# Identification of Transverse Crack in a Cracked Cantilever Beam by

Identification of Transverse Crack in a Cracked Cantilever Beam Using Fuzzy Logic and Kohonen Network A Thesis Submitted in Partial Fulfillment of the Requirements for the Award of the Degree of Master of Technology in Mechanical Engineering (Machine Design & Analysis) by Sasanka Choudhury Roll No.209ME1259 NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA राष्ट्रीय प्रौधोगिकी संस्थान राउरकेऱा PIN-769008 ODISHA, INDIA June 2011 Identification of Transverse Crack in a Cracked Cantilever Beam Using Fuzzy Logic and Kohonen Network A Thesis Submitted in Partial Fulfillment of the Requirements for the Award of the Degree of Master of Technology in Mechanical Engineering (Machine Design & Analysis) by Sasanka Choudhury Roll No.209ME1259 Under the Supervision of Dr. Dayal Ramakrushna Parhi NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA राष्ट्रीय प्रौधोगिकी संस्थान राउरकेऱा PIN-769008 ODISHA, INDIA June 2011 To Bhagawan Shri Sathyasai Baba & My beloved Parents National Institute of Technology Rourkela Certificate This is to certify that the thesis entitled, “Identification of Transverse Crack in a Cracked Cantilever Beam using Fuzzy Logic and Kohonen Network” submitted by Mr. Sasanka Choudhury to National Institute of Technology Rourkela, is a record of bonafide research work carried out by him under my supervision and is worthy of consideration for the award of the degree of Masters of Technology in Mechanical Engineering with specialization in Machine Design and Analysis. The embodiment of this thesis has not been submitted to any other University and/or Institute for the award of any degree or diploma. Date: Dr. Dayal Ramakrushna Parhi Professor Dept. of Mechanical Engineering National Institute of Technology Rourkela-769008 ACKNOWLEDGEMENT First, I offer my humble pranam on the lotus feet of Bhagawan Sri Sathya Sai Baba for his subtle support in all difficulties through out my life. Its my wishes to express my gratitude towards my parents for their eternal effort. I would like to express my greatest appreciation to my supervisor Dr. Dayal Ramakrushna Parhi for his encouragement and guidance, timely suggestions during the completion of the project. The discussions held with him during the research work, which helped me a lot in materializing this dissertation work and also to grow up my knowledge. My sincere thank also to Prof. R. K. Sahoo and Prof. N. Kavi, for their valuable suggestions and comments. I mention the help and support received from my dearest friends. The guidance and support received from all the members who contributed and who else part of this project, whose names could have not been mentioned here, is highly acknowledged. I owe my indebtedness to my younger brother Satish and elder brother Sanjib for their moral support and encouragement. Last but not least, I deeply appreciate the help and guidance of Prof. Sarat Kumar Patra in every time I need during my stay at NIT Rourkela. Sasanka Choudhury i ABSTRACT The issue of crack detection and diagnosis has gained wide spread industrial interest. Crack/damage affects the industrial economic growth. Generally damage in a structural element may occur due to normal operations, accidents, deterioration or severe natural events such as earth quake or storms. Damage can be analyzed through visual inspection or by the method of measuring frequency, mode shape and structural damping. Damage detection by visual inspection is a time consuming method and measuring of mode shape as well as structural deflection is difficult rather than measuring frequency. As Nondestructive method for the detection of crack is favorable as compared to destructive methods. So, our analysis has been made on the basis of non-destructive methods with the consideration of natural frequency. Here the crack is transverse surface crack. In the current analysis, methodologies have been developed for damage detection of a cracked cantilever beam using analytical, fuzzy logic, kohonen network as well as experimental. Theoretical analysis has been carried out to calculate the natural frequency with the consideration of mass and stiffness matrices. The data obtained from theoretical analysis has been fed to fuzzy controller as well as the kohonen competitive learning network. The Fuzzy Controller uses the different membership functions as input as well as output. The input parameters to the Fuzzy Controller are the first three natural frequencies. The output parameters of the fuzzy controller are the relative crack depth and relative crack location. Several Fuzzy rules have been trained to obtain the results for relative crack depth and relative crack location. Kohonen network is nothing but a competitive learning network is used here for the detection of crack depth and location. It is processed through a vector quantization algorithm. A comparative study has been made between fuzzy logic technique and Kohonen network technique after experimental verification. It has been observed that the process of kohonen network can predict the depth and location accurately as close to fuzzy logic technique. ii CONTENTS Acknowledgement................................................................................................................... i Abstract .................................................................................................................................. ii Contents ................................................................................................................................. iii List of Tables ......................................................................................................................... vi List of Figures ...................................................................................................................... vii Nomenclature ........................................................................................................................ ix Chapter 1 Introduction .......................................................................................................... 1 1.1. Theme of Thesis .......................................................................................................... 2 1.2. Motivation of Work .................................................................................................... 4 1.3. Thesis Layout ............................................................................................................. 5 Chapter 2 Literature Survey ................................................................................................. 7 2.1. Introduction ................................................................................................................. 8 2.2. Overview .................................................................................................................... 8 2.3. Methodologies uses in the area of research on crack detection .................................. 9 Chapter 3 Theoretical vibration analysis for identification of crack ............................. 21 3.1. Introduction ............................................................................................................... 22 3.2. Local flexibility of a cracked cantilever beam under axial load and bending moment ......................................................................................................................................... 22 3.3. Analysis of vibration characteristics of a cracked cantilever beam........................... 25 3.3.1. Analysis of free vibration .............................................................................. 25 3.3.2. Analysis of forced vibration .......................................................................... 27 3.4. Finite element formulation ........................................................................................ 27 3.4.1. Theory............................................................................................................ 27 3.4.2. Governing equations ...................................................................................... 28 3.4.3. Process of crack detection ............................................................................. 30 Chapter 4 Analysis of Fuzzy Inference System for Identification of Crack .................. 31 4.1. Introduction ............................................................................................................... 32 iii 4.2. Fuzzy sets and membership functions ....................................................................... 33 4.3. Fuzzy inference system ............................................................................................. 33 4.3.1. Fuzzy linguistic variables .............................................................................. 34 4.3.2. Fuzzy controller/ Fuzzy If-then rule .............................................................. 