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Abstract
The dynamics of a cracked, simply supported uniform beam is treated for either bending or axial vibrations. The crack is simulated by an equivalent spring, connecting the two segments of the beam. Analysis of this approximate model results in algebraic equations which relate the natural frequencies to beam and crack characteristics. These expressions are then applied to studying the inverse problem—identification of crack location from frequency measurements. It is found that the only information required for accurate crack identification is the variation of the first two natural frequencies due to the crack, with no other information needed concerning the beam geometry or material and the crack depth or shape. The proposed method is confirmed by comparing it with results of numerical finite element calculations.
... This property is confirmed in [10] and shows that it is possible to address independently the tasks of localization and quantification of single cracks. It was successfully used to find the location of a crack [11,12]. ...
... Based on the ML-EHB, we can define a model of the beam with the first crack as a healthy beam with higher density. The eigenfrequencies ( , ) 11 i c d f can be found according to Equation (12) and the densities with Equation (15). Now, we can repeat the calculus by involving Equation (12) and considering that the second crack affects the healthy beam with frequency ...
... Taking advantage of the fact that we can easily create a database that contains the eigenfrequencies for many damage scenarios, we have developed a damage detection method based on multimodal analysis. From the database we calculate the RFSs for several weak-axis bending vibration modes according to Equation (11). These constitute the damage patterns (DP) which are known a priori because they contain information only about the curvatures of the healthy beam and the severities dependent on the crack depths. ...
Identifying cracks in the incipient state is essential to prevent the failure of engineering structures. Detection methods relying on the analysis of the changes in modal parameters are widely used because of the advantages they present. In our previous research, we found that eigenfrequencies were capable of indicating the position and depth of damage when sufficient vibration modes were considered. The damage indicator we developed was based on the relative frequency shifts (RFS). To calculate the RFSs for various positions and depths of a crack, we established a mathematical relation that involved the squared modal curvatures in the healthy state and the deflection of the healthy and damaged beam under dead mass, respectively. In this study, we propose to calculate the RFS for beams with several cracks by applying the superposition principle. We demonstrate that this is possible if the cracks are far enough from each other. In fact, if the cracks are close to each other, the superposition method does not work and we distinguish two cases: (i) when the cracks affect the same beam face, the frequency drop is less than the sum of the individual frequency drops, and (ii) on the contrary, cracks on opposite sides cause a decrease in frequency, which is greater than the sum of the frequency drop due to individual damage. When the RFS curves are known, crack assessment becomes an optimization problem, the cost function being the distance between the measured RFSs and all possible RFSs for several vibration modes. Thus, the RFS constitutes a benchmark that characterizes damage using only the eigenfrequencies. We can accurately locate multiple cracks and estimate their severity through experiments and thus prove the reliability of the proposed method.
... This property is confirmed in [10] and shows that it is possible to address independently the tasks of localization and quantification of single cracks. It was successfully used to find the location of a crack [11,12]. ...
... Based on the ML-EHB, we can define a model of the beam with the first crack as a healthy beam with higher density. The eigenfrequencies ( , ) 11 i c d f can be found according to Eq. (12) and the densities with Eq. (15). Now, we can repeat the calculus by involving Eq. (12) and considering the second crack affects the healthy beam with frequency ...
... Taking advantage of the fact that we can easily create a database that contains the eigenfrequencies for many damage scenarios, we have developed a damage detection method based on multimodal analysis. From the database we calculate the RFSs for several weak-axis bending vibration modes according to Eq. (11). These constitute the damage patterns (DP) which are known a priori because they contain information only about the curvatures of the healthy beam and the severities dependent on the crack depths. ...
Identifying cracks in the incipient state is essential to prevent the failure of engineering structures. Detection methods relying on the analysis of the changes in modal parameters are widely used because of the advantages they present. In our previous research, we have found that eigenfrequencies were capable of indicating the position and depth of damage when sufficient vibration modes were considered. The damage indicator we developed was based on the relative frequency shifts (RFS). To calculate the RFSs for various positions and depths of a crack, we established a mathematical relation that involved the squared modal curvatures in the healthy state and the deflection of the healthy and damaged beam under dead mass, respectively. In this study, we propose to calculate the RFS for beams with several cracks by applying the superposition principle. We demonstrate that this is possible if the cracks are far enough from each other. In fact, if the cracks are close to each other, the superposition method does not work and we distinguish two cases: (i) when the cracks affect the same beam face, the frequency drop is less than the sum of the individual frequency drops, and (ii) on the contrary, cracks on opposite sides cause a decrease in frequency, which is greater than the sum of the frequency drop due to individual damage. When the RFS curves are known, crack assessment becomes an optimization problem, the cost function being the distance between the measured RFSs and all possible RFSs for several vibration modes. Thus, the RFS constitutes a benchmark that characterizes damage using only the eigenfrequencies. We can accurately locate multiple cracks and estimate their severity trough experiments and thus prove the reliability of the proposed method.
... This method shows good results, though further study was recommended. Similar works performed by Morassi (1993) and Narkis et al. (1994). Salawu (1997) also worked on frequency-based damage detection, and they authors addressed the inverse problem. ...
... Initial test is performed on the structure without any load while the others are performed after adding uniformly distributed imposed loads on other floors. Major drawback of the technique is the requirement to apply large, imposed loads for large structures that might not be cost-effective or practical (Narkis, 1994) (Salawu, 1997). Some researchers suggested that changes in natural frequency alone will not be suffice for damage localization in structure because cracks at two different locations and may create similar frequency change for the structure. ...
Vibration-based techniques for Structural Health Monitoring (SHM) utilize the dynamic response of a structure measured using a set of sensors to identify the modal properties and potential structural deterioration or damage. Signal processing tools are widely used for analyzing and diagnosing these response signals obtained from a structure. Any changes in the dynamic characteristics of a structure indicates deterioration in the structure. However, a direct comparison of the vibration signals or modal properties at different periods of time may not be sufficient to identify the damages and their locations. Therefore, it is essential to analyze the vibration signals to extract the morphologies of the changes in these response signals and correlate them with the possible location of structural damage. Anomalies can exist in a structure’s response in the form of damage due to loss of stiffness in the structural and non-structural elements and these anomalies alter a structure’s behaviors and modal properties. For some decade researchers have been using experimental and operational modal analysis to estimate modal parameters to measure structural damages in experimental and operational conditions respectively employing vibration-based response of structures. In case of ambient vibration data, processing and estimating the exact location of damage are complex for being non-stationary and non-linear. In this work, a novel method is developed where response signals obtained from structures are decomposed into Intrinsic Mode Functions (IMF) using Empirical Mode Decomposition (EMD) technique. Those IMFs are then processed with Hilbert-Huang transform (HHT) to obtain their corresponding Hilbert Spectra (HS), which allows the estimation of the time-varying instantaneous properties of those response signals. Then for system identification, Singular Value Decomposition (SVD) is performed to obtain the Singular Hilbert Spectra (SHS), and for damage detection, Marginal Hilbert Spectrum (MHS) is estimated. This leads to estimation of coefficients to calculate associated damage indices (DI). Simultaneously, modal frequencies of the structure are obtained from time-frequency-amplitude domain plot of the Singular Hilbert Spectra (SHS). Subsequently, Finite Element (FE) model is constructed for the healthy and damaged structure and tuned moment of inertia of the assembled members, which correlate with the natural frequencies obtained during testing. Joint displacements obtained from the linear modal analysis of the healthy and damaged structures lead to estimation of Displacement Mode Shape (DMS) and Curvature Mode Shape (CMS), and their absolute difference between the healthy and damage structure help to locate damage from the model. Thus, a hybrid method comprising MHS-based damage index along with numerical model-based joint curvature mode shape is proposed. This proposed method is verified using experimental tests conducted on: (a) a cantilever steel beam, (b) five-storey scaled frame, and (c) an eight-storey full scale steel building. The present thesis shows that the results obtained from system identification and damage detection manifest the advantages of the HHT-based technique for health monitoring of structures. It is observed from the results that the damage can be localized effectively in single and multiple damage cases from the DI values obtained using the proposed method. Simultaneously, modal frequencies are estimated as a by-product of the damage detection algorithms. The present investigation indicates that MHS is a promising tool for SHM to assess any deterioration or anomaly by comparing with a baseline model. Data-driven techniques for SHM has been mostly limited to small scale tests in laboratories. To address the challenges in application of data-driven methods for real life structures, it is necessary to incorporate physics-based analysis of structure along with the non-model-based method. For data-driven methods it is absolutely necessary to employ modern instrumentation methodologies to obtain vibration-response from structures. To highlight the present work and advantages of the modern wireless sensors for vibration measurement, two different case studies are presented: (a) modal analysis of pre-cast concrete shear wall with wired Piezotronics sensor, and (b) modal analysis of sixteen-storey Concordia University EV building with modern wireless sensors and data acquisition (DAQ) wirelessly connected with android device. The wireless vibration sensors are found to be quite easy and flexible to use for capturing the frequency response of a structure to detect a range of modal frequencies. However, cost of those sensors could pose a challenge and positioning of optimum of sensors is important in testing of tall buildings or long span bridges. To mitigate the problem of limited number of sensors, multi set up and roving sensors methods were used in the past. That could be time consuming and error prone. Further research is required to make the process more efficient. Manual estimation of frequency band width from the SHS plot, complexity in estimating DI from MHS and time expensive manual iteration in the numerical modeling are some of the limitations for the current work.
