importance to avoid sudden catastrophic failure. It warns users about the impending state of the system. At the initiation of a crack or some other system faults, the system may generate a time-varying state of crack under ambient vibration. It represents the nonlinear breathing phenomena of crack. An assessment of this degree of nonlinearity can be utilized for the detection, localization, and quantification of breathing cracks. Appropriate modeling of such cracks is thus necessary to capture distinctive nonlinear features. Recognizing this importance, various methods of modeling and nonlinear system identification which have been employed in the past for the detection of breathing crack are reviewed. The present study also explores some of the available vibration as well as acoustic-based damage identification techniques, chronologically connecting their evolutions. It summarizes the advantages and limitations of the methods to inspect potential future applications. The future scopes drawn from this review are highlighted to pave the path of wide-spread applications of nonlinear features of crack.
... It is assumed herein also that the membrane force vanishes. Substituting now (15) in (10) leads to the fourth-order equation ...
... The jump of the slope of the deflection of the nano-arch is coupled with the generalized stresses at the cross-section with defects. Following Dimarogonas [15], Anifantis and Dimarogonas [3] the quantities θ j will be treated as generalized displacements. In the linear elastic fracture mechanics, the generalized stresses P i and generalized displacements u i (i = 1, ..., 6) are coupled as ...
... The boundary conditions corresponding to the free edge are presented by (11) and (12). Making use of (15) and (37) ...
The free vibrations of elastic circular arches made of a nano-material are considered. A method of determination of eigenfrequencies of nano-arches weakened with stable cracks is developed making use of the concept of the massless spring and Eringen's nonlocal theory of elasticity. The aim of the paper is to evaluate the sensitivity of eigenfrequencies on the geometrical and physical parameters of the nano-arch.
... Many studies on vibration analysis of cracked beams, columns, and frame structures are in the literature [1][2][3][4][5][6]. Du et al. [7] used the transfer matrix method for the free vibration analysis of an axially loaded variable cross-section Euler-Bernoulli beam with many fxed members. ...
... In the presence of a crack width b, local fexibility [2] is defned as ...
In this study, the effects of cracks in the columns of a single degree of freedom (SDOF) steel structure model on the system responses due to the decrease in the equivalent spring coefficient were investigated. The crack in the columns was modeled as an edge crack, and the crack was thought to behave like a spring. The crack is considered as the mode I crack type (opening mode) specified in linear fracture mechanics. In order to examine the responses of the system under the influence of different excitations, the changes in displacement, acceleration, and power spectral density responses under the influence of harmonic excitation at a frequency equal to the fundamental frequency of the SDOF model and six different earthquake excitations were examined.
... To include the cracks in the proposed model, it is necessary to decompose the pillar into n þ 1 sections at the damaged cross-sections. The sections, numbered from bottom to top, are assumed to be connected by rotational and translational springs, as shown in Figure 1 (Dimarogonas, 1996;Ferna´ndez-Sa´ez and Navarro, 2002;Freund and Herrmann, 1976;Liu et al., 2008;Loya et al., 2009;Sayyad et al., 2013;Sayyad and Kumar, 2012;Yokoyama and Chen, 1998). As the pillar undergoes buckling, it is reasonable to assume that the configuration resulting from compressive forces causes the edge cracks within the pillar to open up. ...
... ., n: Figure 1. A miniaturized pillar with n edge cracks supported by an elastic foundation under the action of the compressive load P. Rotational and translational springs are used to model the influence of cracks (Dimarogonas, 1996;Fernández-Sáez and Navarro, 2002;Freund and Herrmann, 1976;Liu et al., 2008;Loya et al., 2009;Sayyad and Kumar, 2012;Sayyad et al., 2013;Yokoyama and Chen, 1998). ...
The buckling instability of micro- and nanopillars can be an issue when designing intelligent miniaturized devices and characterizing composite materials reinforced with small-scale beam-like particles. Analytical modeling of the buckling of miniaturized pillars is especially important due to the difficulties in conducting experiments. Here, a well-posed stress-driven nonlocal model is developed, which allows the calculation of the critical loads and buckling configurations of the miniaturized pillars on an elastic foundation and with arbitrary numbers of edge cracks. The discontinuities in bending slopes and deflection at the damaged cross-sections due to the edge cracks are captured through the incorporation of both rotational and translational springs. A comprehensive analysis is conducted to investigate the instability of pillars containing a range of one to four cracks. This analysis reveals interesting effects regarding the influence of crack location, nonlocality, and elastic foundation on the initial and subsequent critical loads and associated buckling configurations. The main findings are: (i) the shielding and amplification effects related to a system of cracks become more significant as the dimensions of pillars reduce, (ii) the influence of the shear force at the damaged cross-section related to the translational spring must not be neglected when dealing with higher modes of buckling and long cracks, (iii) an elastic foundation decreases the effects of the cracks and size dependency on the buckling loads, and (iv) the effects of the edge cracks on the critical loads and buckling configurations of the miniaturized pillars are highly dependent on the boundary conditions.
... The presence of a crack in beam-like structures leads to changes in modal parameters such as natural frequencies, modal damping, and mode shapes [3][4][5][6][7][8][9] . A good amount of literature survey is presented on crack localization techniques [10][11][12] . The major challenge in the crack localization is to get the high spatial resolution of beam deflection measurement. ...
The structures under fatigue loading are fault prone. The damage reduces the local stiffness. This local stiffness leads to a slope discontinuity in the structure's elastic line. Localizing the local discontinuity reveals the location of the damage. Wavelet transform is a powerful tool to localize a local slope discontinuity in a signal. The major challenges in the localization of damage in a beam are obtaining the high spatial resolution beam deflection and eliminating the border distortion. The high spatial resolution shrinks the border distortion as well as gives more localized crack detection. The reduced border distortion leads to the detection of cracks very close to the ends of the beam. In the present work, finite element analysis is used for getting the simulated beam deflection. The lifting wavelet is used for the localization of cracks in the beam. The lifting wavelet has certain advantages over the classical wavelet. The lifting wavelet possesses perfect reconstruction and a narrower border distortion zone. A comparative study is presented between the discrete wavelet transform and the lifting wavelet transform for localizing the crack. The ability of lifting wavelet is tested for different noise conditions and multiple crack localization. A photographic method is used to get the high-resolution of experimental beam deflection of stainless-steel material.
... Following the existing literature evidences, a Finite Element (FE) numerical study is thus presented in this paper to assess the convenience of non-destructive dynamic techniques for the mechanical characterization and for the early damage / deterioration detection of EVA bonding interlayers for PV modules (Fig. 2). The methodology takes advantage of classical structural health monitoring (SHM) procedures for civil structures and machineries (Pander et al. 2011, Limongelli et al. 2021, Dimarogonas 1996, where the on-site experimental analysis in the frequency domain is usually carried out to detect and characterize the vibration modes of the system object of study, and thus to assess the effect of single components and possible influencing parameters. The approach follows earlier applications to laminated glass systems and composite elements, where in-field and/or historic structures have been efficiently characterized under various loading and boundary conditions, including walkways in unfavourable environment and operational configurations (Bedon 2019a), facades (Bedon et al. 2022), fractured pedestrian systems (Bedon et al. 2021) and beams with delamination (Bedon 2019b). ...