35 4.3.3. Creating Fuzzy rules ...................................................................................... 35 4.3.4. Fuzzification .................................................................................................. 36 4.3.5. Defuzzification of output distribution ........................................................... 36 4.4. Fuzzy mechanism used for localization and identification of crack ......................... 37 4.5. Function of fuzzy controller for localization and identification of crack .................. 39 4.5.1. Discussion...................................................................................................... 45 4.5.2. Comparison of Results .................................................................................. 52 4.5.3. Summary........................................................................................................ 52 4.6. Why fuzzy logic is used ............................................................................................ 52 Chapter 5 Analysis of Kohonen Network for identification of crack .............................. 54 5.1. Introduction ............................................................................................................... 55 5.2. Essential processes of kohonen network/ the SOM training Algorithm ................... 56 5.2.1. Initialization ................................................................................................... 56 5.2.2. Competitive Process ...................................................................................... 57 5.2.3. Co-operative Process ..................................................................................... 58 5.2.4. Synaptic adaptation Process .......................................................................... 58 5.3. Mechanism ................................................................................................................ 59 5.3.1. Competition Mechanism ............................................................................... 59 5.3.2. Co-operative Mechanism............................................................................... 59 5.3.3. Adaptive Mechanism ..................................................................................... 60 5.4. Flow chart of Kohonen network ............................................................................... 61 5.5. Comparison of Results .............................................................................................. 62 5.5.1. Discussion...................................................................................................... 62 5.5.2. Summary........................................................................................................ 62 Chapter 6 Experimental Setup for identification of crack ............................................... 63 6.1. Introduction ............................................................................................................... 64 6.2. Experimental Setup ................................................................................................... 64 6.2.1. Instruments Used ........................................................................................... 65 iv 6.2.2. Description .................................................................................................... 68 6.3. Discussion.................................................................................................................. 68 Chapter 7 Results and Discussions ..................................................................................... 69 7.1. Introduction ............................................................................................................... 70 7.2. Discussions ................................................................................................................ 70 7.3. Comparison of Results .............................................................................................. 72 7.4. Characteristic Curves................................................................................................. 72 Chapter 8 Conclusions ......................................................................................................... 75 8.1. Conclusions ............................................................................................................... 76 8.2. Applications ............................................................................................................... 77 8.3. Scope for Future Work .............................................................................................. 77 References ............................................................................................................................ 78 Publications .......................................................................................................................... 83 v LIST OF TABLES Table: 4.1 Linguistic terms used for fuzzy membership functions ...................................... 46 Table: 4.2 Fuzzy rules for fuzzy inference system ............................................................... 47 Table: 4.3 Comparison of results between theoretical analysis and different fuzzy controller analysis ................................................................................................................ 52 Table: 5.1 Comparison of results between theoretical analysis and Kohonen Network technique.............................................................................................................. 62 Table: 6.1 Instruments used in the experimental analysis .................................................... 66 Table: 7.1 Comparison of results between theoretical analysis, experimental analysis, fuzzy controller analysis and kohonen network technique............................................ 72 vi LIST OF FIGURES Figure: 3.1 Geometry of cracked cantilever beam ................................................................. 22 Figure: 3.2 Beam model ......................................................................................................... 25 Figure: 3.3 Representation of a single crack cantilever beam ............................................... 30 Figure: 4.1 Schematic diagram of operation of a fuzzy inference system ........................... 34 Figure: 4.2 Fuzzy controller architecture ............................................................................. 36 Figure: 4.3 Schematic diagram of fuzzy inference system ................................................... 39 Figure: 4.4 (a)Triangular Membership functions for relative natural frequency for first mode of vibration ................................................................................................ 40 Figure: 4.4 (b) Triangular Membership functions for relative natural frequency for second mode of vibration ................................................................................................ 40 Figure: 4.4 (c) Triangular Membership functions for relative natural frequency for third mode of vibration ................................................................................................ 40 Figure: 4.4 (d) Triangular Membership functions for relative crack location ...................... 41 Figure: 4.4 (e) Triangular Membership functions for relative crack depth .......................... 41 Figure: 4.5 (a) Trapezoidal Membership functions for relative natural frequency for first mode of vibration ................................................................................................ 41 Figure: 4.5 (b) Trapezoidal Membership functions for relative natural frequency for second mode of vibration ................................................................................................ 42 Figure: 4.5 (c) Trapezoidal Membership functions for relative natural frequency for third mode of vibration ................................................................................................ 42 Figure: 4.5 (d) Trapezoidal Membership functions for relative crack location.................... 42 Figure: 4.5 (e) Trapezoidal Membership functions for relative crack depth ........................ 42 Figure: 4.6 (a) Gaussian Membership functions for relative natural frequency for first mode of vibration .......................................................................................................... 43 Figure: 4.6 (b) Gaussian Membership functions for relative natural frequency for second mode of vibration ................................................................................................ 43 Figure: 4.6 (c) Gaussian Membership functions for relative natural frequency for third mode of vibration ............................................................................................... 43 Figure: 4.6 (d) Gaussian Membership functions for relative crack location ........................ 