... Numerous studies have focused on the analysis of the dynamic behavior of a cracked rotating shaft. Changes in the behavior of the mechanical elements have sometimes been used to detect and identify defects in the components by solving an inverse problem [2][3][4][5][6][7][8][9][10][11][12]. ...
... For this study, a set of cracked shafts with a length of L = 0.900 m and a diameter of D = 0.020 m was tested. The tested shafts were made of aluminum, with a Youngś Modulus E = 72 GPa, Poisson ratio ν = 0.33, and density ρ = 2700 kg/m 3 ...
In this work, a procedure to obtain an accurate value of the critical speed of a cracked shaft is presented. The method is based on the transversal displacements of the cracked section when the shaft is rotating at submultiples of the critical speed. The SERR (Strain Energy Ralease Rate) theory and the CCL (Crack Closure Line) approach are used to analyse the proposed methodology for considering the behaviour of the crack. In order to obtain the best information and to define the procedure, the orbits and the frequency spectra at different subcritical rotational speed intervals are analyzed by means of the Fast Fourier Transform. The comparison of the maximum values of the FFT peaks within the intervals allows the subcritical speed to be determined, along with the value of the critical speed. When verified, the proposed procedure is applied to shafts with the same geometry and material and with cracks of increasing depth. The results show that the critical speed diminishes with the severity of the crack, as expected. A comparison is made between the critical speed obtained using the vertical and the horizontal displacements, finding no remarkable differences, meaning that in practical applications only one sensor for one of the displacements (in the vertical or horizontal direction) is needed to determine the critical speed. This is one of the main contributions of the paper, as it means that the orbits of the shaft are not needed. Finally, after this study we can conclude that the best results are achieved when the critical speed is obtained using data displacement in only one direction within the intervals around 12 or 13 of the critical speed.
... It is assumed that the effect of the crack damage is local to its immediate neighbourhood and can be modelled as a rotational spring. 64,65 At the crack location, the damaged upper beam is divided into two separate subbeams connected by a rotational spring whose stiffness depends on the parameters of the beam and the crack geometry (Narkis 63 ). Figure 1 shows the model considered in this paper. ...
... However, the slope of the displacement at the crack location is discontinuous. The crack is modelled by an equivalent rotational spring with stiffness being dependent on beam parameters, crack geometry and nondimensional crack flexibility, θ (Narkis 63 ). The boundary conditions have been summed up as follows: w l ð0, tÞ ¼0 w r ðL, tÞ ¼ 0 wð0, tÞ ¼ 0 wðL, tÞ ¼ 0 w l ða, tÞ ¼ w r ða, tÞ w 00 l ða,tÞ ¼ w 00 r ða, tÞ, w 00 l ð0, tÞ ¼ 0 w 00 r ðL, tÞ ¼ 0 w 00 ð0, tÞ ¼ 0 w 00 ðL, tÞ ¼ 0 w 000 l ða, tÞ ¼ w 000 r ða, tÞ w 0 r ða, tÞ À w 0 l ða, tÞ ¼ Lθw 00 r ða, tÞ: ...
In this paper, the dynamic response of a damaged double-beam system traversed by a moving load is studied, including passive control using multiple tuned mass dampers. The double-beam system is composed of two homogeneous isotropic Euler–Bernoulli beams connected by a viscoelastic layer. The damaged upper beam is simulated using a double-sided open crack replaced by an equivalent rotational spring between two beam segments, and the lower primary beam is subjected to a moving load. The load is represented by a moving Dirac delta function and by a quarter car model, respectively. Road surface roughness (RSR) is classified as per ISO 8606:1995(E). The effect of vehicle speed of the moving oscillator and variable RSR profiles on the dynamics of this damaged double Euler–Bernoulli beam system for different crack-depth ratios (CDRs) at various crack locations is studied. It is observed that coupling of two beams leads to a vehicular effect on the damaged beam, even when no vehicle on it is present. The effects of single and multiple tuned mass dampers to control the vibrational responses of the primary beam due to damage on the secondary beam is studied next. The performance of tuned mass dampers to reduce the transverse vibrations of the damaged double-beam system and of the quarter car is investigated. The paper links the coupling between the two levels of double beam with the inertial coupling of the vehicle to the double-beam system.
... So far, many analytical and numerical methods have been reported to predict the modal frequencies of cracked 1-D waveguides [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Most applications using both methods have been focused on predicting modal frequencies associated with the first few modes (such as, from the 1 st to the 3 rd mode), which are insensitive to incipient cracks because the first few of the lower modes are available through conventional modal experiments such as impact hammer modal testing and shaker modal testing. ...
... Using the propagation matrix described in Eq. (15) in [30], six equations are obtained: where , and represent a characteristic matrix, identity matrix, and wave vector, respectively. ...
This paper presents a generic frequency equation and its approximate solution for a one-dimensional (1-D) waveguide with incipient open cracks. Extending the principle of phase closure for an intact 1-D waveguide to that for a cracked waveguide, the proposed approach is applicable to any type of cracked 1-D waveguide with arbitrary support conditions. The closed-form frequency equation is simply expressed in terms of phase shifts at the incipient cracks and at both ends of the waveguide, which is physically intuitive and easy to understand from a wave perspective. The validity and implication of the proposed frequency equation is demonstrated for the case of Timoshenko beam with multiple incipient cracks through comparisons with the numerical results from the Timoshenko 1-D model and finite element analysis and with the experimental data.
... (2) It is assumed that the effect of damage on the mass properties of the structure is negligible, and we assume that the structure under consideration is undamped. (3) e crack-induced structural member damage is not continuous in the finite-element model, this is reasonable in the previous study [26][27][28][29], and furthermore we could change the scale of element size in the numerical model. ...
... where the element in row 1 and column 7 of S 3 is 1; the element in row 2 and column 8 of S 3 is 1; the element in row 3 and column 9 of S 3 is 1; the element in row 4 and column 10 of S 3 is 1; the element in row 5 and column 11 of S 3 is 1; the element in row 6 and column 12 of S 3 is 1; and rest of the elements of S is zero. Substitute equation (27) into equation (24), the strain energy can be deduced as Substitute equation (25) into equation (29), then the stiffness matrix of element 3 under the global coordination can be expressed by the overall displacement vector as ...
Damage identification based on the change of dynamic properties is an issue worthy of attention in structure safety assessment, nevertheless, only a small number of discontinuous members in existing structure are damaged under service condition, and the most remaining members are in good condition. According to this feather, we developed an effective damage location and situation assessment algorithm based on residual mode vector with the first mode information of targeted structure, which utilized the quantitative relationship between first natural modes of global structure with the change of the element stiffness. Firstly, the element damage location is determined with exploitation of the sparseness of element stiffness matrices based on the discontinuity of damaged members. Then, according to the distribution characteristics of the corresponding residual mode vector, the nodal equilibrium equation about the damage parameter is established based on the residual mode vector, and the damage coefficients of structural elements are evaluated with the proposed equations. Two numerical examples are given to verify the proposed algorithm. The results showed that the proposed damage identification method is consistent with the preset damage. It can even accurately identify large-degree damages. The proposed algorithm only required the first-order modal information of the target structures and held few requirements of analysis resource; hence when compared with existing methods, it has obvious advantages for structural damage identification.