In engineering applications, the frequency analysis represents a first and practical step to collect relevant parameters for structural and mechanical diagnostics. Any possible material / component degradation and deterioration can be prematurely detected by frequency modifications that exceed a certain alert value. In this paper, the attention is given to the dynamic mechanical analysis of commercial photovoltaic (PV) modules, in which the solar cells are typically encapsulated in thin viscoelastic interlayers made of Ethylene-Vinyl Acetate (EVA), which are primarily responsible for the load-bearing capacity of the sandwich PV system. As a major effect of ageing, ambient conditions, non-uniform / cyclic thermal gradients, humidity and even extreme mechanical / thermal loads, the rigidity of these films can largely modify and decrease, thus possibly affecting the mechanical capacity of the PV module, and even exposing the solar cells to fault. Knowledge of the effective bonding level is an important step for diagnostic purposes. In this regard, the present study is based on a preliminary but extensive parametric Finite Element (FE) numerical investigation of full-scale commercial PV modules of typical use in buildings. The attention is given – for PV module arrangements of technical interest – to the effect of EVA stiffness in terms of vibration modes and especially frequency sensitivity. As shown, when compared to newly installed PV modules, any kind of stiffness decrease is associated to major frequency modifications for the composite system, and in the worst configuration, such a frequency scatter can decrease down to -40% the original condition. Such a marked stiffness decrease would be implicitly associated to a weak mechanical performance of the sandwich section, with major stress peaks and deflections in the PV system, even under ordinary loads. The presented results, in this sense, suggest that major consequences can be prevented and minimized by monitoring the vibration frequency of PV modules.
... Following the above considerations and the existing literature evidences, an experimental and Finite Element (FE) numerical study is thus presented in this paper to assess the convenience of non-destructive dynamic techniques for the mechanical characterization and for the early damage / deterioration detection of EVA bonding interlayers for PV modules (Fig. 3). The methodology takes advantage of classical structural health monitoring (SHM) procedures for civil structures and machineries [36][37][38], where the on-site experimental analysis in the frequency domain is carried out to detect and characterize the vibration modes of the system object of study, and thus to assess the effect of single components and possible influencing parameters. The approach follows earlier applications to laminated glass systems and composite elements, where in-field and/or historic structures have been efficiently characterized under various loading and boundary conditions, including walkways in unfavourable environment and operational configurations [39], historic facades [40], fractured pedestrian systems [41], structural beams with delamination [42], and even film-retrofitted glass members investigated under severe post-fracture configurations [43,44]. ...
In engineering applications, the frequency analysis represents a first and practical step to collect relevant parameters for structural and mechanical diagnostics. Any possible material / component degradation and deterioration can be prematurely detected by frequency modifications that exceed a certain alert value. In this paper, the attention is given to the dynamic mechanical analysis of commercial photovoltaic (PV) modules, in which the solar cells are typically encapsulated in thin viscoelastic interlayers made of Ethylene-Vinyl Acetate (EVA), which are primarily responsible for the load-bearing capacity of the sandwich PV system. As a major effect of ageing, ambient conditions, non-uniform / cyclic thermal gradients, humidity and even extreme mechanical / thermal loads, the rigidity of these films can largely modify and decrease, thus possibly affecting the mechanical capacity of the PV module, and even exposing the solar cells to fault. Knowledge of the effective bonding level is an important step for diagnostic purposes. In this regard, the present study is based on a preliminary non-destructive experimental analysis, and on an extensive parametric Finite Element (FE) numerical investigation of full-scale commercial PV modules of typical use in buildings. The attention is given-for PV module arrangements of technical interest-to the effect of EVA stiffness in terms of vibration modes and frequency sensitivity. As shown, especially compared to newly installed PV modules, any kind of stiffness decrease is associated to major frequency modifications for the composite system, and in the worst configuration, such a frequency scatter can decrease down to − 40% the original condition. Such a marked stiffness decrease would be implicitly associated to a weak mechanical performance of the sandwich section, with major stress peaks and deflections in the PV system, even under ordinary loads. The presented results, in this sense, suggest that major consequences can be prevented and minimized by monitoring the vibration frequency of PV modules.
... Аналіз останніх досліджень та публікацій. Аналіз вібрації валів парових турбін зосереджується головним чином на поперечних вібраціях, особливо при проходженні критичних швидкостей обертання валів [8,9]. Проте, як було показано в багатьох дослідженнях, наприклад [10-12], крутильні коливання валопроводів турбін можуть бути достатньо інтенсивними, щоб викликати їхнє втомне пошкодження і, як наслідок, катастрофічне руйнування. ...
During operation, a steam turbine shaft is subjected to a wide range of thermomechanical and thermochemical loading. Long-term operation of steam turbines has one of its consequences the formation of fatigue cracks of various types in their structural elements. Evidence of this is a number of accidents and catastrophic destruction of steam turbines. Potential reasons of damage in turbine shafts are all technological operations used in the process of manufacture (forging, turning, and milling, heat treatment), since they are accompanied with plastic deformation of material. It accumulates during long-term cyclic deformation and turns into local damage of a fatigue crack type. In addition, cracking in turbine shafts is caused by the presence of stress concentrators. One of the reasons for the long-term accumulation of fatigue damage in steam turbines elements is torsional vibrations of the shafting caused by many reasons. Among these reasons are a short circuit on the turbogenerator, the connection of the turbogenerator to an electrical network with inaccurate synchronization, as well as the dynamic instability of the system turbogenerator-electrical network. A short circuit in a turbine generator is a rare phenomenon that under certain conditions can cause serious damage or even complete destruction of the turbine but does not cause the accumulation of fatigue damage in the material. At the same time, the source of multiple excitations of torsional vibrations of turbine shafting, which operates throughout their service life and can cause fatigue damage the material, is the connection of the turbogenerator to an electrical network. A methodology is proposed to study the process of circumferential crack growth in the turbine shafting because of many such connections. The methodology is based on the use of a 3D finite element model of the steam turbine K-200-130 shafting and a fracture mechanics approach. There was demonstrated that the inaccuracy of turbogenerator connection to an electric network is the most influential factor affecting the crack growth and the conditions for the crack to reach a critical size has been evaluated. In the problem being considered damping capacity of the rotor steel influences the intensity of crack growth insignificantly.
... Some of these characteristics have been tested experimentally as well. The effectiveness of each method is proportional to the crack depth (Dimarogonas, 1996;Sabnavis, Kirk, Kasarda & Quinn, 2004;Sinou & Lees, 2007;Sinou, 2008;Machorro-López, Adams, Gómez-Mancilla & Gul, 2009;AL-Shudeifat, Butcher & Stern, 2010;Varney & Green, 2012). Sinou & Lees (2007) addressed the influence of crack opening and closing on dynamic response during operation. ...
Shafts are often subjected to difficult operating conditions in high-performance rotating equipment such as compressors, steam and gas turbines, generators and pumps. As a result, shafts are susceptible to fatigue failures due to transverse cracks. In this study, vibration monitoring and orbital paths observation were used to detect the presence of a flaw in a shaft. Two types of flaws were tested: a straight slot, and a fatigue crack. For both flaw types, specimens of different depths were examined in order to assess the detection capability. A new approach to examine vibrations at the critical speed is proposed; this speed is chosen because of the strong connection to the basics of the physical problem. Orbital paths are suggested as means for fault detection as well. The presence of a straight slot in the shaft was found to be related to a decrease in the natural frequency and to a decrease in amplitude of the first order at critical speed. For the fatigue crack, a consistent trend in critical speed and in amplitude was not seen as crack depth grew. A new method to detect the change in the shaft natural frequency is proposed. The combination of two indicators, change in critical speed and change in amplitude at critical speed, are suggested for classification of flaw size. For the straight slot case, the method proposed was able to distinguish between different fault depths.