43 Figure: 4.6 (e) Gaussian Membership functions for relative crack depth ............................ 44 vii Figure: 4.7 (a) Hybrid Membership functions for relative natural frequency for first mode of vibration .......................................................................................................... 44 Figure: 4.7 (b) Hybrid Membership functions for relative natural frequency for second mode of vibration ................................................................................................ 44 Figure: 4.7 (c) Hybrid Membership functions for relative natural frequency for third mode of vibration .......................................................................................................... 44 Figure: 4.7 (d) Hybrid Membership functions for relative crack location ........................... 45 Figure: 4.7 (e) Hybrid Membership functions for relative crack depth................................ 45 Figure: 4.8 Resultant values of relative crack depth and relative crack location of triangular membership function when Rules 3 and 12 of Table: 4.2 are activated ............. 48 Figure: 4.9 Resultant values of relative crack depth and relative crack location of trapezoidal membership function when Rules 3 and 12 of Table: 4.2 is activated ............................................................................................................................. 49 Figure: 4.10 Resultant values of relative crack depth and relative crack location of Gaussian membership function when Rules 3 and 12 of Table: 4.2 are activated ............. 50 Figure: 4.11 Resultant values of relative crack depth and relative crack location of hybrid membership function when Rules 3 and 12 of Table: 4.2 are activated ............. 51 Figure: 5.1 Possible Architectures in Kohonen Network ..................................................... 56 Figure: 5.2 Process of Initialization in Kohonen Network ................................................... 57 Figure: 5.3 Process of Competition in Kohonen Network ................................................... 57 Figure: 5.4 Process of Co-operative in Kohonen Network .................................................. 58 Figure: 5.5 Flow chart showing the processes of Kohonen Network .................................. 61 Figure: 6.1 (a) Pictorial view of complete assembly of Experimental setup1 ...................... 65 Figure: 6.1 (b) Pictorial view of complete assembly of Experimental setup2 ..................... 65 Figure: 7.1 Relative first natural frequencies versus relative crack depth............................ 72 Figure: 7.2 Relative first natural frequencies versus relative crack location........................ 73 Figure: 7.3 Relative second natural frequencies versus relative crack depth ....................... 73 Figure: 7.4 Relative second natural frequencies versus relative crack location ................... 73 Figure: 7.5 Relative third natural frequencies versus relative crack depth .......................... 74 Figure: 7.6 Relative third natural frequencies versus relative crack location ...................... 74 viii NOMENCLATURE a1 = Depth of crack A = Cross-sectional area of the beam Ai i = 1to 12 = Unknown coefficients of matrix A B = Width of the beam B1 = Vector of exciting motion Cu E = ( )1 / 2 Cy =( E = Young‟s modulus of elasticity of the beam material fnf = Relative first natural frequency Fi i = 1, 2 = Experimentally determined function i, j = Variables J = Strain-energy release rate EI 1 / 2 ) K1,i i = 1, 2 = Stress intensity factors for Pi loads L Cu Ku = Ky L2 = Cy Kij = Local flexibility matrix elements L = Length of the beam L1 = Location (length) of the crack from fixed end Mi i=1, 4 = Compliance constant Pi i=1, 2 = Axial force (i=1), bending moment (i=2) Q = Stiff-ness matrix for free vibration. Q1 = Stiff-ness matrix for forced vibration rcd = Relative crack depth rcl = Relative crack location 1/ 2 ix snf = Relative second natural frequency tnf = Relative third natural frequency ui i=1, 2 = Normal functions (longitudinal) ui(x) x = Co-ordinate of the beam y = Co-ordinate of the beam Y0 = Amplitude of the exciting vibration yi i=1, 2 = Normal functions (transverse) yi(x) W = Depth of the beam ω = Natural circular frequency β = Relative crack location = Aρ ρ = Mass-density of the beam ξ1 = Relative crack depth V = Aggregate (union) = Minimum (min) operation = For every argj max = Argument of maximum value argj min x wj = Argument of minimum value L1 L a1 W = Input vector = Weight Vector h j,i ( x ) d 2 j,i = exp 2 2 h j,i ( x ) =Neighborhood Function dj,i = Lateral distance between the winning neuron „i‟ and excited neuron j =Width of Gaussian function ( t ) t 0 exp 1 1 = Time Constant t = Number of iterations x t = Learning Rate 2 = Time Constant BMU = Best Matching Unit xi INTRODUCTION 1.1. Theme of Thesis 1.2. Motivation of Work 1.3. Thesis Layout National Institute of Technology Rourkela 1 Chapter 1 Introduction CHAPTER 1 Introduction 1.1. Theme of Thesis Damage is one of the important aspects in structural analysis because of safety reason as well as economic growth of the industries. Generally damage in a structural element may occur due to normal operations, accidents, deterioration or severe natural events such as earth quake or storms. To achieve their industrial goal, now a days the plants as well as industries are running round the clock fully. During operation, all structures are subjected to degenerative effects that may cause initiation of structural defects such as cracks which, as time progresses, lead to the catastrophic failure or breakdown of the structure. Thus, the importance of inspection in the quality assurance of manufactured products is well understood. To avoid the unexpected or sudden failure earlier crack detection is essential. Taking this ideology into consideration crack detection is one of the most important domains for many researchers. This is basically appears in the vibrating structures while undergoes operations. The most common structural defect is the existence of a crack in machine member. The presence of a crack could not only cause a local variation in the stiffness but it could affect the mechanical behavior of the entire structure to a considerable extent. It has been observed that for damage/crack detection non-destructive testing is preferable over destructive testing. Many researchers have carried out different nondestructive methodologies for crack detection but it has been observed that the vibration based method is fast and inexpensive for crack/damage identification. Vibration- based methods can be classified into two categories: linear and nonlinear approaches. Linear approaches detect the presence of cracks in a target object by monitoring changes in the resonant frequencies in the mode shapes or in the damping factors. Depending on the assumptions, the type of analysis, the overall beam characteristics and the kind of loading or excitation, a number of research papers containing a variety of different approaches have been reported in the relevant literature. Damage can be analyzed through visual inspection or by the method of measuring frequency, mode shape and structural damping. Damage detection by visual inspection is a time consuming method and measuring of mode shape as National Institute of Technology Rourkela 2 Chapter 1 Introduction well as structural deflection is difficult rather than measuring frequency. The presence of crack induces local flexibility, which affects the dynamic behavior of the whole structure as a result the reduction occurs in natural frequency and mode shape. By considering the changes in those parameters crack can be identified in terms of crack depth and crack location. Many researches have been carried out their research works using open crack models, which means they have considered that a crack remains open during vibration. The assumption of an open crack leads to a constant shift of natural frequencies of vibration. Numerous methodologies investigated over last few decades, however, indicate that a real fatigue crack opens and closes during vibration. It exhibits non-linear behavior due to the variation of the stiffness which occurs during the response cycle. As a result, a breathing crack gives rise to natural frequencies falling between those corresponding to the open and closed states. Therefore, if an always open crack is assumed, the decrease in experimental natural frequencies will lead to an underestimation of the crack depth. Beams are one of the most commonly used structural elements in several engineering applications and experience a wide variety of static and dynamic loads. Cracks may develop in beam-like structures due to such loads. Considering the crack as a significant form of such damage, its modeling is an important step in studying the behavior of damaged structures. Knowing the effect of crack on stiffness, the beam or shaft can be modeled