... Liang et al. [19] detected cracks in fixed-fixed and beam structures on the basis of changes in amplitudes and standard observations. Narkis [20] has used an approximate analytical solution to compute the standard observations of a cracked cantilever beam. Narkis [20] concluded that the variation in the initial standard observations caused by the crack is the only information required. ...
... Narkis [20] has used an approximate analytical solution to compute the standard observations of a cracked cantilever beam. Narkis [20] concluded that the variation in the initial standard observations caused by the crack is the only information required. Dado [21] detected the crack depth and location with respect to predefined beam structures with the help of a direct mathematical model for detecting beam cracks where the standard observations were used as input observations for the first two bending vibration modes. ...
Implementation of improved instruments is used to detect damage in an accurate manner and fully analyze its characteristics. An aluminum beam has been used in this work to identify cracks by using a vibration technique. The simulation of frequency response feature was conducted using a finite element model to provide average measures of intensities of vibration. Two forms of wavelet packet transform (WPT) entropies Shannon and log energy were applied to identify the position, width, and size of the crack. The results showed that with an increase in crack depth, the amplitude also increased at certain crack sizes and for all crack positions. For two crack depths of 1.6 mm and 0.16 mm having the same crack size and position 12 mm and 60 mm, respectively, a 4.5% increase in amplitude was observed at a crack depth of 1.6 mm. Moreover, the amplitude varied inversely with the position. A 12.6% increase in amplitude was observed at a crack depth of 1.6 mm rather than 0.16 mm, while both depths occurred at the same crack position (75 mm) and size (20 mm). Experimental validation was performed on a cantilever beam with one crack. The maximum absolute error found was 7.5% for the crack position and 9.1% for the crack size. With the increase in crack depth, the obtained results decrease the stiffness of a beam in a single crack case.
... Chan Ghee Koh [6] evaluated the hardness index of each floor to diagnose damage of frame structure. Narkis Y. [7] locates cracks in the beam structure. Hassiotis S. [8] use the global planning method with finite element method to solve the problem of structural identification. ...
This paper studies a method to identify the elastic fixed stiffness of the frame structure. The model of the problem is three dimensional structure, linear elastic deformation, pile - soil link is replaced by elastic fixed with stiffness. The problem will be solve by the penalty function method - the minimum of the objective function (which is the total squared error between the measured value and the calculated values particular) - combined with the finite element method. The numerical calculations show that the model, algorithm and calculation program are reliable. The program can be used to identify the elastic fixed stiffness of the frame structure in three dimension, serving to determine the actual working state of the structure, to propose solutions for reinforcement, repairing, improving bearing capacity, prolonging the life of the structure.
... Consider an open crack of length a located at a distance β = (b/L) from the lower support, as sketched in Figure 2 (left). Following the method proposed by Freund and Herrmann in 1976 [21] and followed by many other authors [22][23][24][25][26][27], the cracked column is considered as two segments connected by elastic rotational springs, as shown in Figure 2 (right), whose stiffness depends on the crack depth and the geometry of the cracked section. The discontinuity in the deflection slope of the column at the cracked section, ∆θ, is proportional to the bending moment transmitted by that section, M(b), as follows: ...
This work analyses the buckling behaviour of cracked Euler–Bernoulli columns immersed in a Winkler elastic medium, obtaining their buckling loads. For this purpose, the beam is modelled as two segments connected in the cracked section by a mass-less rotational spring. Its rotation is proportional to the bending moment transmitted through the cracked section, considering the discontinuity of the rotation due to bending. The differential equations for the buckling behaviour are solved by applying the corresponding boundary conditions, as well as the compatibility and jump conditions of the cracked section. The proposed methodology allows calculating the buckling load as a function of the type of support, the parameter defining the elastic soil, the crack position and the initial length of the crack. The results obtained are compared with those published by other authors in works that deal with the problem in a partial way, showing the interaction and importance of the parameters considered in the buckling loads of the system.
... Most previous research uses the open crack model while comparatively few studies have been conducted applying the breathing crack model. Narkis [9] simulated the cracks as an equivalent rotating spring and studied the dynamics and identification problems of a uniform cracked, simply supported beam. Dado [10] reported a comprehensive crack identification algorithm for beams under different support conditions using the local flexibility approach. ...
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.
... Most previous research uses the open crack model while comparatively few studies have been conducted applying the breathing crack model. Narkis [35] simulated the cracks as an equivalent rotating spring and studied the dynamics and identification problems of a uniform cracked, simply supported beam. Dado [15] reported a comprehensive crack identification algorithm for beams under different support conditions using the local flexibility approach. ...
The contribution provided in this paper is to investigate the dynamic and buckling response of bidirectional graded material beams (BDFB) with transverse cracks, considering different boundary conditions. The stretch effect by means of a normal deformation in conjunction with the shear deformation influence is integrated for accurate results. The material properties of the beam are supposed to be dependent on the gradation pattern through the width and thickness directions via power-law form. Lagrange’s principle is employed in order to extract the governing equations of motion. The stiffness of the cracked beam element is computed based on the shrinking of the beam’s cross section. The Differential quadrature finite elements method is used as an effective and accurate tool to determine the buckling loads and natural frequencies of the BDFG beam. The numerical results are evaluated with those from earlier studies. Finally, several studies examples were done to examine the impact of the power-law gradation index, crack depth, and position for various boundary conditions on the critical buckling of the beam and natural frequencies. These investigations highlight the advantages of the bidirectional FG beam over the unidirectional FG and pure metallic beams under the presence of a crack.
... The theoretical basis of the frequency-based method for damage detection is the so-called characteristic equation that relates natural frequencies to damage parameters. Different forms of the characteristic equation were conducted in [15][16][17][18] for a beam-like structure with a single crack. Then, the equation was established for beams with multiple cracks [19][20][21][22]. ...
An explicit expression of natural frequencies through crack parameters is derived for multiple cracked beams with simply supported boundaries using the Rayleigh quotient. The obtained expression provides not only a simple tool for calculating natural frequencies of multiple cracked beams, but also allows employing the so-called crack scanning method for detecting multiple cracks in simply supported beams from measured natural frequencies. A numerical example demonstrates that the crack scanning method, in combination with the Rayleigh quotient, enables consistent identification of cracks with 1% relative depth.
... Another model of local crack on beam, that was treated as a singularity of beam stiffness, has been adopted in [19,20] and used for obtaining closed form solution of vibration modes for cracked beams. The spring model of concentrated crack was successfully employed for solving numerous problems of vibration in cracked Euler-Bernoulli [21][22][23][24][25][26] and Timoshenko [27][28][29][30][31] beams. The closed-form solutions for the vibration mode of multiple cracked homogeneous beams have been obtained in [26,31] respectively for Euler-Bernoulli and Timoshenko beams. ...
This paper presents a unified approach to vibration analysis of functionally graded beams with transverse open-edge cracks based on the so-called vibration shape obtained as a general solution of vibration equations in the frequency domain. The crack is modeled by a pair of translational and rotational springs of stiffness computed from the crack depth in dependence upon functionally graded material parameters. The frequency-dependent vibration shape functions allow one not only to obtain the closed-form solution of both free and forced vibrations for multiple cracked FGM beams but also to develop the well-known methods such as Transfer Matrix Method or Dynamic Stiffness Method for analysis of FGM framed structures. The proposed theoretical developments have been illustrated by their application for modal analysis and frequency response analysis of multi-span and multistep beams.
... Preliminary studies contributed to laying the theoretical background on frequency-based methods for the detection of structural damage [3][4][5][6][7][8][9][10][11][12]. Besides, numerous works studied the influence of slots, cracks, notches, and other geometrical changes in modifying the frequency properties of damaged elements [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. This hypothesis was subsequently considered in several types of structures such as beams with multisupport boundary conditions [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] (i.e., free-free, simply supported, fixed-fixed, fixed-free); slender beams [44]; beams with varying depth [45]; composite beams [46][47][48]; beams on an elastic foundation [49]; beams with dissipative boundary conditions [50]; Timoshenko beams [51]; frame structures [52][53][54][55]; offshore platforms [56][57][58]; trusses [59]; bridges [60][61][62][63];and more recently, double-hinged parabolic and circular arches [64][65][66]. ...