... Peyman [9] outlined a comprehensive list of approaches to anticipate the alteration of dynamic properties as a function of the location and size of the crack. Dimarogoneas [10] reviewed analytical, numerical and experimental studies on the recognition of cracks based on dynamic feature variations. Van [11] developed a hybrid beam element based on first-order deformation theory for the analysis of static bending behavior in FG sandwich beams. ...
This article examines the impact of inclined transverse cracking on the natural frequency of Euler-Bernoulli-model imperfect FG beams on an elastic foundation, taking into account the pinned-
pinned and clamped-clamped boundary conditions. The equations are diffused using the classic finite element method (h-version). Material properties are considered to vary in both directions of the beam; width and thickness, using the power-law formula. The approximate porosity model is adopted with a uniform distribution. Cracked element stiffness is determined by reducing the cross-section of a bi-directional FG beam. The numerical results are compared with those of previous convergence studies for dimensionless fundamental frequencies. Case studies were carried out to evaluate the effect of the porosity values, the crack depth, the crack location, slope angle, and the Winkler-elastic foundation parameters on the first three natural frequencies of a beam with different support conditions and material gradient. In addition, the results demonstrated that the two-way distribution function plays an important role in the convergence of frequencies.
... To this end, early detection of cracks can help the maintenance process and increase safety in different mechanical applications. A review of using vibration-based damage detection techniques has been presented by [5] [6]. While the crack localization and quantification using the modal characteristics methods have been reviewed by [7] [8]. ...
... The aforementioned FG-GPLRC structures are often operating in complex environments, including the subjection to different dynamic loadings in various engineering applications, and it is challenging to completely avoid the occurrence of structural damage. The presence of cracks in an engineering structure may significantly reduce the local stiffness and strength of the structure and affect structural performance accordingly [31,32]. Several studies have been devoted to the structural behavior analysis of cracked FG-GPLRC beam. ...
In this paper, we investigate the damped nonlinear vibration of cracked functionally graded (FG) graphene platelets (GPLs)-reinforced composite (FG-GPLRC) dielectric beam. The effective material properties of the composites are evaluated by effective medium theory (EMT) and rule of mixture. Governing equations incorporating damping and dielectric properties are derived from an energy method with the framework of Timoshenko beam theory and nonlinear von Kármán strain–displacement relationship. Stress intensity factor (SIF) of cracked FG-GPLRC beam at the crack tip is obtained via finite element method (FEM). Differential quadrature (DQ) and direct iterated methods are utilized to discretize and solve the nonlinear system. Accuracy and convergence of the model and the solution are verified. An extensive numerical study is performed to examine the effects of crack location and depth, damping and attributes of GPL and the applied electric field on the nonlinear vibration behavior of the cracked FG-GPLRC beam. It is found that the frequency ratios of cracked FG-GPLRC beams are more sensitive to the applied electric field when the crack with larger depth is located close to the mid-span. The cracked FG-GPLRC beams with FG distribution profiles exhibit better stability.
... The dynamic behavior of the spinning beam-disk and the characteristics of different nonlinear behaviors have been extensively investigated using different models and methods in order to optimize the structural system and avoid malfunctions [5][6][7][8]. Paez Chavez et al. [9] modeled the Jeffcott rotor containing bearing clearances and analyzed the bifurcation phenomena and trajectory paths of the rotor system. Mamandi et al. [10] derived the nonlinear control coupled differential equations of motion for beams under bending rotation, longitudinal and transverse displacement of the curved section using Hamilton's principle. ...
In this study, an attempt is made to model and investigate the dynamic behavior of the spinning Timoshenko beam-disk with nonlinear elastic boundaries, in which an unbalanced concentrated mass and axial loads are considered. In order to satisfy the elastic boundary conditions of the spinning beam-disk containing translational and rotational stiffnesses as well as nonlinear stiffnesses, an improved version of Fourier series is employed for the admission function construction. Nonlinear dynamic behavior of the spinning beam-disk and its boundary supporting system are described based on energy principle, and the system governing equations of spinning beam-disk are formulated by the Lagrange equation of the second type. The time-domain response is then obtained by solving the system dynamic equations through Runge–Kutta method, while the reliability of the current model is verified through the comparison with those predicted by harmonic balance method. Then, the effect of sweep direction and nonlinear elastic boundary parameters on system dynamic behavior of the spinning Timoshenko beam-disk is investigated and addressed. The results show that the dynamic responses of the spinning beam-disk with nonlinear elastic boundary are sensitive to the initial values of calculation, and the nonlinear elastic boundary parameters make the spinning beam-disk exhibit complex dynamic behavior. Analysis of Poincare points in the phase diagram can better determine the dynamic behavior of spinning beam-disk, and a set of suitable nonlinear elastic boundary parameters can suppress the complex dynamic response of the spinning beam-disk.
... simulation of actual stator and rotor blades. characterize the open crack model [2]. ...
As one of the most important parts in the engine, the structure and state of the rotating blade directly affect the normal performance of the aero engine. In order to monitor engine crack failure and ensure flight safety, it is necessary to carry out research on the dynamic modeling of the cracked blade and breathing crack induced vibration mechanisms. This paper summarizes the current research status on the dynamics of cracked blade and the related topics mainly include four aspects: crack propagation path, mechanical model of open and breathing cracks, dynamic modeling methods of cracked blades such as lumped mass model, semi-analytical model and finite element model, and dynamic characteristics of cracked blades. The review will provide valuable references for future studies on dynamics and fault diagnosis of cracked blade in aero engine.
... The Finite Element Method (FEM) is the most widely used method for solving engineering and computational model problems. The FEM is a numerical method for solving partial differential equations with two or three variables in space [11]. FEA based simulations are valuable tools because they avoid the need for multiple physical prototype creation and testing for different high fidelity situations [12]. ...
Complex structures can develop cracks and defects over time, which can compromise their long-term performance and safety. Structural Health Monitoring (SHM) systems are essential for detecting and measuring these defects by monitoring the load and deformation of the solid materials. This paper presents a simulation study of the frequency and strength of solid cylindrical bars made of aluminum and steel under different loads and crack conditions. Finite Element Method (FEM) and COMSOL Multiphysics software are used to perform the simulation, and a resonance model is used to analyze the results. The study investigates how cracks affect the frequency and deformation of the bars, and how different materials respond to load and bending. The results show that frequency varies linearly with load, cracks decrease the stiffness and increase the frequency at the crack location, and aluminum bars deform more than steel bars. The paper concludes that steel bars are more resistant to load and bending than aluminum bars for both cracked and uncracked case. Finally, it is found that steel bars are more resistant to load and bending than aluminum bars for both cracked and uncracked case.
... The load (forces and moments) direction based on a transverse crack representation is shown in Fig. 3. The additional flexibility matrix because of the crack, on the basis of the fracture mechanics approach, is generally used as a crack model [22,25]. The time-dependent additive crack has the following form ...