using either Euler-Bernoulli or Timoshenko beam theories. The beam boundary conditions are used along with the crack compatibility relations to derive the characteristic equation relating the natural frequency, the crack depth and location with the other beam properties. In the past few years the problem of health monitoring and fault detection of structures has received considerable deliberation. The changes can be considered as an indication of the health of the structure. Subsequently, these methods of fault detection are based on the comparison of the vibrant response of the healthy structure with the active response of the deserted structure. The evaluation is carried out through some algorithm, which employs the modal data of the healthy and deserted structure. Therefore, the fault detection problem is in need of the modal data for the healthy structure, the modal data for the deserted structure and the algorithm that uses these data and provides information about the state of the National Institute of Technology Rourkela 3 Chapter 1 Introduction structure. Each of these items has its own aspects and associated problems that affect the results of the fault detection. The most common structural defect is the existence of a crack. Cracks are present in structures due to various reasons. The presence of a crack could not only cause a local variation in the stiffness but it could affect the mechanical behavior of the entire structure to a considerable extent. Cracks present in vibrating/rotating components could lead to catastrophic failure. They may also occur due to mechanical defects. Another group of cracks are initiated during the manufacturing processes. Generally they are small in sizes. Such small cracks are known to propagate due to fluctuating stress conditions. If these propagating cracks remain undetected and reach their critical size, then a sudden structural failure may occur. Hence it is possible to use natural frequency measurements to detect cracks. To help in a continuous safety assessment of a machine or structure it is very necessary to constantly assess the health of its critical components. This calls for a continuous assessment of changes in their static and/or dynamic behavior. The development of a crack does not necessarily make a component instantly useless, but it is a signal that its behavior has to be monitored more carefully. Such monitoring can play a significant role in assuring an uninterrupted operation in service by the component. A direct procedure is difficult for crack identification and unsuitable in some particular cases, since they require minutely detailed periodic inspections, which are very costly. In order to avoid these costs, researchers are working on more efficient procedure in crack detection through vibration analysis. 1.2. Motivation of Work The main objective of the present research work is to develop an organized structure for damage detection of a cracked cantilever beam using fuzzy logic technique as well as kohonen network technique. These techniques are usually called the intelligent techniques because the techniques can be processed without human intervention. It is necessary that structures must safely work during its service life but, damages initiate a breakdown period on the structures which directly affect the industrial growth. It is a recognized fact that dynamic behavior of structures changes due to presence of crack. It has been observed that the presence of cracks in structures or in machine members lead to operational problem as well as premature failure. A number of researchers around the earth National Institute of Technology Rourkela 4 Chapter 1 Introduction are working on structural dynamics and particularly on dynamic characteristics of structures with the presence of crack. The information available on dynamic characteristics of crack is not so exhaustive for real application. Therefore an effort has been given to formulate some Intelligent Techniques for localization and identification of crack in cantilever beam structures. Due to presence of cracks the dynamic characteristics of structure changes. The change in dynamic behavior has been utilized as one of the criteria of fault diagnosis for structures. Major characteristics of the structure which undergo change due to presence of crack are; natural frequencies, the amplitude responses due to vibration and the mode shapes. The phases of the process plan for the present investigation are as follows: Theoretical expressions have been developed for free and forced Vibration analysis of the single cracked cantilever beam for the detection of crack depth and crack location. In all these theoretical approach transverse crack has been analyzed. Experimental Analysis has been performed to obtain the relative values of first, second and third modal natural frequencies. The data obtained from the theoretical as well as experimental analysis has been trained to fuzzy controller for designing the rule base for the detection of crack depth and crack location. Using the same data the kohonen competitive learning network has been developed for the localization and identification of crack. Finally a comparative study has been made between fuzzy logic technique and Kohonen network technique after experimental verification. 1.3. Thesis Layout The research work has been delineated in this thesis by dividing eight chapters. Following the Introduction presented in the Chapter 1, the Chapter 2 followed the literature survey which contains the previous studies had been made in the analysis of cracked structure using vibrational techniques, finite element analysis, fuzzy logic techniques, neuro genetic techniques and the application of kohonen network. Chapter 3 analyses the theoretical expressions obtained for free and forced vibration for computing the localization and identification of crack of a single crack cantilever beam. In this analysis the transverse crack has been taken into consideration. National Institute of Technology Rourkela 5 Chapter 1 Introduction Chapter 4 defines the concept of the fuzzy logic and outlines the methodology used to design an intelligent fuzzy logic controller to train the data obtain from the theoretical as well as experimental analysis for prediction of relative crack location and relative crack depth. A comparative study has been made between fuzzy logic technique and Kohonen network technique after experimental verification. Chapter 5 defines the concept of kohonen network. The data obtained from theoretical analysis has been fed to the kohonen competitive learning network. Kohonen network is nothing but a competitive learning network is used here for the detection of crack depth and location. It is processed through a vector quantization algorithm. Chapter 6 outlines the details of the developed experimental set-up for vibration analysis along with the specifications. Chapter 7 sketches the results and discussion of the chapters mentioned above. Chapter 8 highlights the conclusions drawn from the research work and the scope of the future work. The reference taken for the research work and paper published related to the research area has been listed in the last section of the chapter. National Institute of Technology Rourkela 6 LITERTURE SURVEY 2.1. Introduction 2.2. Overview 2.3. Methodologies uses in the area of research on crack detection National Institute of Technology Rourkela 7 Chapter 2 Literature Survey CHAPTER 2 Literature Survey 2.1. Introduction Now a day‟s crack detection is one of the most important areas of the research. Most of the researchers are doing their research work related to crack detection using various techniques. This chapter highlights the various methodologies uses by researchers in their era of research in the last few decades. The area of research basically includes the theoretical approach, experimental verification and the intelligent techniques. 