An Efficient Method for Crack Identification in Simply Supported Euler–Bernoulli Beams

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
L. Rubio
Keyword(s):  
Inverse Problem ◽  
Mass Density ◽  
Optimization Technique ◽  
Least Square ◽  
Closed Form Expression ◽  
Crack Identification ◽  
Cracked Beam ◽  
Identification Procedure ◽  
Simply Supported ◽  
Euler Bernoulli Beam

An effective crack identification procedure has been developed based on the dynamic behavior of a Euler–Bernoulli cracked beam. It is very well known that the presence of a crack in a structure produces a change in its frequency response that can be used to determine the crack properties (position and size) solving what is called an inverse problem. In this work, such an inverse problem has been solved by the use of the classical optimization technique of minimizing the least square criterion applied to the closed-form expression for the frequencies obtained through the perturbation method. The advantage of this method with respect to the ones derived previously is that the knowledge of the material and its properties (Young’s modulus and mass density) is not necessary, not even the behavior of the uncracked element. The methodology has been successfully applied to a simply supported Euler–Bernoulli beam.

scholarly journals Comparative study of direct and inverse problems of cracked beams

2018 ◽  
Vol 149 ◽  
pp. 02015
Author(s):  
Chettah Mahieddine ◽  
Lassoued Rachid
Keyword(s):  
Inverse Problem ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Natural Frequencies ◽  
Cracked Beam ◽  
Simply Supported Beam ◽  
Rotational Spring ◽  
Simply Supported ◽  
Cracked Structures

In recent decades, the analysis and evaluation of the cracked structures were hot spots in several engineering fields and has been the subject of great interest with important and comprehensive surveys covering various methodologies and applications, in order to obtain reliable and effective methods to maintain the safety and performance of structures on a proactive basis. The presence of a crack, not only causes a local variation in the structural parameters (e.g., the stiffness of a beam) at its location, but it also has a global effect which affects the overall dynamic behavior of the structure (such as the natural frequencies). For this reason, the dynamic characterization of the cracked structures can be used to detect damage from non-destructive testing. The objective of this paper is to compare the accuracy and ability of two methods to correctly predict the results for both direct problem to find natural frequencies and inverse problem to find crack’s locations and depths of a cracked simply supported beam. Several cases of crack depths and crack locations are investigated. The crack is supposed to remain open. The Euler–Bernoulli beam theory is employed to model the cracked beam and the crack is represented as a rotational spring with a sectional flexibility. In the first method, the transfer matrix method is used; the cracked beam is modeled as two uniform sub-segments connected by a rotational spring located at the cracked section. In the second method which is based on the Rayleigh’s method, the mode shape of the cracked beam is constructed by adding a cubic polynomial function to that of the undamaged beam. By applying the compatibility conditions at crack’s location and the corresponding boundary conditions, the general forms of characteristic equations for this cracked system are obtained. The two methods are then utilized to determine the locations and depths by using any two natural frequencies of a cracked simply supported beam obtained from measurements as inputs. The two approaches are compared with a number of numerical examples for simply supported beams including one crack. The theoretical results show that the accuracy of the Rayleigh’s method to predict natural frequencies decreases for higher modes when crack depth increases. It is also found that for the inverse problem, the transfer matrix method show a good agreement with those obtained from previous works done in this field.

Behavior of cracked Euler–Bernoulli beam and inverse problem for assessing crack severity

2021 ◽  
pp. 095745652110557
Author(s):  
Ehab Samir Mohamed Mohamed Soliman
Keyword(s):  
Inverse Problem ◽  
Mode Shape ◽  
Natural Frequencies ◽  
Crack Depth ◽  
Von Mises Stress ◽  
Simply Supported ◽  
Crack Location ◽  
Bending Mode ◽  
Von Mises ◽  
Euler Bernoulli Beam

In this present study, natural frequencies of the first two modes of bending vibration for the cracked simply supported Euler–Bernoulli beam is determined using finite element analysis (FEA). FEA natural frequencies for the cracked beam are used to investigate the behavior of the cracked beam and also used in the inverse problem of crack depth detection. Dynamic behavior of the cracked simply supported beam is observed, and it is found that normalized mode shape at crack location has great effect on amount of decreasing of natural frequencies. When normalized mode shape at crack location is increased, then natural frequencies decrease. In this study, pattern of mode shape played a vital role in decreasing or increasing natural frequencies. At the midpoint of the beam, there is largest bending moment in first bending mode and there is nodal point in second bending mode. Harmonic analysis for the cracked simply supported beam is carried out to find von Mises stress responses and appearance of peaks at frequency of first bending mode is noticed in graphs of von Mises stress response, expressing high values of von Mises stress at crack tip. Inverse problem of assessing the crack depth is performed using results of FEA first mode frequency ratio and published experimental results and the method showed good results in case of high crack depth ratios.

scholarly journals Application of the topological sensitivity method to the reconstruction of plasma equilibrium domain in a tokamak