... Indeed, the literature in this field covers a wide range of applications, ranging from mechanical to aerospace and civil structures. Applications to rotating machinery, roller bearings, large-span bridges, aqueducts and monuments are present [4][5][6][7]. ...
This paper presents an approach to damage identification in beams by modal curvatures based on the use of beamforming algorithms. These processors have been successfully used in acoustics for the last thirty years to solve the inverse problems encountered in source recognition and image reconstruction, based on ultrasonic waves. In addition, beamformers apply to a broader range of problems in which the forward solutions are computable and measurable. This especially concerns the field of structural vibrations, where the use of such estimators has not received attention to date. In this paper, modal curvatures will play the role of replica vectors of the imaging field. The choice to use modal curvatures is motivated by means of numerical studies and experimental tests on a steel beam. Furthermore, we compare the performance of the Bartlett and minimum variance distortionless response (MVDR) beamformers with an estimator based on the simple minimization of the difference between model and measured data. The results suggest that the application of the MVDR beamformer is highly effective, especially in cases of slight damage between two sensors. MVDR enables both damage localization and quantification.
... Because energy methods are becoming important in the dynamic analysis of structures, good estimates of energy in 3D frame structures become a necessity. One application of the structural potential energy is to use it in Structural Health Monitoring techniques (SHM), as methods of nondestructive testing and monitoring [7,9,24], well suited for relatively large-scale damages. The so-called "vibration-based structural damage identification" methods are widely used [18]. ...
In contrast to the kinetic energy, which can be easily estimated using measurements of the velocities at certain points in a frame structure, the structural potential energy cannot be easily estimated because it involves the spatial derivative of the displacement vector. This difficulty can be eased by using spectral formulations based on the spectral finite element method (SEM). The main benefit of SEM is related to its dynamic stiffness matrix, which is calculated from the analytical solution directly in the frequency domain. Based on these types of solutions, mathematical expressions tailored for bar, beam, and shaft elements are presented for the numerical estimation of potential energy in 3D frame structures. Specific expressions are given for Bernoulli–Euler and Timoshenko beams. Numerical simulations are performed to validate the proposed expressions, and then they are benchmarked against finite element results. The accuracy of these novel expressions is demonstrated and satisfactory results are obtained.
... Indeed, literature in this field covers a wide range of applications ranging from mechanical to aerospace and civil structures. Applications to rotating machinery, roller bearings, large-span bridges, aqueducts and monuments are present [2][3][4][5]. ...
This paper presents an approach to damage identification in beams by modal curvatures based on the use of beamforming algorithms. These processors have been successfully used in acoustics for the last thirty years to solve the inverse problems encountered in source recognition and image reconstruction, based on ultrasonic waves. In addition, beamformers apply to a broader range of problems in which the forward solutions are computable and measurable, especially regarding the field of structural vibrations, where the use of such estimators has not received attention to date. In this paper, modal curvatures will play the role of the replica vectors of the imaging field. By means of numerical studies and experimental tests on a steel beam, we motivate the choice of modal curvatures as observed quantities. Furthermore, we compare the performance of the Bartlett and minimum variance distortionless beamformers (MVDR) with an estimator based on the simple minimization of the difference between model and measured data. The results suggest that the application of the MVDR beamformer is highly effective, especially in cases of slight damage between two sensors. MVDR enabled both damage localization, and quantification.
... The above-mentioned FG-GNPRC structures often operate in complex environments and bear diverse dynamic loads in a variety of engineering applications and it is challenging to completely avoid the occurrence of structural damage. The presence of cracks in an engineering structure may significantly reduce the local stiffness and strength of the structure and significantly affect the performance of the structure [22]. Several works have been devoted to the analysis of structural behavior of cracked FG-GNPRC beam. ...
Functionally graded (FG) graphene nanoplatelets (GNPs) reinforced composite (FG-GNPRC) is widely used in various engineering fields due to its high-performance and multifunctional features. Cracks generated in complex environments have a significant impact on the structural behavior of FG-GNPRC structures. This paper studies the damped nonlinear dynamics of cracked FG-GNPRC dielectric beam subjected to mechanical excitation and electrical field. The effective material properties of the composites are determined by the effective medium theory (EMT). Based on Timoshenko beam theory and nonlinear von Kármán strain-displacement relationship, the governing equations which incorporate damping and dielectric properties are derived by using energy method. Stress intensity factor (SIF) of cracked FG-GNPRC beam at the crack tip is calculated via finite element method. Differential quadrature (DQ) and incremental harmonic balance (IHB) methods combined with arc-length algorithm are utilized to discretize and solve the nonlinear system. After validation of the model and solution, the effects of crack depth and location, damping, FG distribution and the attributes of GNP and the electric field on the dynamic response of the system are comprehensively investigated. The study indicates that when an electric field is applied, the amplitude ratio of cracked FG-GNPRC beam with profile V is the largest when the crack is close to the edge, whereas profile A exhibits the largest value when the crack is near the mid-span. In addition, the generated bending moment enables profiles A and V to transit into profile U at a certain voltage or GNP aspect ratio due to the existence of electrostatic stress.
... As vibrations are nonetheless a propagation of energy in the structure, thus techniques that involve energy calculation are adopted to investigate damage occurrence. Dimarogonas [1] has given a review on the vibration of cracked structures. Damage detection that is based on change of natural frequencies may be used as an indicator of damage but hardly gives information on its location [2]. ...
The dynamic response of a straight beam is used to identify the occurrence of damage. Apparently there is an advantage in using natural frequencies changes as an indicator for damage occurrence, but some trade-offs exist. The first part of the paper is showing these trade-offs by studying a model of an intact straight rectangular cross-section beam built in ANSYS, and identifying its modal parameters. Then a transverse crack is assumed to exist across the beam length and the modal analysis is repeated. Three scenarios of crack sizes are analyzed. These have showed a change in natural frequencies together with new mode shapes arising. Moreover, there is a relation between crack location and the dynamic modal response of the beam, mainly mode shapes. Although the analysis based only on natural frequencies is easier to conduct and the occurrence of crack is clearly observed, the trade-off is that the location of the crack is not identified. In the second part of the paper another more effective crack detection methodology is presented. It is based on calculating the strain energy of each of the finite elements used in the model. This methodology is based on collecting vibrational data of the healthy structure, i.e., before any cracks exist, then the same data are collected when cracks have supposedly occurred. The two sets of data are used to calculate ratios of modal strain energies before and after crack occurrence. These ratios are then used to locate the cracked element. The methodology is implemented in a Matlab code, where a 2-D beam finite element formulations, with two DOF per node, are used. Various degrees of damage Journal of Engineering Research (University of Tripoli) Issue (34) September 2022 46 scenarios are assumed to investigate the validity of the methodology, by observing the change in modal strain energy. The obtained results show the effectiveness of the proposed methodology in detecting crack occurrence, its location, as well as providing information on its relative severity.
... Metal beams are used in many areas such as machinery industry, shipbuilding, automotive industry. Since the crack structure and depth have a high effect on the physical properties of the material, many studies have been carried out in the literature(Binici, 2005;Chondros et al., 1998;Dimarogonas, 1996;Gounaris et al., 1996;Shifrin and Ruotolo, 1999;Zheng and Fan, 2001). The modal behavior of beams with crack structure was investigated and finite element method was used to calculate the frequency values ...