2.2. Overview Cracks are a potential source of catastrophic failure in mechanical machines and in civil structures and in aerospace engineering. To avoid the failure caused by cracks, many researchers have performed extensive investigations over the years to develop structural integrity monitoring techniques. Most of the techniques are based on vibration measurement and analysis because, in most cases, vibration based methods can offer an effective and convenient way to detect fatigue cracks in structures. It is always require that structures must safely work during its service life, however damage initiates a breakdown period on the structures. It is unanimous that cracks are among the most encountered damage types in structures. Crack in structures may be hazardous due their dynamic loadings. So crack detection plays an important for structural health monitoring applications. Many researchers have used the free and forced vibration techniques for developing procedures for crack detection. The eventual goal of this research is to establish new methodologies which will predict the crack location and crack depth in a dynamically vibrating structure with the help of intelligence technique with considerably less computational time and high precision. This chapter recapitulates the previous works, mostly in computational methods for structures, and discusses the possible ways for research. National Institute of Technology Rourkela 8 Chapter 2 Literature Survey 2.3. Methodologies uses in the area of research on crack detection Das and Parhi [1] have performed an analytical study on fuzzy inference system for detection of crack location and crack depth of a cracked cantilever beam structure using six input parameters to the fuzzy member ship functions. The six input parameters are percentage deviation of first three natural frequencies and first three mode shapes of the cantilever beam. The two output parameters of the fuzzy inference system are relative crack depth and relative crack location. Experimental setup has been developed for verifying the robustness of the developed fuzzy inference system. The developed fuzzy inference system can predict the location and depth of the crack in a close proximity to the real results. Suresh et al. [2] have considered the flexural vibration in a cantilever beam having a transverse surface crack. He has stated that modal frequency parameters are analytically computed for various crack locations and depths using a fracture mechanics based crack model. These computed modal frequencies are used to train a neural network to identify both the crack location and depth presented in this paper. First, the crack location is identified with computed modal frequency parameters. Next, the crack depth is identified with computed modal frequency parameters and the identified crack location. Pawar et al. [3] have performed a composite matrix cracking model, which is implemented in a thin-walled hollow circular cantilever beam using an effective stiffness approach. Using these changes in frequencies due to matrix cracking, a genetic fuzzy system for crack density and crack location detection is generated. The genetic fuzzy system combines the uncertainty representation characteristics of fuzzy logic with the learning ability of genetic algorithm. It is observed that the success rate of the genetic fuzzy system in the presence of noise is dependent on crack density (level of damage). It is found that the genetic fuzzy system shows excellent damage detection and isolation performance, and is robust to presence of noise in data. Das and Parhi [4] have investigated the detection of the cracks in beam structures using the fuzzy Gaussian inference technique. Fuzzy-logic controller here used six input parameters and two output parameters. The input parameters to the fuzzy controller are the relative divergence of the first three natural frequencies and first three mode shapes in dimensionless forms. The output parameters of the fuzzy controller are the relative crack depth and relative crack location in dimensionless forms. The strain-energy release rate has been used for calculating the local stiffnesses of the beam for a mode-I type of the crack. Several boundary conditions are outlined that take into National Institute of Technology Rourkela 9 Chapter 2 Literature Survey account the crack location. The developed fuzzy controller can predict the location and depth of the crack in close proximity with the real results. Taghi et al. [5] have proposed a method in which damage in a cracked structure was analyzed using genetic algorithm technique. For modeling the cracked-beam structure an analytical model of a cracked cantilever beam was utilized and natural frequencies were obtained through numerical methods. A genetic algorithm is utilized to monitor the possible changes in the natural frequencies of the structure. The identification of the crack location and depth in the cantilever beam was formulated as an optimization problem. Bakhary et al. [6] applied Artificial Neural Network for damage detection. In his investigation an ANN model was created by applying Rosenblueth‟s point estimate method verified by Monte Carlo simulation, the statistics of the stiffness parameters were estimated. The probability of damage existence (PDE) was then calculated based on the probability density function of the existence of undamaged and damaged states. The developed approach was applied to detect simulated damage in a numerical steel portal frame model and also in a laboratory tested concrete slab. The effects of using different severity levels and noise levels on the damage detection results are discussed. Maity and Saha [7] have presented a method called damage assessment in structures from changes in static parameter using neural network. The basic strategy applied in this study was to train a neural network to recognize the behavior of the undamaged structure as well as of the structure with various possible damaged states. When this trained network was subjected to the measured response; it was able to detect any existing damage. The idea was applied on a simple cantilever beam. Strain and displacement were used as possible candidates for damage identification by a back propagation neural network and the superiority of strain over displacement for identification of damage has been observed. Structural damage detection using neural network with learning rate improvement performed by Fang et al. [8] In this study, he has been explore the structural damage detection using frequency response functions (FRFs) as input data to the back-propagation neural network (BPNN).Neural network based damage detection generally consists of a training phase and a recognition phase. Error back-propagation algorithm incorporating gradient method can be applied to train the neural network, whereas the training efficiency heavily depends on the learning rate. While various training algorithms, such as the dynamic steepest descent (DSD) algorithm and the fuzzy steepest descent (FSD) algorithm, National Institute of Technology Rourkela 10 Chapter 2 Literature Survey have shown promising features (such as improving the learning convergence speed), their performance is hinged upon the proper selection of certain control parameters and control strategy. In this paper, a tunable steepest descent (TSD) algorithm using heuristics approach, which improves the convergence speed significantly without sacrificing the algorithm simplicity and the computational effort, is investigated. The analysis results on a cantilevered beam show that, in all considered damage cases (i.e., trained damage cases and unseen damage cases, single damage cases and multiple-damage cases), the neural network can assess damage conditions with very good accuracy. Saridakis [9] applied neural networks, genetic algorithms and fuzzy logic for the identification of cracks in shafts by using coupled response measurements. In his research the dynamic behavior of a shaft with two transverse cracks characterized by three measures: position, depth and relative angle. Both cracks were considered to lie along arbitrary angular positions with respect to the longitudinal axis of the shaft and at some distance from the clamped end. A local compliance matrix of two degrees of freedom (bending in both the horizontal and the vertical planes) was used to model each crack. The calculation of the compliance matrix was based on established stress intensity factor expressions and was performed for all rotation angles through a function that incorporated the crack depth and position as parameters. Towards this goal, five different objective functions were proposed and validated; two of these were based on fuzzy logic. More computational intelligence was added through a genetic algorithm, which was used to find the characteristics of the cracks through artificial neural networks that approximate the analytical model. Both the genetic algorithm and the neural networks contributed to a remarkable reduction of the computational time without any significant loss of accuracy. The final results showed that the proposed methodology may constitute an efficient tool for real-time crack identification. An approach for crack detection in beam like structures using RBF (Radial Basis Function) neural network have been performed by Huijian et al. [10] with an experimental validation. In the particular research a crack damage detection algorithm was presented using a combination of global and local vibration-based analysis data as input in artificial neural networks (ANNs) for location and severity prediction of crack damage in beam like structures. Finite element analysis has been used to obtain the dynamic characteristics of intact and damaged cantilever steel beams for the first three natural modes. In the experimental analysis, several steel beams with six distributed surface bonded electrical strain gauges and an accelerometer mounted at the tip have been used to obtain modal