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohamed Abdelwahed ◽  
Nejmeddine Chorfi ◽  
Maatoug Hassine ◽  
Imen Kallel
Keyword(s):  
Inverse Problem ◽  
Asymptotic Expansion ◽  
Numerical Algorithm ◽  
Shape Function ◽  
Optimization Technique ◽  
Plasma Equilibrium ◽  
Topological Sensitivity ◽  
Sensitivity Method

AbstractThe topological sensitivity method is an optimization technique used in different inverse problem solutions. In this work, we adapt this method to the identification of plasma domain in a Tokamak. An asymptotic expansion of a considered shape function is established and used to solve this inverse problem. Finally, a numerical algorithm is developed and tested in different configurations.

scholarly journals Vibration Parameters Estimation by Blade Tip-Timing in Mistuned Bladed Disks in Presence of Close Resonances

2020 ◽  
Vol 10 (17) ◽  
pp. 5930
Author(s):  
Saeed Bornassi ◽  
Christian Maria Firrone ◽  
Teresa Maria Berruti
Keyword(s):  
Frequency Response ◽  
Optimization Technique ◽  
Numerical Test ◽  
Parameters Estimation ◽  
Least Square ◽  
Response Curves ◽  
Bladed Disks ◽  
Bladed Disk ◽  
Resonance Frequencies ◽  
Blade Tip

The present paper is focused on the post processing of the data coming from the Blade Tip-Timing (BTT) sensors in the case where two very close peaks are present in the frequency response of the vibrating system. This type of dynamic response with two very close peaks can occur quite often in bladed disks. It is related to the fact that the bladed disk is not perfectly cyclic symmetric and the so called “mistuning” is present. A method based on the fitting of the BTT sensors data by means of a 2 degrees of freedom (2DOF) dynamic model is proposed. Nonlinear least square optimization technique is employed for identification of the vibration characteristics. A numerical test case based on a lump parameter model of a bladed disk assembly is used to simulate different response curves and the corresponding sensors signals. The Frequency Response Function (FRF) constructed at the resonance region is compared with the traditional Sine fitting results, the resonance frequencies and damping values estimated by the fitting procedure are also reported. Accurate predictions are achieved and the results demonstrate the considerable capacity of the 2DOF method to be used as a standalone or as a complement to the standard Sine fitting method.

TWO-DIMENSIONAL DC RESISTIVITY MAPPING FOR SUBSURFACE INVESTIGATION USING SOFT COMPUTING APPROACHES

2015 ◽  
Vol 77 (17) ◽  
Author(s):  
Herman Wahid ◽  
Mohd. Hakimi Othman ◽  
Ruzairi Abdul Rahim
Keyword(s):  
Inverse Problem ◽  
Synthetic Data ◽  
Least Square Method ◽  
Least Square ◽  
Dc Resistivity ◽  
Two Dimensional ◽  
Actual Model ◽  
Subsurface Investigation ◽  
Artificial Neural Network Ann ◽  
Mlp Model

In geophysical subsurface surveys, difficulty to interpret measurement of data obtain from the equipment are risen. Data provided by the equipment did not indicate subsurface condition specifically and deviates from the expected standard due to numerous features. Generally, the data that obtained from the laws of physics computation is known as forward problem. And the process of obtaining the data from sets of measurements and reconstruct the model is known as inverse problem. Researchers have proposed multiple estimation techniques to cater the inverse problem and provide estimation that close to actual model. In this work, we investigate the feasibility of using artificial neural network (ANN) in solving two- dimensional (2-D) direct current (DC) resistivity mapping for subsurface investigation, in which the algorithms are based on the radial basis function (RBF) model and the multi-layer perceptron (MLP) model. Conventional approach of least square (LS) method is used as a benchmark and comparative study with the proposed algorithms. In order to train the proposed algorithms, several synthetic data are generated using RES2DMOD software based on hybrid Wenner-Schlumberger configurations. Results are compared between the proposed algorithms and least square method in term of its effectiveness and error variations to the actual values. It is discovered that the proposed algorithms have offered better performance in term minimum error difference to the actual model, as compared to least square method. Simulation results demonstrate that proposed algorithms can solve the inverse problem and it can be illustrated by means of the 2-D graphical mapping.

scholarly journals Solution of inverse heat conduction equation with the use of Chebyshev polynomials

2016 ◽  
Vol 37 (4) ◽  
pp. 73-88 ◽  
Author(s):  
Magda Joachimiak ◽  
Andrzej Frąckowiak ◽  
Michał Ciałkowski
Keyword(s):  
Inverse Problem ◽  
Chebyshev Polynomials ◽  
Direct Problem ◽  
Least Square Method ◽  
Least Square ◽  
Random Disturbance ◽  
Inverse Heat Conduction ◽  
Chebyshev Nodes ◽  
The Difference ◽  
The Stability