Breast cancer is a common and life-threatening disease that affects every aspect of women's lives worldwide. Early detection of breast abnormalities plays a critical role in improving patient outcomes. Deep learning techniques can be considered as promising approaches for automatic breast cancer detection using mammographic images based on the success rates of previous studies. This study aims to evaluate the performance of different deep learning architectures on the MIAS dataset for breast cancer detection. The architectures considered include VGGNet, DenseNet, ResNet, and MobileNet. Performance metrics such as accuracy, sensitivity, specificity, and precision are utilized to assess the effectiveness of each architecture. The results demonstrate the superior performance of VGGNet and DenseNet, achieving accuracy rates of 92.5% and 91.8%, respectively. ResNet and MobileNet also exhibit substantial potential with accuracies of 89.3% and 88.6%, respectively. These findings contribute to the growing body of knowledge in medical image analysis and highlight the effectiveness of deep learning architectures in breast cancer detection. Further investigations can focus on optimizing these architectures and exploring ensemble methods to enhance their performance.
... Cracks and other defects cause additional structural compliance, which can be calculated by using the method developed in [6] and [3] and has been applied by many authors in the study of vibration problems [12,14,15]. ...
The current study presents the natural vibrations of single-walled carbon nanotubes (SWCNTs) using the nonlocal theory of elasticity. The study will help to develop nanodevices. The study focuses on the natural vibrations of curved or arch-like carbon nanotubes (CNTs). It is considered that the CNT has a crack-like defect and is simply supported (SS) at both ends. To model the equations of motion for the CNTs, a curved nanobeam approach is adopted, and the governing equations are solved using the separation of variables method. This research investigates the natural frequencies of three types of CNTs: armchair, zigzag, and chiral. A qualitative validation study demonstrates that the obtained results align with those published in the literature. Notably, the study reveals that small-scale effects and defects in the nanotubes influence the natural frequencies of CNTs.
... Chondros and Dimarogonas [5] provided a detailed overview of approaches to predict the change in dynamic characteristics as function of crack location and size. Dimarogonas [6] reviewed analytical, numerical, and experimental studies on crack recognition based on dynamic feature changes. Doebling et al [7] reviewed methods for detecting, locating, and estimating the severity of damage in structural systems by examining changes in measured vibration responses. ...
In this article, we analyze the effect of transverse cracks on the natural frequencies of a Euler-Bernoulli functional gradient beam. The studied beam was discretized into finite elements and the global matrices of the motion equation are determined by applying the Lagrange equation to the beam kinetic and deformation energies. The material properties are considered to vary in the directions of the beam thickness, the gradation is described by the power-law distribution, the stiffness of the cracked element is determined based on the reduction of the beam cross-section. The numerical results obtained are compared with those available in the previous study. Finally, case studies were carried out to analyses the influence of the power law index, the depth and the opposition of the crack on the natural frequencies of the beam for different boundary conditions; these studies demonstrate the advantage of the FGM beam over the purely metal beam.
... To represent the degree of damage, Young's modulus of concrete is used as the parameter to be updated. If the damage type is cracking, the cracked structures are known to show nonlinear vibration [8]. However, in some cases when the nonlinearity is not severe, cracks are also modeled simply as the stiffness reduction. ...
In this paper, we present a damage identification method for small damages based on topology optimization and Lasso regularization. In particular, this work extends the applicability of the previously developed damage identification method using frequency response functions and topology optimization, by conducting rigorous parametric studies in terms of damping, measurement noise, and damage size. It is shown that the presented method successfully identifies small damaged regions with a reasonable accuracy. To evaluate the effectiveness of the proposed method, we applied the method to identify the damages in cantilevered plates that are subject to static or dynamic loads. The method succeeded in detecting the locations and shapes of damages more accurately than the method without Lasso regularization. Furthermore, in most cases we have considered, spurious damages generated during the optimization were successfully suppressed.
... Since vibration signals can effectively reflect the operating conditions of bearing, they have been widely applied to the bearing fault diagnosis [2]. When the vibration signals are acquired in the start-stop Manuscript process of rotating machinery, they are nonstationary, but they contain richer fault features than those acquired under the stable condition [3]. Unfortunately, the traditional signal processing methods based on the fast Fourier transform are not suitable for analyzing nonstationary signals. ...
Tacholess order tracking is a commonly-used technique for the fault diagnosis of bearing with time-varying speed, in which time-frequency analysis (TFA) is a key step for estimating the rotation frequency. The current TFA methods are either weak in energy concentration or have high computational complexity. To this end, an adaptive fast chirplet transform (AFCT) based on the adaptive optimal search angle band is proposed in this paper. The proposed method uses the modulation operator of synchrosqueezed transform to adaptively optimize the search band of the frequency modulation parameters, which overcomes the problem of low computational efficiency in chirplet transform (CT) and its variants. Moreover, a novel time-frequency energy concentration evaluation index mean-energy-to-peak-ration (MEPR) is proposed. The simulation result shows that the proposed method has both good time-frequency concentration performance and low computational complexity. Especially, its calculation speed is much faster than other TFA methods with high time-frequency resolution. The proposed method is successfully applied to estimate the rotation frequencies from the fault vibration signals from a test rig and a civil aircraft engine. The comparative results verify the comprehensive advantage of the proposed method in ridge extraction accuracy and computational efficiency compared to the existing classical TFA methods. With the proposed method, the fault features of rolling bearings under time-varying speed can be effectively extracted.
The paper considers the problem of determining the natural frequencies and eigenwaves of transverse vibrations for the Bernoulli–Euler equation with variable coefficients. Such problems arise both in the case of complex geometry of a vibrating solid and in the case of functionally graded materials or the accumulation of damage in a classical elastic material. Solutions of boundary value problems are constructed using the expansion in Peano series. Under broad assumptions, the uniform convergence of Peano series is shown and estimates of the residual terms are obtained. Examples of the numerical implementation of the proposed procedure are given for bending vibrations of a rod with certain parameters of a variable cross section (geometric heterogeneity) and elastic modulus distribution (physical heterogeneity). Numerical examples are focused on assessing the geometric and elastic properties of samples in an experimental study of the fatigue strength of alloys during high-frequency cyclic tests based on the general principle of point resonant loading. The method proposed for solving problems of resonant vibrations for the Bernoulli–Euler equation can be used in the design of new promising cyclic test schemes and mathematical modeling of fatigue failure processes under high-frequency resonant vibrations.
The current research study utilized fuzzy logic to detect transverse cracks in industry-driven woven glass fiber/epoxy composite beam. Laminated composites, with their increasing prevalence in engineering applications, demand reliable methods for detecting structural defects such as cracks. Traditional crack detection approaches often face challenges to accommodate the inherent complexities of composite materials. The motivation behind employing fuzzy logic for crack detection in cantilever-laminated composite beams using frequency response stems from the imperative need for robust and adaptive structural health monitoring techniques. Considering a cantilever boundary condition for the glass/epoxy beam, the free vibration frequencies were numerically evaluated on ABAQUS platform. The frequency data were simulated as input data in a Mamdani fuzzy inference system (FIS) developed in MATLAB, using various standalone fuzzy sets (Triangular, Trapezoidal, Gaussian and Gbell). The output of the FIS scale results to determine the crack severity in terms of location and depth. Using a fast Fourier transform analyzer, experimental validation is conducted for cantilever composite beams with controlled damage scenarios to assess the effectiveness and accuracy of the developed fuzzy logic-based crack detection methodology. The results showcase the system’s potential to provide crack detection in composite structures, contributing to the advancement of structural health monitoring techniques. The integration of fuzzy logic in crack detection presents a promising avenue for enhancing the reliability of damage assessment in composite materials, offering valuable insights for the broader field of structural health monitoring.