National Institute of Technology Rourkela 11 Chapter 2 Literature Survey parameters such as resonant frequencies and strain mode shapes. Finally, the Radial Basis Function ANNs have been trained using the data obtained from the numerical damage case to predicate the severity and localization of the crack damage. Buezasa et al. [11] has been investigating the crack detection in structural elements by means of a generic algorithm optimization method. The present study deals with two and three dimensional models to handle the dynamics of a structural element with a transverse breathing crack. The methodology is not restricted to beam-like structures since it may be applied to any arbitrary shaped 3D element. The crack is simulated as a notch or a wedge with a unilateral Signorini‟s contact model. The contact may be partial or total. All the simulations are carried out using the partial differential solver of the general purpose, finite element code FlexPDE. A genetic algorithm (GA) optimization method is successfully employed for the crack detection. The dynamic responses at some points of the damaged structures are compared with the solution of the computational (FE) model using least squares for each proposed crack depth and location. An objective function arises which is then optimized to obtain an estimate of both parameters. Physical experiments were performed with a cantilever damaged beam and the resulting data used as input in the detection algorithm. Panigrahi et al. [12] has firstly formulate of an objective function for the genetic search optimization procedure along with the residual force method are presented for the identification of macroscopic structural damage in an uniform strength beam. Two cases have been investigated here. In the first case the width is varied keeping the strength of beam uniform throughout and in the second case both width and depth are varied to represent a special case of uniform strength beam. The developed model requires experimentally determined data as input and detects the location and extent of the damage in the beam. Here, experimental data are simulated numerically by using finite element models of structures with inclusion of random noise on the vibration characteristics. It has been shown that the damage may be identified for the said problems with a good accuracy. Chou et al. [13] stated that, the problem is initially formulated as an optimization problem, which is then solved by using genetic algorithm (GA). Static measurements of displacements at few degrees of freedom (DOFs) are used to identify the changes of the characteristic properties of structural members such as Young's modulus and cross-sectional area, which are indicated by the difference of measured and computed responses. In order to avoid structural analyses in fitness evaluation, the displacements at unmeasured DOFs are also determined by GA. The proposed method is able to detect the approximate location of National Institute of Technology Rourkela 12 Chapter 2 Literature Survey the damage, even when practical considerations limit the number of on-site measurements to only a few. Suh et al. [14] have presented a method to identify the location and depth of a crack on a structure by using hybrid neuro-genetic technique. Feed-forward multi-layer neural networks trained by back-propagation are used to learn the input (the location and depth of a crack)–output (the structural eigen frequencies) relation of the structural system. With this trained neural network, genetic algorithm is used to identify the crack location and depth minimizing the difference from the measured frequencies. The problem of crack identification in a beam when clamp rigidity is unknown performed by Horibe and Asano [15]. The identification method is based on the genetic algorithm (GA) and the proposed method is verified by numerical simulation. Sahoo and Maity [16] stated that artificial neural networks (ANN) have been proved to be an effective alternative for solving the inverse problems because of the patternmatching capability. But there is no specific recommendation on suitable design of network for different structures and generally the parameters are selected by trial and error, which restricts the approach context dependent. A hybrid neuro-genetic algorithm is proposed in order to automate the design of neural network for different type of structures. The neural network is trained considering the frequency and strain as input parameter and the location and amount of damage as output parameter. Damage detection methods of structures based on changes in their vibration properties have been widely employed during the last two decades. Existing methods include those based on examination of changes in natural frequencies, mode shapes or mode shape curvatures. Doebling et al. [17] published a stateof the- art review on vibration based damage identification methods. Messina et al. [18] used the sensitivity and a statistical-based method to structural damage detection. Kosmatka and Ricles [19] presented the modal vibration characterization method using the vibratory residual forces and weighted sensitivity analysis. Ratcliffe [20] performed the frequency and curvature based experiments. Vestroni and Capecchi [21] presented the method for concentrated damage detection based on natural frequency measurement. Gawronski and Sawicki [22] used the method based on modal and sensor norms. Hu et al. [23] presented a method using quadratic programming. Law et al. [24] presented a method for large-scale structures using super-elements with the concept of damage detection orientation modelling. Sahin and Shenoi [25] have presented a damage detection algorithm using a combination of global and local vibration-based data as input to artificial neural networks (ANNs) for location and severity prediction of the damage. National Institute of Technology Rourkela 13 Chapter 2 Literature Survey Sadettin Orhan [26] has been performed an analysis of free and forced vibration of a cracked beam in order to identify the crack in a cantilever. Single- and two-edge cracks were evaluated. The study results suggest that free vibration analysis provides suitable information for the detection of single and two cracks, whereas forced vibration can detect only the single crack condition. However, dynamic response of the forced vibration better describes changes in crack depth and location than the free vibration in which the difference between natural frequencies corresponding to a change in crack depth and location only is a minor effect. Mei et al. [27] has been presented his research work on wave vibration analysis of an axially loaded cracked Timoshenko beam. It includes the effects of axial loading, shear deformation and rotary inertia. From wave standpoint, vibrations propagate, reflect and transmit in a structure. The transmission and reflection matrices for various discontinuities on an axially loaded Timoshenko beam are derived. Such discontinuities include cracks, boundaries and changes in section. The matrix relations between the injected waves and externally applied forces and moments are also derived. These matrices are combined to provide a concise and systematic approach to both free and forced vibration analyses of complex axially loaded Timoshenko beams with discontinuities such as cracks and sectional changes. The systematic method is illustrated using several numerical examples. Viola et al. [28] have investigated the changes in the magnitude of natural frequencies and modal response introduced by the presence of a crack on an axially loaded uniform Timoshenko beam using a particular member theory. A new and convenient procedure based on the coupling of dynamic stiffness matrix and line-spring element is introduced to model the cracked beam. The application of the theory is demonstrated by two illustrative examples of bending–torsion coupled beams with different end conditions, for which the influence of axial force, shear deformation and rotatory inertia on the natural frequencies is studied. Moreover, a parametric study to investigate the effect of the crack on the modal characteristics of the beam is conducted. A theoretical and experimental dynamic behavior of different multi-beams systems containing a transverse crack has been presented by Saavedra and Cuitino [29]. The additional flexibility that the crack generates in its vicinity is evaluated using the strain energy density function given by the linear fracture mechanic theory. Based on this flexibility, a new cracked finite element stiffness matrix is deduced, which can be used subsequently in the FEM analysis of crack systems. The proposed element is used to evaluate the dynamic response of a cracked free–free beam and a U- National Institute of Technology Rourkela 14 Chapter 2 Literature Survey frame when a harmonic force is applied. The resulting parametrically excited system is nonlinear and the equations of motion are solved using the Hilbert, Hughes and Taylor integration method implemented using a Matlab software platform. Kisa and Gurel [30] have investigated the, a novel numerical technique applicable to analyses the free vibration analysis of uniform and stepped cracked beams with circular cross section. In this approach in which the finite element and component mode synthesis methods are used together, the beam is detached into parts from the crack section. These substructures are joined by using the flexibility matrices taking into account the interaction forces derived by virtue of fracture mechanics theory as the inverse of the compliance matrix found with the appropriate stress intensity factors and strain energy release rate expressions. To reveal the accuracy and effectiveness of the offered method, a number of numerical examples are given for free vibration analysis of beams with transverse nonpropagating open cracks. Numerical results showing good agreement with the results of other available studies, address the effects of the location and depth of the cracks on the natural frequencies and mode shapes of the cracked beams. An identification procedure to determine the crack characteristics (location and size of the crack) from dynamic measurements has been developed and tested by Shen and Taylor [31]. 