Vibration-based damage detection techniques, and particularly frequency and modal-based methods, address the problem of localization and quantification of damage in a structure by using observed changes in its dynamic properties. Most of these procedures are based on an optimal criterion in which the stiffness distribution of an element into a structural system is iteratively updated in order to match the computed natural frequencies with the measured ones at a certain level of deterioration. This paper provides a comprehensive review of the optimization methodologies based on random search procedures that have been employed to find practical solutions in the forward and inverse frequency and modal-based Structural Health Monitoring (SHM) problem. Over the last three decades, the application of machine learning approaches and population-based metaheuristics to solve real-engineering optimization problems has grown massively, especially in the SHM field. Therefore, the literature review provided in this paper is organized into three main sections: (1) Machine learning techniques; (2) Population-based metaheuristics; and (3) coupled methodologies. Finally, the purpose of this paper is to provide the reader with a wider understanding and a critical point of view of frequency and modal-based techniques for structural damage detection, and the various computational methods that have been employed to solve the problem.
... The combined use of both models allowed global vibration characteristics to be used to identify multiple discrete cracks with sufficient local detail, such as the location and depth of cracks. Subsequently, the natural frequencies of a cracked simple supported uniform were found by an approximate analytical solution examining both bending and axial vibration [16]. The author stated that the analysis of the variation data of just two natural frequencies is enough for the identification of crack location. ...
Infrastructure development is a common feature of emerging countries. As a result, the design and construction of complex structures susceptible to damage is becoming increasingly common. Over the years, multiple advances in Structural Health Monitoring (SHM) haven allowed researchers and engineers to detect, locate, and quantify structural damage in critical components of civil engineering structures. Frequency-based methods have demonstrated their reliability in multiple numerical and experimental applications. This study presents a brief chronological literature review of methodologies based on frequency analysis that have been used in the detection ofstructural damage over the last forty years. It is worth noting that the paper focuses on computer-aided techniques such as artificial neuronal networks (ANN), genetic algorithms (GAs), and metaheuristics that have been employed to solve the inverse damage detection problem.
... The combined use of both models allowed global vibration characteristics to be used to identify multiple discrete cracks with sufficient local detail, such as the location and depth of cracks. Subsequently, the natural frequencies of a cracked simple supported uniform were found by an approximate analytical solution examining both bending and axial vibration [16]. The author stated that the analysis of the variation data of just two natural frequencies is enough for the identification of crack location. ...
Las cargas que generan los eventos sísmicos y los fuertes vientos son fuerzas de la naturaleza que someten a las obras civiles a situaciones extremas, lo que provoca eventualmente la falla de las estructuras y en muchas ocasiones, la pérdida de vidas humanas. Para enfrentar estas fuerzas de carácter aleatorio y de difícil predicción, la ingeniería estructural plantea normativas de diseño y construcción de obligatorio cumplimiento en la mayoría de los países del mundo, que permiten que las estructuras puedan resistir de manera adecuada las fuerzas impuestas. Y como la historia lo ha demostrado, algunas veces un buen diseño no es suficiente, por lo que la ingeniería sismoresistente desarrolla nuevas metodologías y dispositivos que ayuden a proteger aún más a las estructuras cuando se ven sometidas a acciones como los sismos y los vientos. Para afrontar estos retos, aparecen mecanismos como los amortiguadores y controladores, agrupados como dispositivos pasivos, activos, semiactivos e híbridos, con diseños innovadores que contribuyen en gran medida a dar mayor seguridad y confianza a nuestras obras civiles. En este artículo se presenta una visión general de los amortiguadores de masa sintonizada, su desarrollo histórico, modelos mecánicos y analíticos, alcances, fortalezas y debilidades.
... As the local stiffness of a beam decreases in the presence of a crack without the change of mass, the modal frequencies of the cracked beam are lower than those of a pristine one. Most previous theoretical and experimental studies have focused on predicting the reduction of modal frequencies associated with the first few bending modes available through conventional modal experiments such as impact hammer modal testing and shaker modal testing [9][10][11][12][13][14][15][16][17]. The potential of using the modal frequency has been undermined owing to the low sensitivity of these bending mode frequencies to small cracks. ...
There is a premise that the reduction of the bending modal frequency does not occur in the presence of a transverse crack on the bending node of a beam. This study investigated the modal behaviors of a beam with a transverse crack on a bending node and revealed that this premise does not hold, particularly for a deep crack on a high-frequency bending node. Mode II crack (sliding crack mode) of fracture mechanics, which has not been considered in most previous studies, was found to cause the reduction of the modal frequency of a beam with a crack on a bending node. A new frequency equation that accounts for mode II crack was devised. The implications of the new frequency equation were demonstrated through the normalized modal frequency shift factor. Further, the validity of the proposed frequency equation was demonstrated by comparing the quantitative results with finite element analysis results and experimental data.
... There are two types of condition evaluation indicators, namely, vibration-based and static-based indicators. Frequency [4][5][6][7], mode shape [8][9][10], frequency response function [11,12], and modal strain energy [13] are examples of vibration-based damage indicators. In several domains, vibration-based indicators have had variable degrees of effective-ness, although they are typically thought to be more sensitive to environmental changes (such as temperature). ...
A novel damage detection approach is proposed in this study for a continuous girder bridge in which support reaction influence lines (ILs) are adopted. First, the relationship between the local damage of a continuous girder bridge and a damage index, based on support reaction ILs, is established through analytical derivation. Subsequently, the sensitivity of a support reaction IL-based damage index is analyzed using Dempster-Shafer(D-S) evidence theory, and it shows that the support reaction IL-based damage index is more noise-resistant if more support reaction ILs from a variety of locations are used. Three case studies (a simple numerical study of a two-span continuous beam, a laboratory experimental study of a two-span aluminum beam, and a complicated numerical study of a continuous girder bridge in Xiamen) have been conducted to validate the effectiveness of the proposed method in different damage scenarios, including single damage and multiple damages. Satisfactory damage identification results can be obtained even in high-level measurement noise conditions, showing that the proposed approach offers a promising field detection technique for identifying local structural damages in continuous girder bridges.
... Theoretical models considering the dynamics of cracked Euler-Bernoulli beams 45 are numerous [32,33,34,35,36,37]. Commonly, the beam is considered as a set of segments connected by massless rotational springs, whose stiffness is related to the crack length by the Fracture Mechanics theory, placed at the cracks positions. ...
This work analyzes the dynamic behavior of cracked Timoshenko beams immersed in a Winkler elastic medium, obtaining their natural frequencies of bending vibration. For this purpose, the beam is modeled as two segments connected in the cracked section by two massless springs, one extensional and another one rotational. Their stiffnesses are proportional to the shear force and bending moment transmitted through the cracked section, respectively, considering the discontinuity of vertical displacement and rotation due to bending. The differential equations for the free vibration are solved by applying the corresponding boundary conditions and the compatibility conditions of the cracked section. The proposed methodology allows calculating the natural frequencies of vibration as a function of the type of support, the parameter defining the stiffness of the elastic soil, the crack position and the initial length of the crack. The results obtained are compared with those published by other authors who model the crack in a simplified way, showing the interaction and importance of the parameters considered in the natural frequencies of the system.
... Cawley and Adams [18] proposed a method based on the frequency shift that identifies the position of the damage in a plane structure. Narkis [19] analyzed the inverse problem for identification of crack position from frequency measurements. Silva and Gomes [20] proposed a technique using the frequency shift coefficient (FSC) to detect the crack size and position. ...
Health surveillance in industries is an important prospect to ensure safety and prevent sudden collapses. Vibration Based Structure Health Monitoring (VBSHM) is being used continuously for structures and machine diagnostics in industry. Changes in natural frequencies are frequently used as an input parameter for VBSHM. In this paper, the Frequency Shift Coefficient (FSC) is used for the assessment of various numerical damaged cases. An FSC-based algorithm is employed in order to estimate the positions and severity of damages using only the natural frequencies of healthy and unknown (damaged) structures. The study focuses on cantilever beams. By considering the minimization of FSC, damage positions and severity are obtained. Artificially damaged cases are assessed by changes in its positions, the number of damages and the size of damages along with the various parts of the cantilever beam. The study is further investigated by considering the effect of uncertainty on natural frequencies (0.1%, 0.2% and 0.3%) in damaged cases, and the algorithm is used to estimate the position and severity of the damage. The outcomes and efficiency of the proposed FSC based method are evaluated in order to locate and quantify damages. The efficiency of the algorithm is demonstrated by locating and quantifying double damages in a real cantilever steel beam using vibration measurements.