Vibrations of beams containing cracks are usually investigated by using open or breathing crack models. Neither one of these approaches considers the plasticity induced crack closure (PICC) phenomena which occurs in fatigue crack growth. In this study, PICC is integrated into the vibration analysis of a cracked beam. First, fatigue crack growth at a notch is simulated with elastoplastic finite element analysis. Residual deformations, stresses and the contact pressures on crack faces are obtained. In the second phase, the model which includes these quantities is used for forced vibration analysis under harmonic loading by using implicit finite element method. For comparison, analyses are also running without PICC. Simulation results indicate that PICC has a significant effect on vibrations of cracked beams. For low load amplitude levels, cracked beam behaves like an intact beam. Nonlinear behavior appears only at rather high amplitude levels. It is concluded that PICC could suppress the crack breathing behavior and this could make it difficult to detect a fatigue crack in a beam from its nonlinear vibration response.
The present work aims to investigate the fatigue performance, by detecting changes in the first-order natural frequency, of single-lap joint specimens subjected to dynamic loadings. A particular attention is given to the evaluation of the effect of tightening torque on fatigue crack initiation life and fatigue crack initiation position. The results show that the joint with the tightening torque of 10 Nm has the highest fatigue life among all investigated joints. The analyses further reveal that this is mainly attributed to the prolonged crack initiation life resulted from the suppressed crack initiation at the joint contacting surface due to the high fretting fatigue resistant produced by the appropriate level of tightening torque.
The existence of a crack in a shaft causes a local slope discontinuity in the shaft elastic line. The crack in a shaft can be located by detecting this local slope discontinuity in the shaft elastic line. Wavelet transform is a powerful technique for extracting the local information from the signal. One drawback of discrete wavelet transform (DWT) is that it analyzes the signal at the coarser level). Wavelet packet transform (WPT) overcomes this difficulty by analyzing the signal at finer level. In DWT, only the output low-pass filter is taken for further decomposition. However, in WPT, the output of low passes, as well as high passes filters, are taken for further level of decomposition. In this way, the WPT performs a complete decomposition at each level and hence can reach a higher resolution in the high-frequency zone. In the present work, WPT is used to locate the crack along the shaft length. Another critical issue for wavelet-based crack detection is spatial measurement resolution. In the present work, the photographic image-based technique is used to obtain the high spatial measurement resolution of shaft deflection. The algorithm is also employed for two simultaneous crack detection.
The cracks present in a mechanical/structural member alter flexibility and thereby change modal parameters. The one-dimensional open cracks in the multi-span Euler–Bernoulli beam are modeled as rotational spring. The supports are modeled as a combination of rotational and linear springs. A computer code is developed to obtain numerical results from the formulation. The effect of crack locations, crack depths, several supports, and cracks on various modal parameters are presented for isotropic multi-span cracked beams.
The Federal Highway Administration (FHWA) recognizes a need for cost-effective and practical structural health monitoring (SHM) techniques to manage the bridge network. System identification (SI) techniques are the first step in developing SHM frameworks to extract bridge dynamic properties (natural frequencies, damping ratios, and mode shapes). One promising indirect SI technique uses accelerometer-instrumented vehicles to measure the dynamic properties of bridges, such that a few instrumented vehicles could economically monitor a large network of bridges. This study proposes an autonomous framework called autonomous peak picking variational mode decomposition (APPVMD), which leverages an ensemble of signal processing techniques and heuristic models to autonomously extract bridge frequencies from instrumented vehicles. The proposed framework does not require prior information or model-informed training. The framework is tested on four vehicles and 39 unique bridge classes using finite element (FE) models to assess the feasibility of deploying it into practice. The results of this study indicate that the algorithm can make successful bridge frequency extractions in more than 69.2% of the bridge cases using any vehicle and a maximum value of 97.4% using an SUV, while considering the effects of road surface roughness. The influence of bridge stiffness, mass, and span length on the algorithm's success rate is discussed. It is concluded that flexural rigidity and span length play a significant role in the accuracy of APPVMD. The combined effects of bridge inertia and damping cause heavier bridges to have a higher amplitude signal for a longer duration, increasing the algorithm's successful detection rate of bridge frequency. Next, the authors examine the effect of reducing the tire stiffness in the truck model on the success of the APPVMD algorithm. The customized truck outperformed the original truck and the SUV by successfully extracting the bridge frequencies of 97.4% of the bridge cases and by having more bridge frequencies predicted in a higher confidence region than the SUV. In addition, the customized truck model is used to verify the APPVMD's robustness to large levels of sensor noise. Even at significant levels of white noise (1% SNR), as the success rate for bridge frequency extraction using the customized truck only drops to 84.6%. Finally, the influence of vehicle speed on the APPVMD algorithm's effectiveness is studied. Generally, slower speeds resulted in more successful predictions of bridge frequency, but minor improvements in the accuracy of the frequency predictions were observed at slightly increased velocities, 13.4 m/s (30 mph) and 17.9 m/s (40 mph), but a reduction in accuracy was observed at higher velocities, 22.3 m/s (50 mph).
Cracking frequently encountered in railway bridges can lead to significant changes in bridge dynamic characteristics and further dynamic performance of train–bridge interaction (TBI) system. It is desirable to develop an effective method to understand the dynamic behavior of coupled train and cracked bridge systems. This paper proposes a framework for coupled vibration analysis of train and cracked bridge systems using multiscale finite element modeling. A multiscale bridge model adopting frictional surface–surface contact to characterize nonlinear breathing phenomenon of the crack is incorporated into the three-dimensional coupled TBI model. Subsequently, a multiloop numerical solution strategy is formulated to solve the established dynamic equation of coupled train and cracked bridge systems. The proposed framework is used to analyze the cracked beams presented in the literature, and the obtained and literature data are compared. Moreover, an actual heavy-haul railway bridge is taken as an illustrative example, in which vehicle dynamic response, bridge vibration, and train running safety index are calculated considering various crack types, train speeds, and track conditions. The results show that apart from the bridge displacement and acceleration, the presence of crack can have a large influence on the vehicle acceleration response, derailment factor, and offload factor. Differences larger than ±10% in the maximum dynamic indices associated with the intact and cracked bridges are observed, indicating the necessity of considering the crack damage in train–bridge coupled dynamic analysis.