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. Crack detection in beam-like structures has been presented by Rosales et al. [32]. Two approaches are herein presented: The solution of the inverse problem with a power series technique (PST) and the use of artificial neural networks (ANNs). Cracks in a cantilever Bernoulli Euler (BE) beam and a rotating beam are detected by means of an algorithm that solves the governing vibration problem of the beam with the PST. The ANNs technique does not need a previous model, but a training set of data is required. It is applied to the crack detection in the cantilever beam with a transverse crack. The first methodology is very simple and straightforward, though no optimization is included. It yields relative small errors in both the location and depth detection. When using one network for the detection of the two parameters, the ANNs behave adequately. National Institute of Technology Rourkela 15 Chapter 2 Literature Survey However better results are found when one ANN is used for each parameter. Finally, a combination between the two techniques is suggested. Gounaris and Dimarogonas [33] performed a finite element of a cracked prismatic beam for structural analysis. A surface crack on a beam section introduces a local flexibility to the structural member. In order to model the structure for FEM analysis, a finite element for the cracked prismatic beam is developed. Strain energy concentration arguments lead to the development of a compliance matrix for the behavior of the beam in the vicinity of the crack. This matrix is used to develop the stiffness matrix for the cracked beam element and the consistent mass matrix. The element developed can be used in any appropriate matrix analysis of structures program. The element was used to evaluate the dynamic response of a cracked cantilever beam to harmonic point force excitation. Resonant frequencies and vibration amplitudes are considerably affected by the existence of moderate cracks. Onard et al. [34] has been presented free-vibration behavior of a cracked cantilever beam and crack detection .This study is based on cracks that occurred in metal beams obtained under controlled fatigue-crack propagation. The beams were clamped in a heavy vise and struck in order to obtain a clean impulse modal response. Spectrograms of the freedecay responses showed a time drift of the frequency and damping: the usual hypothesis of constant modal parameters is no longer appropriate, since the latter are revealed to be a function of the amplitude. Signal processing such as the worm transform and phase spectrogram methods have been developed with enough accuracy to display the behavior of an uncracked beam where a slight non-linear stiffness is generated by the clamping. Moreover, extracted worms show that the second mode of a beam with a deep crack is modulated in frequency by the "first mode. In fact, the dominant mode opens and closes the crack, thereby modulating the beam stiffness, which affects higher modal frequencies. With deep cracks, three vibration states are observed: one where the crack is alternately fully open and fully closed, a second with a crack partially opened, and a third with alternating force acting on a closed crack. In the latter case, the peak force is smaller than the intrinsic closure load of the crack. The "first state is difficult for a small crack to reach since highamplitude excitation is required to fully open the crack. For crack detection purposes, the damping criterion, harmonic distortion criterion and bispectrum appear less sensitive to small cracks than the phase spectrogram and coherent-modulated power. National Institute of Technology Rourkela 16 Chapter 2 Literature Survey Detection of the Crack in Cantilever Structures Using Fuzzy Gaussian Inference Technique has been introduced by Das and Parhi [35]. Detection of the cracks in beam structures using the fuzzy Gaussian inference technique has been investigated by Das and Parhi. Fuzzy-logic controller here used six input parameters and two output parameters. The input parameters to the fuzzy controller are the relative divergence of the first three natural frequencies and first three mode shapes in dimensionless forms. The output parameters of the fuzzy controller are the relative crack depth and relative crack location in dimensionless forms. The strain-energy release rate has been used for calculating the local stiffness of the beam for a mode-I type of the crack. Several boundary conditions are outlined that take into account the crack location. The developed fuzzy controller can predict the location and depth of the crack in close proximity with the real results. Chandrasekhar et al. [36] has stated that geometric and measurement uncertainty cause considerable problem in damage assessment which can be alleviated by using a fuzzy logicbased approach for damage detection. Curvature damage factor (CDF) of a tapered cantilever beam is used as damage indicators. Monte Carlo simulation (MCS) is used to study the changes in the damage indicator due to uncertainty in the geometric properties of the beam. Variation in these CDF measures due to randomness in structural parameter, further contaminated with measurement noise, are used for developing and testing a fuzzy logic system (FLS). Results show that the method correctly identifies both single and multiple damages in the structure. Das and Parhi [37] applied hybrid artificial intelligence technique for fault detection in a cracked cantilever beam. The hybrid technique used here uses a fuzzy-neuro controller. Here the fuzzy-neuro controller has two parts. The first part is comprised of the fuzzy controller, and the second part is comprised of the neural controller. The input parameters of the neural segment of the fuzzy-neuro controller are relative deviation of the first three natural frequencies and relative values of percentage deviation for the first three mode shapes, along with the initial outputs of the fuzzy controller. The output parameters of the fuzzy-neuro controller are final relative crack depth and final relative crack location. The results of the developed fuzzy-neuro controller and experimental method are in good agreement. Hasanzadeh et al. [38] have been proposed an aligning method, which is formalized by a fuzzy recursive least square algorithm as a learning methodology for electromagnetic alternative current field measurement (ACFM) probe signals of a crack. This method along with a set of fuzzy linguistic rules, including adequate adaptation of National Institute of Technology Rourkela 17 Chapter 2 Literature Survey different crack shapes for combining knowledge and data whenever the superposition theory can be utilized, provides a means to compensate for the lack of sufficient samples in available crack databases. Salahshoor et al. [39] has been described that the issue of fault detection and diagnosis has gained widespread industrial interest in process condition monitoring applications. They have presented an innovative data-driven fault detection and diagnosis methodology on the basis of a distributed configuration of three adaptive neurofuzzy inference system (ANFIS) classifiers for an industrial 440 MW power plant steam turbine with once-through Benson type boiler. Each ANFIS classifier has been developed for a dedicated category of four steam turbine faults. They have also conducted experimental studies to realize such fault categorization scheme. Kohonen et al. [40] have been proposed that the Self Organizing Map (SOM) method is a new powerful software tool for the visualization of high dimensional data. They explain that SOM