AbstractA direct problem and an inverse problem for the Laplace’s equation was solved in this paper. Solution to the direct problem in a rectangle was sought in a form of finite linear combinations of Chebyshev polynomials. Calculations were made for a grid consisting of Chebyshev nodes, what allows us to use orthogonal properties of Chebyshev polynomials. Temperature distributions on the boundary for the inverse problem were determined using minimization of the functional being the measure of the difference between the measured and calculated values of temperature (boundary inverse problem). For the quasi-Cauchy problem, the distance between set values of temperature and heat flux on the boundary was minimized using the least square method. Influence of the value of random disturbance to the temperature measurement, of measurement points (distance from the boundary, where the temperature is not known) arrangement as well as of the thermocouple installation error on the stability of the inverse problem was analyzed.

A Comparison of Two Solution Techniques for the Inverse Problem of Simultaneously Estimating the Spatial Variations of Diffusion Coefficients and Source Terms

2003 ◽  
Author(s):  
Marcelo J. Colac¸o ◽  
Helcio R. B. Orlande ◽  
George S. Dulikravich ◽  
Fabio A. Rodrigues
Keyword(s):  
Inverse Problem ◽  
Gradient Method ◽  
Optimization Technique ◽  
Adjoint Problem ◽  
Diffusion Problem ◽  
Hybrid Optimization ◽  
Simultaneous Estimation ◽  
Random Errors ◽  
Flow Detection ◽  
Solution Techniques

This work deals with the simultaneous estimation of the spatially varying diffusion coefficient and of the source term distribution in a one-dimensional nonlinear diffusion problem. This work can be physically associated with the detection of material non-homogeneities such as inclusions, obstacles or cracks, heat conduction, groundwater flow detection, and tomography. Two solution techniques are applied in this paper to the inverse problem under consideration, namely: the conjugate gradient method with adjoint problem and a hybrid optimization algorithm. The hybrid optimization technique incorporates several of the most popular optimization modules; the Davidon-Fletcher-Powell (DFP) gradient method, a genetic algorithm (GA), the Nelder-Mead (NM) simplex method, quasi-Newton algorithm of Pshenichny-Danilin (LM), differential evolution (DE), and sequential quadratic programming (SQP). The accuracy of the two solution approaches was examined by using simulated transient measurements containing random errors in the inverse analysis.

ON DETECTION OF DEFECTS IN BEAMS AND TRUSSES FROM DYNAMIC RESPONSES

2000 ◽  
Vol 24 (1A) ◽  
pp. 1-31
Author(s):  
S. Lukasiewicz ◽  
K. Palka
Keyword(s):  
Bending Stiffness ◽  
Least Square ◽  
Dynamic Responses ◽  
Axial Stiffness ◽  
Mathematical Form ◽  
The Finite Element Method ◽  
Identification Procedure ◽  
Identification Method ◽  
Model Equations ◽  
Source Of Information

This paper presents an identification method to detect cracks and corroded members in vibrating structures. The mathematical identification procedure based on the least square technique uses the measured dynamic response of a structure as the source of information. The application of the Finite Element Method (FEM) for the representation of all constraints and model equations allows presentation of the identification process in a simple and very efficient mathematical form. Propagation of cracks and other failures of the members cause changes in the bending and axial stiffness of the members. One can detect the crack by observing the change in the bending stiffness caused by the closing and opening of the crack in two different configurations. The proposed identification method provides highly precise calculated results which allows detection of small changes in the bending stiffness of the members resulting from cracks and corrosion. The method was tested on simulated experimental data.

Effect of a breathing crack on the damping changes in nonlinear vibrations of a cracked beam: Experimental and theoretical investigations

2020 ◽  
pp. 107754632096031
Author(s):  
Masoud Kharazan ◽  
Saied Irani ◽  
Mohammad Ali Noorian ◽  
Mohammad Reza Salimi
Keyword(s):  
Nonlinear Vibrations ◽  
Crack Depth ◽  
Beam Theory ◽  
Experimental Tests ◽  
Damping Coefficient ◽  
Crack Identification ◽  
Breathing Crack ◽  
Cracked Beam ◽  
Identification Method ◽  
Nonlinear Crack

The attempts to identify damping changes in a cracked beam can improve the accuracy of the nonlinear crack identification method. For the purpose of this aim, a parametric nonlinear equation of motion is obtained using the Euler–Bernoulli beam theory and parametric nonlinear breathing crack assumptions. Several experiments were conducted to identify the effect of breathing cracks on changing the damping value in nonlinear vibrations of a cracked beam. Experimental tests have revealed that increasing the crack depth and the level of excitation enlarges the damping coefficient in a vibrating beam. For this reason, the effects of the excitation force and crack depth on the structural damping coefficient are investigated. The obtained results indicated that considering the nonlinear response of a cracked beam and measuring the value of the damping changes can significantly improve the accuracy of the nonlinear crack identification method.

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