The present investigation aims to develop a general probabilistic approach for the prediction of potential damage configurations of structure under modal oscillations. The multiple-crack distribution of the as-predicted damage configurations were characterized by the associated probability measure in terms of crack patterns, crack locations and crack depths. It is postulated that the structure is damaged at a prevailing modal response and the cracking process is sequential. The first passage probability of critical structural parameter exceeding the characteristic material allowable was evaluated to predict the mean time to failure, crack initiation, and crack propagation. In conjunction with the multiple-crack interactive stress relief effects, the probabilistic distributions of damage configuration were established for some representative modal shapes. The analysis results indicate that the damage configurations are highly affected by the modal shape wherein modal stress peak regions provide the most probable locations for cracking. Multiple-crack configuration is more likely to occur at higher mode of vibration. A single crack pattern is dominant at lower mode of vibration.
Applying the theory of Lyapunov exponents for nonsmooth dynamical systems, chaotic motions and strange attractors are found in the case of a cracked rotor. To detect the crack and establish a clear relation between shaft cracks in turbo rotors and induced phenomena in vibrations measured in bearings, a model-based method is applied. Based on a fictitious model of the time behaviour of the nonlinearities, a state observer of an extended dynamical system is designed resulting in estimates of the nonlinear effects.
This paper investigates the stability and the stability degree of a flexible cracked rotor supported on different kinds of journal bearings. It is found that no matter what kind of bearings is used, the unstable zones caused by rotor crack locate always within the speed ratio 2N(1-△Kξ4)<Ω<2N when gravity parameter Wg > 1.0; and locate always within the speed ratio 2ΩαN(1-△Kξ4)<Ω<2ΩαN when Wg < 0.1, where ΔKξ is the crack stiffness ratio, N = 1, 2, 3, 4, 5 … and Ωα=(1+2α2α)1/2. When 0.1 < Wg < 1.0, there is a region, where no unstable zones caused by rotor crack exist.
Outside the crack ridge zones, the rotor crack has almost no influence on system’s stability and stability degree; while within the crack ridge zones, the stability and stability degree depend both on the crack and system’s parameters. In some cases, the system may still be stable even the crack is very large. For small gravity parameter (Wg < 0.1), the mass ratio α has large influence on the position of unstable region, but its influence on the stability degree is small. The influence of fixed Sommerfeld number So on the crack stability degree is small although So has large influence on the stability degree of uncracked rotor.
The nonstationary behavior of a cracked rotating shaft with a disk is analyzed. The shaft is accelerated or decelerated through a critical speed, and both flexural and torsional responses are examined. A breathing crack is considered, which is either completely open or completely closed at any given time. Forces due to eccentricity, gravity, and internal and external damping are included. Galerkin’s method is utilized, and the resulting bilinear equations are integrated numerically. The effects of acceleration and deceleration rate, internal and external damping, crack location and depth, and location, mass, and eccentricity of the disk are investigated. Application of the results to detection of cracks is discussed.
A new method is proposed to detect and locate a crack in a rotor system. A vector quantity ‘Residue’ is defined using the measured vibration response of the rotor system and the modeled system matrices. It is shown that the non-zero elements of ‘Residue’ correspond to the nodes encompassing the element carrying the crack. Although, the derivation of ‘Residue’ is based on the linear model of the rotor system, the response due to the non-linearity caused by the breathing of the crack causes only a small amount of contamination. To investigate the robustness of the proposed method, a sensitivity analysis is conducted. It is found that the method could detect a crack of as small depth as 4% of the shaft diameter. The effects of light damping and the order of finite element discretization on the detection result are also investigated, and are found to be small.
Examples of extensive damages to turbine and generator rotors operating in the speed range of up to 120% of the nominal speed are reviewed and categorized according to the failure causes. The latest achievements in the area of damage research are illustrated and some typical damage cases are analyzed. Suggestions for damage prevention are presented.
The Griffith-model of brittle fracture of elastic solids and the model by Irwin and Orowan for the brittle fracture of elastic-plastic solids predict the propagation of cracks on the basis of energy supplied by the work of externally impressed forces and by the change of strain energy. Previous discussions have neglected the three-dimensional viewpoint and the body forces. Since the Irwin method of fracture-strength analysis has become of increased interest, especially with respect to rotor fracture, a general analysis of energy supply is presented. The analysis uses Clapeyron’s theorem and Betti’s reciprocal theorem. One of the results is that the energy supplied for crack extension equals the strain energy of the difference of the two stress fields before and after crack extension.
A rotating blade with a single transverse crack is considered. The proposed model is a torsionally rigid, non-pretwisted Bernoulli-Euler beam of extremely different bending stiffnesses. Two fields are connected by a local spring element characterizing the reduced stiffness of the crack region. The governing nonlinear boundary value problem is solved in two steps: first, the stationary pre-deformation due to the centrifugal forces is calculated and subsequently, the linearized equations of motion describing the superimposed small vibrations are analysed. Quantitative results concerning the stationary deformation and the natural frequencies of the coupled, free extensionalflexural vibrations are presented if crack depth and location are varied.
A method is described for non-destructively assessing the integrity of structures using measurmenets of the structural natural frequencies. It is shown how measurements made at a single point in the structure can be used to detect, locate and roughly to quantify damage. One set of frequencies is measured before the structure is put into service, subsequent measurements being used to test whether the structure is still sound. Alternatively, measurements at an intermediate 0 state may be used as a baseline and the progress of damage monitored from this stage. The presence of damage can be detected simply from changes in the natural frequencies of the structure. Results are presented for a straight bar, an automobile camshaft, fiber composite plates and honeycomb panels obtained from the aerospace industry. It is shown to be possible to detect and locate a variety of forms of damage including fatigue cracks and impact damage in fiber composite structures which is barely visible yet causes severe reductions in the strength of the laminate.
Mechanisms simulating rotor joint restoring moments different from the common axisymmetric elastic hinge are derived and their effects on the dynamics of a complete turbomachine are calculated by impressing equivalent perturbing moments. A mechanism for locking subsynchronous whirl to a fractional frequency is described and supporting experimental observations are discussed. Sample analysis shows the possibility of self-sustained synchronous whirl due to preload asymmetry.
A theoretical analysis of the dynamics of a rotor-bearing system with a transversely cracked rotor is presented. The rotating assembly is modeled using finite rotating shaft elements and the presence of a crack is taken into account by a rotating stiffness variation. This stiffness variation is a function of the rotor’s bending curvature at the crack location and is represented by a Fourier series expansion. The resulting parametrically excited system is nonlinear and is analyzed using a perturbation method coupled with an iteration procedure. The system equations are written in terms of complex variables and an associated computer code has been developed for simulation studies. Results obtained by this analysis procedure are compared with previous analytical and experimental work presented by Grabowski.
A crack in a vibrating beam produces a change in stiffness depending upon the crack is open or close. If the crack opens and closes with the vibration, a non-linear behavior takes place. The presence of closing cracks can be detected by analyzing the spectrum response for a given forcing function, even through the change in natural frequency associated with the crack is very small.
In the past the dynamic behaviour of rotors containing cracks has been studied mostly from the theoretical view-point. This paper deals with the comparison of analytical and experimental results of the dynamics of vibration of a rotor excited by means of an artificial crack. The experimental results are found to be in good agreement with those for the crack model used in the analysis. The general possibility of determining a crack by extended vibration control is indicated.
The purpose of this demonstration is to observe the behavior of a rotor with a single crack. This is presented in oscilloscope orbits, 2X POLAR PLOTS, and spectrum cascade diagrams. Crack detection must be observed in all three formats and monitored over a period of time or start-ups. The change in the 2X behavior pattern provides an early warning that a crack potentially exists and that it is likely propagating toward total rotor fracture.