... When the deformation is vibrational, the given data usually includes either a vector-valued time series relating to the vibrational deformation of the body or modal characteristics of the vibration such as natural frequencies and mode shapes. In this case, the damage identification problem is a dynamic problem [5,[34][35][36][37][38][39][40]. In several practical cases, a static load is easier to arrange than a (controlled) dynamic load. ...
This paper considers a statistical method for damage identification of simply-supported elastic beams from static data. The problem is cast as an inverse problem and analyzed in Bayesian inversion framework. The main goal is to obtain a probabilistic model for the spatial distribution of the damage across a beam given limited and noisy data on the static beam response. A principal distinction of the proposed method is its ability to identify not only an estimate of the damage occurred to the beam but also the uncertainty (error) associated with such an estimate. Moreover, the proposed method uses simple static data instead of dynamic or spectral data, unlike the majority of research work published in the literature. This is a challenge as static data is a rather limited source of information about the underlying physical system compared to dynamic data. Finally, the proposed method does not involves any computationally expensive high-dimensional integration algorithms unlike the existing methods. According to the numerical analysis conducted in this paper, the result obtained by using the proposed damage identification method lies within one standard deviation interval.
... For the detection of damages, a mathematical model of the structure along with the experimental modal parameters of the structure has to be used. Changes in the natural frequencies [22][23][24][25], mode shapes and their derivatives [26][27][28][29], measured dynamic flexibility [30][31][32] or frequency response function [33][34][35][36], these three are the basis for the identification approach. It is easy to measure the natural frequency with great precision, and it is also the most common used dynamic parameter for damage detection. ...
Finite element model updating (FEMU) is a technique to improve the analytical finite element (FE) model of any structure from its experimental modal test data. The main purpose to apply FEMU on structures is to remove the uncertainties or errors present in the analytical FE model. The main objective of this paper is to present a review on the various FEMU techniques which can be applied to remove the uncertainties present in the FE model of the actual engineering structures. Applications of various FEMU techniques on the metallic and the composite structures have been discussed in this review paper. FEMU is applied on the metallic and the composite structures to remove the error present in their FE models. The main objective of the FEMU is to accurately predict the modal analysis characteristics such as the spatial-model, modal-model and the response model of the structures. The uncertainties present in the analytical or simulated FE model of any structure may be due to its material properties, dimensions and most probably due to the uncertainties present in the boundary conditions of the structure. However, to provide a sufficient strength to this review paper, the different updating methods are applied on the three degree of freedom spring mass system, on a 1-D aluminum beam, 2-D aluminum panel and on a graphite-epoxy composite material laminate. It is found that the updating algorithms are fast and reliable enough to remove error present in the numerical or simulated FE model of the structures and deliver the accurate estimation of the spatial-model, modal-model and response model of the different material structures.
... Table 1 compares the first natural frequency, which is typically available from such analyses, for comparison. An open crack in the beam is considered and modelled as a smeared crack (Bovsunovsky & Matveev, 2000), a rotational beam model (Narkis, 1994) for localized crack and a continuum model (ref) driven by the Hu-Washizu-Barr (Christides & Barr, 1984) functional. It is observed that the variations in natural frequency can be small and can vary based on how the crack is modeled or is present. ...
... Because of the response spectrum analysis restriction to apply nonlinear response of a complex structure, nonlinear time-history analysis should be the issue regarding the degradation of different elements of the structure, load pattern characteristics resulting from ground motion intensity, and also the parameters induced in the nonlinear dynamic analysis. Besides the nonlinear time history analysis allows to evaluate the effect of supplementary energy-dissipation devices introduced in structural systems [34][35][36][37]. ...
The infill walls are usually considered as nonstructural elements and, thus, are not taken into account in analytical models. However, numerous researches have shown that they can significantly affect the seismic response of the structures. The aim of the present study is to examine the role of masonry infill on the damage response of steel frame without and with various types of openings systems subjected to nonlinear static analysis and nonlinear time history analysis. For the purposes of the above investigation, a comprehensive assessment is conducted using twelve typical types of steel frame without masonry, with full masonry and with different heights and widths of openings. The results revealed that the influence of the successive earthquake phenomenon on the structural damage is larger for the infill buildings compared to the bare structures. Furthermore, when buildings with masonry infill are analyzed for seismic sequences, it is of great importance to account for the orientation of the seismic motion. The nonlinear static response indicated that the opening area has an influence on the maximal strength, the ductility and the initial rigidity of these frames. But the shape of the opening will not influence the global behavior. Then, the nonlinear time history analysis indicates that the global displacement is greatly decreased and even the behavior of the curve is affected by the earthquake intensity when opening is considered. Doi: 10.28991/cej-2021-03091653 Full Text: PDF
... In particular in the aerospace and oil -shore there are many authors interested in research such as Farrar and Doebling [4], Ghoshal et al. [5], Friswell and Penny [6], Lee and Shin [7], .... Chan Ghee Koh, Lin Ming See, Thambirajah, Balendra [8] evaluated the hardness index of each floor to diagnose damage of frame structure. Narkis Y. [9] locates cracks in the beam structure. Hassiotis S. and Jeong G.D. [10] use the global planning method with finite element method to solve the problem of structural identification. ...
The paper presents the results of solving the problem of identifying the equivalent depth of the linkage of the frame - pile structural system. The computational model of the problem is a frame - pile structural system in the form of three-dimensional frame, linear elastic deformation, pile - soil link is replaced by a hard restraint (fixed) with equivalent restraint depth. The problem is solved by the method of penalty function to combine with the finite element method. Examples of numerical calculations show that models, algorithms and calculation programs can be trusted and acceptable
... The former uses the measured time domain signals such as displacement and acceleration responses, and the latter uses the frequency data such as frequency response function (FRF). Narkis 9 identified crack location in the simply supported beams using the variation in measured first two natural frequencies due to crack. This method is capable of finding crack location only. ...
A novel spectral transfer matrix for a cracked beam element is developed in this article and the same is used to identify the crack parameters on the beam structures. Spectral transfer matrix is developed from trigonometric functions based on the theory of fracture mechanics. This matrix determines the natural frequencies of a structure with crack with better accuracy than any other transfer matrices in the literature. The state vector at a node on the structure is formed which includes the displacement, rotation, internal and external forces, and moments at that node. When the state vector is multiplied with the transfer matrix, the state vector at the adjacent node is obtained. Each element is assumed to have a single open breathing crack with unknown depth and location. Initially, the developed spectral transfer matrix is used to determine the natural frequencies of a known cantilever, and after successful validation, the same is used for crack damage detection. By an inverse approach, crack parameters in each element are identified. The state vector at one node on the structure is obtained by measurement of input and out responses which is known as the initial state vector. Acceleration responses at selected nodes on the structure are measured and the state vectors at those nodes are predicted using spectral transfer matrices. The mean square error between measured and simulated responses is minimized using a heuristic optimization algorithm, with crack depth and location in each element as the optimization variables. Spectral transfer matrix method is applied to two numerical problems with single crack in each element; later, this method is successfully validated experimentally with structures having different boundary conditions. The accuracy in identified crack parameters and the applicability to sub-structures of a large structure are the important aspects of this method.
... One class of method is to model the cracked beam as two parts joined by a rotational spring whose stiffness is related to the crack depth as in [8,9]. For example, Narkis [10] modeled the crack as an equivalent spring joining the two portions of the beam to understand the dynamics of a cracked beam under simply supported boundary conditions. Lin et al. [11] developed an analytical model of a beam with a random number of cracks using a similar approach for modeling the crack using a rotational spring. ...