Steel plants constructed during the high economic growth period (1960's) now have to cope with many structural problems. These are fatigue damage due to many years of use and vibration problems due to poor rigidity. It is very efficient to use structure diagnosis technology applying simulation technology to fatigue crack initiation and propagation prediction, together with FEM analysis and experimental modal analysis. Fatigue crack initiation and propagation prediction system must predict when the crack initiates how this crack propagates and when it will fail. FEM and experimental modal analysis must decide an appropriate method for facility modification using stress and vibration simulation.
SHVAM (Second Harmonic Vibration Amplitude Monitor) is presented. The monitoring of the second harmonic component of the vibration is demonstrated to be a valuable method of early assessing structural degradation. The rationel is shown considering a real case, where extensive Fracture Mechanics, sophisticated NDE and analytical structural analyses were performed together with field direct monitoring.
Applying the theory of Lyapunov exponents for nonsmooth dynamical systems, chaotic motions and strange attractors are found in the case of a cracked rotor. To detect the crack and establish a clear relation between shaft cracks in turbo rotors and induced phenomena in vibrations measured in bearings, a model-based method is applied. Based on a fictitious model of the time behavior of the nonlinearities, a state observer of an extended dynamical system is designed resulting in estimates of the nonlinear effects.
In this paper, the numerical calculation and the method of multiple scales are used to investigate the vibrational behavior of a vertical rotor containing a transverse crack with unbalanced excitation. It is found that resonance will occur when the spin speed of the rotor is close to 2/3ωc or 2ωc (ωc is the critical speed of the rotor without crack). It is believed that the phenomenon of parametric resonance may be a basis for crack diagnosing.
The fundamental frequency of an axisymmetric clamped circular bimodulus thick plate subjected to a combination of a pure bending stress and extensional stress in the plane of the plate is investigated. The governing equations which are obtained by using the average stress method are solved by the Galerkin method. Natural frequencies are compared with the previous results of Irie et al. for ordinary thick plates. The effects of various parameters on the natural frequencies and neutral surface locations are studied. The bimodulus properties are shown to reduce the frequency coefficient OMEGA , significantly.
An analytical algorithm is proposed to represent eigensolutions [λm2, ψm(r)]m = 1∞ of an imperfect structure C containing cracks in terms of crack configuration σc and eigensolutions [ω-n$/2, φn(r)]n = 1∞ of a perfect structure P without the cracks. To illustrate this algorithm on mechanical systems governed by the two-dimensional Helmholtz operator, the Green's identity and Green's function of P are used to represent ψm(r) in terms of an infinite series of φn(r). Substitution of the ψm(r) representation into the Kamke quotient of C and stationarity of the quotient result in a matrix Fredholm integral equation. The eigensolutions of the Fredholm integral equation then predict λm2 and ψm(r) of C. Finally, eigensolutions of two rectangular elastic solids under anti-plane strain vibration are calculated numerically.
Recent occurrences of cracked shafts in major pumps at nuclear power stations 1-5 have generated a great deal of interest and investigation into methods that can be used to detect such a failure. In this article, a thorough analysis of cracked shafts occurring in the Recirculation Pumps at Grand Gulf Nuclear Station (GGNS), Unit 1, will be presented. This experience confirmed many previous observations and expected behavior patterns for cracked shafts. In addition, the data clearly shows the need for continuous monitoring of pump vibrations in order to detect a cracked shaft before catastrophic failure occurs.
Transmissibility changes in a structure caused by damage are investigated as a feasible means to diagnose the structural damage. Transmissibility, for the purposes of this study, is defined as the ratio of the peak acceleration at the response location to the amplitude of the sinusoidal excitation in a planar frame structure. Transmisibilities at different locations in an undamaged structure and in a structure with known damage are computed and the results are compared to relate the changes in transmissibilities to the damage. Transmissibility is a function of the load frequency and has maximums at the modal frequencies of the structure.
This paper considers the development of the probabilistic methodology for the prediction of multiple-crack distribution in a structure of beam elements associated with individual modal oscillations. The probabilistic measure of crack distribution can then be used for the probabilistic diagnosis of crack damage (depth) and its location (spacing) under random loading and to resolve some of the intrinsic uncertainties in the modal theories of fracture diagnosis. The structural system considers some randomness of material strength. The arresting fracture toughness is characterized as a random variable with the appropriate probability distribution. The application of LEFM (Linear Elastic Fracture Mechanics) in connection with the stress relief effect due to the presence of a crack suggests a means of predicting depth and spacing of tension cracks at a given random modal oscillation. The resulting redistributed random bending stresses (moments) will be a measure to compute the subsequent crack state. With postulation that secondary cracking is dominantly affected by its immediately preceding crack, the process of the successive cracking can be treated as a Markov process. The analyses are performed, under these probabilistic assumptions, for the first few representative normal modes of interest. The probability distribution of the overall structural system, therefore, is obtained dependent on a weighted distribution of modes for a particular excitation spectrum.
The change of the dynamic torsional response of a shaft from crack growth can be analysed as a function of the compliance. The aim of this paper is to describe an application of the transfer matrix technique to analyse the dynamic behaviour of a cracked element subjected to torsional vibration, through the concept of compliance. This approach is of interest to engineering students and the design engineer. Finally, in the last half of the article it is shown that the method can be used to detect, locate and quantify damage.
Frequently cracks in turbine rotors were found. Nevertheless, until now it is not really known, how cracks can be recognized early enough so that large consecutive damage can be prevented.
Vibration methods have proved useful as an indirect means of determining geometry and structural integrity. This paper surveys methods for the assessment of geometry and the detection and location of flaws in one-dimensional structures utilising resonant vibrations. A literature survey is presented of previous work and an extensive reference list is provided. Finally a new method of crack detection are indicated.
The feasibility of using vibration analysis to detect the presence of a peripheral crack in a thick-walled pipe was investigated and the results are reported in this paper. A mathematical model of the damaged pipe was developed and used to predict its vibrational characteristics, and these predictions were verified experimentally.
When structural damage occurs in structures due to cracking at critically-stressed sections, it reduces the stiffness and increases the damping at the local section. This change is reflected globally in the structure through changes in its dynamic behaviour, viz., natural frequencies, modal damping, modal shapes and system transfer functions. Acoustic energy is radiated into the surrounding medium from the structural vibration. This acoustic energy carries considerable information about the state of the structure which, when analyzed, could give details about the defects in the structure. A steel vibrating cantilever beam is currently being used as a simple mechanical system upon which quantitative acoustic measurement procedures can be developed for non contact fatigue crack detection in more complicated steel structures. In this paper we focus on the ability to obtain modal information on the state of the cantilever from non-contact acoustic measurements. This paper summarizes the results of modal analysis of a cantilever beam based on measurements with an accelerometer and a force transducer, measurements with a pressure microphone and a force transducer and measurements with a pair of pressure microphones. The results from each method are illustrated for a machined smooth cantilever specimen. Finite element analysis predictions are also included. The long term objective is to combine finite element analysis, modal analysis and acoustic measurement techniques into a global monitoring capability for fatigue crack detection in steel structures. This paper shows the typical confidence level that has been achieved in extracting modal shape and damping information from acoustic measurements versus more conventional mechanical acceleration measurements.