PurposeThis work aims to investigate a cracked cantilever beam subjected to a moving point force using the Discrete Element Method (DEM).Contribution and MethodA novel approach to mathematically include a moving force in discrete element formulation of a cracked beam is the main contribution of this paper. The local reduction in the stiffness of the cantilever beam due to the presence of a crack has been accounted for by a popularly used result from fracture mechanics. Hence, the present work provides an alternative approach to numerically evaluate the forced vibration of cracked beams under the application of a moving point force.Conclusion
The methodology has been verified by comparing some of the results obtained here to those obtained using an already published analytical method. In the end, the effects of crack length, crack location, and force-velocity on the dynamic behavior of the cracked beam are studied using the proposed methodology. The proposed method can provide an effective alternative for the analysis of cracked beams subjected to a moving point force.
... Muchos estudios sobre vibraciones transversales de vigas dañadas se basaron en la teoría de Bernoulli-Euler (Narkis, 1994;Shen y Pierre, 1990;Aydin, 2008;Rezaei et al., 2016). En el caso de fisuras por fatiga, el daño se describió como una reducción localizada de la rigidez flexional y también como un resorte de giro interpuesto en la estructura (Rosales et al., 2009). ...
Palabras clave: vigas dañadas, rigidez seccional, viga Timoshenko-Vlasov, fisura de fatiga. Resumen. Se desarrolla una metodología para la determinación de las propiedades seccionales de vigas tipo Timoshenko-Vlasov con secciones generales y presentando un patrón de daño arbitrario. Se basa en la formulación de un problema estático particular de una estructura dañada mediante un modelo tridimensional que se resuelve por el método de elementos finitos. Luego, se compatibiliza la cinemática con la de una viga tipo Timoshenko-Vlasov y se determinan a partir de cálculos estáticos las propiedades seccionales equivalentes de la viga dañada. Una vez obtenidas, el modelo unidimensional resultante puede emplearse ventajosamente para el estudio dinámico y su aplicación en la identificación de daños.
Structural damage identification plays an important role in the performance evaluation and maintenance of existing structures and post-disaster damaged buildings. On the basis of summarizing and analyzing the commonly used damage models of concrete structures and the description of structural safety state in domestic and foreign codes, a method of damage index fitting and safety state identification for reinforced concrete flexural members based on BP neural network is proposed in this study. Two independent BP neural network damage models for damage index fitting and safety state recognition, respectively, are established using Matlab. For the damage index fitting, this method takes the crack characteristic parameters as the input of the network, and the damage index calculated by the dual-variable damage model based on stiffness and energy is regarded as the output. For the safety state recognition, the proposed method takes the crack characteristic parameters as the input and the safety state classification result following the FEMA-356 code serves as the output. Thereby, the mapping relationship between the crack characteristic parameters and reinforced concrete member damage can be established. Compared with the traditional damage assessment methods, this proposed method has the advantages of accuracy, promptness and convenience, and it enriches the technical means of structural health monitoring.
This study presents a multi-objective optimization method for designing a vibration absorber to reduce the vibrations of a cracked Euler–Bernoulli beam with flexible support under moving forces. Using the assumption of an open crack, the crack is modeled as a decrease in cross-sectional flexibility. After adding an absorber to the beam, the effect of cracks with different intensities on its vibration behavior was examined. First, the dynamic response of the cracked beam has been determined under the influence of different speeds of the moving force. Genetic algorithms have been used to optimize the parameters of the absorber and to examine the effect of mass and damping constant on its efficiency. Despite cracks increasing the dynamic deflection of a beam with an absorber, a cracked beam without an absorber still gains more dynamic deflection than one with an absorber. As a result, vibration absorbers that are designed for a healthy beam are still effective in reducing the dynamic deflection of the beam following the occurrence of cracks and changes in the structure's dynamics.
This paper studies a method to identify the elastic fixed stiffness of the frame structure. The model of the problem is three dimensionals structure, linear elastic deformation, pile - soil link is replaced by elastic fixed with stiffness. The problem will be solved by the penalty function method - the minimum of the objective function (which is the total squared error between the measured value and the calculated values particular) - combined with the finite element method. The numerical calculations show that the model, algorithm and calculation program are reliable. The program can be used to identify the elastic fixed stiffness of the frame structure in three dimensions, serving to determine the actual working state of the structure, to propose solutions for reinforcement, repairing, improving bearing capacity, prolonging the life of the structure.
Modern‐day research in composite material development primarily focuses on tailoring combinations by proportions of constituent materials and monitoring the changes in their targeted properties. In line with this trend, a new class of glass fiber reinforced polymer hybrid composite beam of size 600 mm×50 mm×6 mm is fabricated by adding graphene (average particle size:10 μm), and flyash (average particle size: 60 μm). The dynamic behavior of the hybrid beam by introducing two transverse cracks at different positions with varying crack depths is studied by employing analytical, finite‐element, and experimental approaches. The dimensionless relative natural frequencies of the cracked/faulty hybrid beam obtained by the proposed methods are compared with the intact hybrid beam. Also, a comparison is made for the hybrid beam with a single crack. An increase in relative crack depth resulted in an increase in values of dimensionless compliances. Further, the effect of fiber orientation and lamina stacking sequence on the dynamic parameters of the hybrid beam are also analyzed. The introduction of the second crack induces higher nonlinearity in bending modes of vibration.
This research presented a procedure to automatically identify the modal characteristics of structures to detect and quantify any localized damage of structures in frequency domain. This process uses the acceleration frequency response of the selected degree of freedoms, some natural frequency of the damaged structure and mode shapes of the undamaged Finite Element (FE) model to identify the structural stiffness of the damaged structure containing the damping ratio. The incomplete data in the unmeasured degree of freedom will be updated using a genetic optimization algorithm. The accuracy and efficiency of the proposed method have been tested on a 25 elements truss subjected to a harmonic load with sweep frequency. Optimization results demonstrated that the proposed method is capable of precisely identifying the modal characteristics and stiffness reduction factor of the damaged structure using genetic optimization algorithm along with 5% noise in the input force and response of the system in all studied cases.
Objectives. To study the direct and inverse problem of vibrations of a rectangular rod having a longitudinal notch, to analyze regularities of the behavior of natural frequencies and natural forms of longitudinal vibrations when changing the location and size of the notch, and to develop a method for uniquely identifying the parameters of the longitudinal notch using the natural frequencies of longitudinal vibrations of the rod.
Methods. The rod with a longitudinal notch is modeled as two rods, where the first one does not have a notch, while the second one does. For connection, conjugation conditions are used, in which longitudinal vibrations and deformations are equated. The solution of the inverse problem is based on the construction of a frequency equation under the assumption that the desired parameters are included in the equation. Substituting natural frequencies into this equation, the nonlinear system with respect to unknown parameters is derived. The solution of the latter is the desired notch parameters.
Results. Tables of eigenfrequencies and graphs of eigenforms are given for different notch parameters. The results for different boundary conditions are obtained and analyzed. A method for identifying notch parameters by a finite number of eigenfrequencies is presented. The inverse problem is shown to have two solutions, which are symmetrical about the center of the rod. The unambiguous solution requires eigenfrequencies of the same problem with different boundary conditions at the right end. By adding additional conditions at the ends of the rod, the inverse problem can be solved with new boundary conditions to construct the exact solution and develop an algorithm for checking the uniqueness of the solution.
Conclusions. The developed method can be used to solve the problem of identification of geometric parameters of various parts and structures modeled by rods.
La surveillance de la santé structurelle (SHM) des structures est primordiale pour une utilisation sûre et essentielle pour le développement durable. L'analyse modale opérationnelle (OMA) pour SHM est appropriée pour les structures réelles car elle offre plusieurs avantages : faible coût, utilisation normale des structures et surveillance continue. Cependant, elle présente quelques obstacles majeurs : (i) excitations non mesurées; (ii) problèmes de sous-détermination; (iii) relation entre les endommagements et le changement des paramètres modaux, n'est pas directe; (iv) identification difficile des endommagements multiples. Par conséquent, les objectifs de la thèse sont : (i) faire une analyse sur des méthodes efficaces et populaires pour l'analyse modale opérationnelle et pour l'identification des endommagements ; (ii) proposer des améliorations des méthodes existantes ou de nouvelles méthodes qui peuvent traiter les cas sous-déterminés ; (iii) développer une procédure de détection rapide des endommagements; (iv) introduire une procédure améliorée pour la détection des dommages multiples. Les résultats obtenus dans le cadre de cette thèse peuvent être résumés en quatre contributions principales suivantes :La première contribution est une amélioration de la technique d'identification modale existante basée sur la décomposition PARAllel FACtor (PARAFAC) dans le domaine temporel. Le tenseur de troisième ordre de la covariance des réponses est décomposé en composantes correspondant aux modes structurels ou aux déflections harmoniques. Une longueur minimale des fonctions d'auto-covariance utilisant les périodes naturelles et les taux d'amortissement est suggérée pour distinguer avec précision les harmoniques des modes structurels.La deuxième contribution consiste à développer une nouvelle méthode d'identification modale basée sur la décomposition PARAFAC dans le domaine fréquentiel. En utilisant la décomposition PARAFAC, un tenseur d'ordre 3 en fréquence construit à partir de la densité spectrale de puissance (DSP) des réponses est d'abord décomposé en plusieurs tenseurs d'ordre de rang 1 qui peuvent être des modes structurels ou des composantes harmoniques. La fonction auto-PSD de chaque tenseur d'ordre 3 de rang 1 est ensuite utilisée pour identifier les paramètres modaux tandis que les valeurs de kurtosis spectral sont utilisées pour distinguer les modes structurels et les harmoniques. La performance de la méthode proposée a été étudiée avec un amortissement proportionnel/non-proportionnel, des modes très rapprochés, des cas sous-déterminés et en présence d'excitations harmoniques.La troisième contribution est consacrée à la proposition d'une méthode rapide pour la détection et la quantification d'un changement local simple de la masse et/ou la rigidité de structures de type poutre en utilisant des paramètres modaux identifiés. La relation entre les changements locaux de masse et/ou de rigidité d'une poutre et son décalage de fréquence naturelle et sa déformée modale est examinée et une expression analytique est explicitement donnée. Sur la base de l'expression proposée, une régression linéaire est appliquée pour obtenir des résultats précis du changement de masse et/ou rigidité des poutres.La quatrième contribution vise à étendre la procédure précédente d'identification des endommagements pour de multiples changements locaux de masse et/ou de rigidité. La comparaison entre les décalages de fréquence naturelle obtenus directement à partir de l'expression analytique établie dans la contribution précédente au lieu d'utiliser la méthode des éléments finis et ceux mesurés permet ensuite l'identification des endommagements en utilisant une inférence Bayésienne. L'identification des endommagements proposée devient rapide car elle évite le coût de calcul causé par les simulations FEM.Toutes les contributions ci-dessus ont été validées par des simulations numériques et des tests expérimentaux au laboratoire.
This paper's goal is to build a meshless radial basis function based on the differential quadrature approach that makes use of Hermite interpolation. This method approximates by adding a linear weighted sum of the sought function values defined at scattered nodes, one can calculate derivatives in a governing equation. On other hand the finite element analysis of mesh size is critical because the size of the mesh is closely related to the accuracy and number of mesh required for the meshing of the beam element. The stability of two created algorithms is discussed. The effectiveness of the algorithms is tested numerically, and cantilever beam of various reinforcing behaviours are provided. The proposed methods are discovered to be precise, straightforward, and quick, in order to extract free vibration with various boundary conditions and compare the natural frequency results, the method is applied to a cantilever beam with carbon glass fibre with varied bombyx mori reinforcement (10%, 20%, and 30%) as a numerical test.
Dynamic methods are a powerful tool for studying the behaviour of existing structures and their health conditions. The practical application, however, often raises subtle questions related to the accuracy and completeness of experimental data, the complexity of the mechanical modelling and, ultimately, the inverse nature of the problems that leads to ill-conditioning and non-uniqueness. This chapter addresses some of these aspects, and presents a short overview of the topic, with particular emphasis on dynamic structural identification and damage detection.KeywordsStructural identificationDamage detectionInverse problems
The Rayleigh quotient is an attractive relationship between natural frequency and mode shape of structural vibration. However, it would be useful only for approximately calculating natural frequencies of a structure by using the properly chosen trial functions of the mode shapes in case when both the modal parameters are unknown. This is completely appropriate for the case of cracked structures when an explicit expression of natural frequencies is needed for some purpose such as structural health monitoring. The present report is devoted first to establish Rayleigh quotient for longitudinal vibration in multiple cracked bars and then to involve it for calculating natural frequencies of the cracked structure by using mode shapes of uncracked one. Numerical examples are accomplished for accessing usability of various approximate expressions of the obtained Rayleigh quotient.
The present work deals with the influence of cryogenic coolants LN2 delivered through holes made on flank surface and rake surface of tungsten carbide cutting tool inserts in turning of super duplex stainless steel (SDSS) using in-house developed cryogenic setup. Experiments were conducted with the cryogenically treated tool, cryogenically treated tool with tempering and cryogenic coolant directly passed through a modified cutting tool insert. Results are compared with dry cutting conditions. The cutting conditions are low feed rate/high depth of cut, medium feed rate/medium depth of cut, and high feed rate/low depth of cut. The material removal rate and cutting speed is kept constant under all three cutting conditions. Microstructural study of the tool as received and cryogenically treated is examined using SEM. The population of harder tungsten carbide phase (gamma phase) is found to be more in the cryogenically treated tool. Due to tempering, the hardness of insert is improved by 8% which in turn increased tool life. By direct supply of LN2 through modified cutting tool increased tool life by 23%, more than the cryogenically tempered tool. There are no appreciable changes in the temperature of the cutting tool under dry cutting and cryogenically treated inserts. However, there is a large difference observed in temperature of cutting tool when LN2 is supplied through a modified insert directly, which in turn yielded high tool life.
A presente Bolsa de Iniciação Científica se insere em um projeto maior de uso de redes Neurais Artificiais para a detecção de Danos do prof. Herbert M. Gomes. O intuito deste projeto maior é o de através de medições de características dinâmicas de estruturas, conseguir detectar padrões que indiquem a presença de dano (falha) em estruturas simples. A detecção de dano em estruturas é uma tarefa difícil, e muitas são as metodologias existentes para tratar este problema. Dentre os diversos métodos existentes aqueles que tentam associar mudanças nas características dinâmicas de estruturas com a presença do dano são os que mais têm tido êxito. Dentre estes métodos podem ser destacados aqueles que fazem uso de modelos paramétricos para esta detecção assim como aqueles baseados pura e simplesmente em respostas da estrutura danificada. Independente da vertente de pesquisa existentes, ambas carecem de dados referentes às previsões de variações de características dinâmicas tanto experimentalmente quanto teoricamente. Numericamente inúmeros trabalhos têm mostrado o potencial de diversas técnicas na solução deste problema. A aplicação deste tipo de conhecimento recai sobre uma grande variedade de problemas que vão deste a detecção de trincas e fissuras em partes de braços robóticos até a detecção em tempo real de pré-trincas e/ou fissuras em “raisers” e tubulações industriais.
O projeto prevê a coleta de dados experimentais, tarefa esta associada a outra Bolsa de Iniciação Científica, assim como o estudo, especificamente no caso desta Bolsa, das variações de freqüência teóricas esperadas em estruturas simples como vigas com danos presentes ao longo da mesma. O estudo se restringiu à aplicação, como será visto posteriormente, à Teoria de Vigas de Euler-Bernoulli. Uma Tese de Doutorado está em andamento com este respeito para a aplicação dos desenvolvimentos teóricos aqui encontrados para outras teorias de vigas e a sua posterior comparação com resultados experimentais.
In this paper a novel analytically method for analysing the dynamic behaviour of Timoshenko beam model, resting on Winkler-type elastic soil, under simply-supported boundary condition and with a crack is proposed. The beam is also supposed to be constrained at the ends by elastically flexible springs, with transverse stiffness and rotational stiffness. Applying the Timoshenko beam theory and employing the auxiliary functions, the equation of motion is derived. The natural frequencies are obtained by applying the Euler–Bernoulli method and are derived by the corresponding auxiliary functions of the governing equation of the Euler–Bernoulli beam in free vibration. For different values of soil parameter, taking into account the effects of rotational and shear deformation and considering the presence of crack, typical results are presented, in order to demonstrate the efficiency of the proposed approach. Finally the obtained results are compared with some results available in the literature. It is shown that very good results are obtained. This approach is very effective for the study of vibration problems of Timoshenko beams. The novelty of the proposed approach is that although the auxiliary functions, used to find the solution to the dynamic problem of a Timoshenko beam, are different for the two theories applied, in both cases, the dynamic problem is traced to the study of an Euler - Bernoulli beam subjected to an axial load.
Influence line (IL) has emerged as very promising damage indices for bridge damage detection. This study proposed a method to localize and quantify damage in beam structures by estimating section flexibility change from deflection IL (DIL) change. To this end, the relationship between second derivative of DIL change and flexibility change was established. To remove noise interference in measurement, piecewise quadratic functions were used to fit and replace noisy DIL change curves, wherein the coefficients of quadratic function were determined via a sparse regularization method, considering the sparsity nature of damage that typically takes place in only a limited number of elements. The feasibility and accuracy of the proposed method are verified through numerical examples and laboratory experiments. Through four hypothetical damage scenarios of a simply supported beam with one or two damaged locations, its ability to quantify minor damage and its anti-noise robustness were well verified. Finally, a laboratory experiment on a simply supported aluminum beam illustrated that the location and extent of damage could be successfully identified in the single-damage and double-damage cases. The numerical and experimental results indicate that the proposed method is promising for future damage localization and quantification of bridge structures.
This paper deals with detection of macro-level crack type damage in rectangular E-Glass fiber/Epoxy resin (LY556) laminated composite plates using modal analysis. Composite plate-like structures are widely found in aerospace and automotive structural applications which are susceptible to damages. The formation of cracks in a structure that undergoes vibration may lead to catastrophic events such as structural failure, thus detection of such occurrences is considered necessary. In this research, a novel technique called as node-releasing technique in Finite Element Analysis (FEA), which was not attempted by the earlier researchers, is used to model the perpendicular cracks (the type of damage mostly considered in all the pioneering research works) and also slant cracks (a new type of damage considered in the present work) of various depths and lengths for Unidirectional Laminate (UDL) ([0]S and [45]S) composite layered configurations using commercial FE code Ansys, thus simulating the actual damage scenario. Another novelty of the present work is that the crack is modeled with partial depth along the thickness of the plate, instead of the through the thickness crack which has been of major focus in the literature so far, in order to include the possibility of existence of the crack up to certain layers in the laminated composite structures. The experimental modal analysis is carried out to validate the numerical model. Using central difference approximation method, the modal curvature is determined from the displacement mode shapes which are obtained via finite element analysis. The damage indicators investigated in this paper are Normalized Curvature Damage Factor (NCDF) and modal strain energy-based methods such as Strain Energy Difference (SED) and Damage Index (DI). It is concluded that, all the three damage detection algorithms detect the transverse crack clearly. In addition, the damage indicator NCDF seems to be more effective than the other two, particularly when the detection is for damage inclined to the longitudinal axis of the plate. The proposed method will provide the base data for implementing online structural health monitoring of structures using technologies such as Machine Learning, Artificial Intelligence, etc.
Measurement of flexural vibrations of a cantilever beam with rectangular cross-section having a transverse surface crack extending uniformly along the width of the beam and analytical results are used to relate the measured vibration modes to the crack location and depth. From the measured amplitudes at two points of the structure vibrating at one of its natural modes, the respective vibration frequency and an analytical solution of the dynamic response, the crack location can be found and depth can be estimated with satisfactory accuracy. The method can be used to identify cracks in structures by measuring its modal characteristics. It is a non-destructive testing method for crack identification, and it is applicable to structures for which a structural analysis is available. The main features of the method are as follows: (a) it requires amplitude measurements at two positions of the structure only; (b) it is applicable to all one-dimensional structures; (c) it demands modest computational effort; (d) it is an accurate, simple and easy to handle method having the advantage that the measurements may be carried out in situ with rather simple equipment.
When a structure is subjected to damage, its dynamic response changes, characterized by shifts in the eigenvalues and modifications of the eigenvectors. Considerable effort has been put into investigating the relationship between the damage location, the damage size and the corresponding changes in the eigenparameters. In most cases, emphasis has been on using the shift in eigenvalues as a means of determining the damage location, and the information derived from the change in eigenvectors has largely remained obscure. In this paper a systematic study is presented of the relationship between damage location, damage size and the changes in the eigenvalues and eigenvectors of a cantilever when subjected to damage. A finite element model of a uniform cross sectioned cantilever was chosen to provide data for analysis. The changes in the eigenvalues and eigenvectors are shown to follow a definite trend in relation to the location and extent of damage.
The differential equation and associated boundary conditions for a nominally uniform Bernoulli-Euler beam containing one or more pairs of symmetric cracks are derived. The reduction to one spatial dimension is achieved using integrations over the cross-section after plausible stress, strain, displacement and momentum fields are chosen. In particular the perturbation in the stresses induced by the crack is incorporated through a local function which assumes an exponential decay with distance from the crack and which includes a parameter which can be evaluated by experimental tests. Some experiments on beams containing cuts to simulate cracks are briefly described and the change in the first natural frequency with crack depth is matched closely by the theoretical predictions.
The problem of free vibration of uniform beams containing a local material damage has been studied with the aim of arriving at accurate closed form analytical expressions for the natural frequency for various homogeneous boundary conditions. Elementary beam theory is used, with the local material damage modelled as an effective reduction in Young's modulus, and the exact solution of the transcendental equations for the frequency parameter is obtained for various symmetric and unsymmetric boundary conditions in terms of the identified damage parameters. The numerical results for the natural frequency show that it is possible to arrive at a simple polynomial type of representation which is the same for various boundary conditions and mode numbers. The results also show that the nature of changes in frequency with respect to the damage location is of the same type as the local curvature in the undamaged beams. Finally, the study demonstrates that the analytical expressions derived for representing the damage dependence can also be used, with slight modification, in situations where there is local stiffening.
A method of analysis of the effect of two open cracks upon the frequencies of the natural flexural vibrations in a cantilever beam is presented. Two types of cracks are considered: double-sided, occurring in the case of cyclic loadings, and single-sided, which is principle occur as a result of fluctuating loadings. It is also assumed that the cracks occur in the first mode of fracture: i.e., the opening mode. An algorithm and a numerical example are included.
An identification procedure to determine the crack characteristics (location and size of the crack) from dynamic measurements is developed and tested. This procedure is based on minimization of either the “mean-square” or the “max” measure of difference between measurement data (natural frequencies and mode shapes) and the corresponding predictions obtained from the computational model. Necessary conditions are obtained for both formulations. The method is tested for simulated damage in the form of one-side or symmetric cracks in a simply supported Bernoulli-Euler beam. The sensitivity of the solution of damage identification to the values of parameters that characterize damage is discussed.
An approximate Galerkin solution to the one-dimensional cracked beam theory developed by Christides and Barr for the free bending motion of beams with pairs of symmetric open cracks is suggested. The series of comparison functions considered in the Galerkin procedure consists of the mode shapes of corresponding uncracked beam. The number of terms in the expansion is determined by the covnergence of the natural frequencies and confirmed by studying the stress concentration profile near the crack. This approach allows the determination of the higher natural frequencies and mode shapes of the cracked beam. It is found that the Christides and Barr original solution was not fully converged and that cracks render the convergence of the Galerkin's procedure very slow by affecting the continuity characteristics of the solution of the boundary value problem. To validate the theoretical results, a two-dimensional finite element approach is proposed, which also allows one to determine the parameter that controls the stress concentration profile near the crack tip in the theoretical formulation without requiring the use of experimental results. Very good agreement between the theoretical and finite element results is observed.
Continuum modeling of large space structures is extended to the problem of detecting construction errors in large space structures such as the proposed space station. First-order dynamic sensitivity equations for structures involving eigenfrequencies, modal masses, modal stiffness, and modal damping are presented. Matrix equations relating changes in element parameters to dynamic sensitivities are summarized. The sensitivity equations for the entire dynamical system are rearranged as a system of algebraic equations with unknowns of stiffness losses at selected locations. The feasibility of the formulation is numerically demonstrated on a simply supported Euler-Bernoulli beam with simulated construction defects. The method is next extended to large space structures modeled as equivalent continua with simulated construction defects.