converts complex, no-linear statistical relationship between high dimensional data into simple geometric relationship on a two-dimensional display. It may also be thought to produce some kind of abstraction. These two aspects visualization and abstraction occur in a member of complex engineering task such as process analysis, machine perception, control and communication. Cottrel et al. [41] have been performed an analytical study on Kohonen Network and suggested that the Kohonen algorithm is a powerful tool for data analysis. In that case they define a specific algorithm which provides a simultaneous classification of the observation and of the modalities. Vesanto et al. [42] have been proposed that the Self-Organizing Map (SOM) is a vector quantization method which places the prototype vectors on a regular low-dimensional grid in an ordered fashion. This makes the SOM a powerful visualization tool. Also its performance in terms of computational load is evaluated and compared to a corresponding C program. Kauppinen et al. [43] have been proposed a non-segmenting defect detection technique combined with a self-organizing map (SOM) based classifier and user interface. They have tried to avoid the problems with adaptive detection techniques, and to provide an intuitive user interface for classification, helping in training material collection and labeling, and with a possibility of easily adjusting the class boundaries. The approach is illustrated with examples from wood surface inspection. Many researchers have been used this kohonen network in different area of research but in this paper we have proposed the essential processes as well as the mechanism followed in the Kohonen Network for the detection of crack depth and crack location. National Institute of Technology Rourkela 18 Chapter 2 Literature Survey Hasanzadeh et al. [44] have developed a methodology for sizing of surface cracks by AC field measurement technique. They have developed an aligning method is formalized by a fuzzy recursive least square algorithm as a learning methodology for electromagnetic alternative current field measurement (ACFM) probe signals of a crack (data). This method along with a set of fuzzy linguistic rules, including adequate adaptation of different crack shapes (knowledge) for combining knowledge and data whenever the superposition theory can be utilized, provide a means to compensate for the lack of sufficient samples in available crack databases. Buezas et al. [45] have presented a technique for damage detection in structural element using genetic algorithm optimization method with the consideration of a crack contact model. In this technique the crack is simulated as a notch or a wedge with a unilateral Signorini contact model. The contact can be partial or total. All the simulations are carried out using the general purpose partial differential solver FlexPDE, a finite element (FE) code. A genetic algorithm (GA) optimization method is successfully employed for the crack detection. The dynamic responses at some points of the damaged structures are compared with the solution of the computational (FE) model using least squares for each proposed crack depth and location. An objective function arises which is then optimized to obtain the parameters. The method is developed for bi- and threedimensional models to handle the dynamics of a structural element with a transverse breathing crack. Nobahari and Seyedpoor [46] have developed a technique for Structural damage detection using an efficient correlation-based index and a modified genetic algorithm; they have developed an efficient optimization procedure to detect multiple damage in structural systems. Natural frequency changes of a structure are considered as a criterion for damage presence. In order to evaluate the required natural frequencies, a finite element analysis (FEA) is utilized. A modified genetic algorithm (MGA) with two new operators (health and simulator operators) is presented to accurately detect the locations and extent of the eventual damage. An efficient correlation-based index (ECBI) as the objective function for the optimization algorithm is also introduced. A method for Structural damage detection using fuzzy cognitive maps (FCM) and Hebian learning techniques have been developed by Beena and Ganguli [47]. In their analysis a Structural damage is modeled using the continuum mechanics approach as a loss of stiffness at the damaged location. A finite element model of a cantilever beam is used to calculate the change in the first six beam frequencies because of structural damage. The measurement deviations due to damage are fuzzified and then mapped to a set of faults National Institute of Technology Rourkela 19 Chapter 2 Literature Survey using FCM. The input concepts for the FCM are the frequency deviations and the output of the FCM is at five possible damage locations along the beam. The FCM works quite well for structural damage detection for ideal and noisy data. Further improvement in performance is obtained when an unsupervised neural network approach based on Hebbian learning is used to evolve the FCM. Liu et al. [48] have stated a methodology for structural damage diagnosis using neural network and feature fusion. In their analysis a structure damage diagnosis method combining the wavelet packet decomposition, multi-sensor feature fusion theory and neural network pattern classification was presented. Firstly, vibration signals gathered from sensors were decomposed using orthogonal wavelet. Secondly, the relative energy of decomposed frequency band was calculated. Thirdly, the input feature vectors of neural network classifier were built by fusing wavelet packet relative energy distribution of these sensors. Finally, with the trained classifier, damage diagnosis and assessment was realized. The result indicates that, a much more precise and reliable diagnosis information is obtained and the diagnosis accuracy is improved as well. National Institute of Technology Rourkela 20 THEORETICAL VIBRATION ANALYSIS FOR IDENTIFICATION OF CRACK 3.1. Introduction 3.2. Local flexibility of a cracked cantilever beam under axial load and bending moment 3.3. Analysis of vibration characteristics of a cracked cantilever beam 3.3.1. Analysis of free vibration 3.3.2. Analysis of forced vibration 3.4. Finite element formulation 3.4.1. Theory 3.4.2. Governing equations 3.4.3. Process of crack detection National Institute of Technology Rourkela 21 Chapter 3 Theoretical Vibration Analysis CHAPTER 3 Theoretical Vibration Analysis for Identification of Crack n 3.1. Introduction In this present research work it has been analyzed that the crack can be detected in the various structures through visual inspection or by the method of measuring natural frequency, mode shape and structural damping. As the measurement of natural frequency and mode shape is quite easy as compared to other parameters, so in this chapter a logical approach has been adopted to develop the expression to calculate the natural frequency and the mode shape of the cantilever beam with the presence of a transverse crack and the effect of natural frequency in the presence of crack. Experimental analysis has been done over cracked cantilever beam specimen for validation of the theory established. 3.2. Local flexibility of a cracked cantilever beam under axial load and bending moment A cantilever beam with a transverse surface crack of depth „a1‟ on beam of width „B‟ and height „W‟ is considered for the current research. The beam is subjected to axial force (P 1) and bending moment (P2) (Fig.3.1) which gives coupling with the longitudinal and transverse motion. The presence of crack introduces a local flexibility, which can be defined in matrix form, the dimension of which depends on the degrees of freedom. Here a 2x2 matrix is considered. Y- Axis Z- Axis P2 B a1 da W P1 X- Axis L1 L Figure: 3.1 Geometry of Cracked Cantilever Beam The strain energy release rate at the fractured section can be written as (Tada et al. [49]) National Institute of Technology Rourkela 22 Chapter 3 J = Theoretical Vibration Analysis 1 1 v 2 1 (for plane strain condition); (K I1 K I2 ) 2 , Where E E E 1 (for plane stress condition) E (3.1a) (3.1b) Kl1, Kl2 are the stress intensity factors of mode I (opening of the crack) for load P1 and P2 respectively. The values of stress intensity factors from earlier studies are; K I1 P1 6P2 a a (F1 ( )), K I 2 BW W BW 2 a (F2 ( a )) W (3.2) Where expressions for F1 and F2 are as follows F1 ( a 2W a 0.5 0.752 2.02 (a / W) 0.37 (1 sin(a / 2W)) 3 ), ( tan( )) cos(a / 2W) W a 2W (3.3) a 2W a 0.5 0.923 0.199 (1 sin(a / 2W)) 4 F2 ( ), ( tan( )) cos(a / 2W) W a 2W Let Ut be the strain energy due to the crack. Then from Castigliano‟s theorem, the additional displacement along the force Pi is: ui U t Pi (3.4) 1 U t da J da The strain energy will have the form, U t a 0 0 a1 Where J= a (3.5) U t the strain energy density function. a From (Equations 3.4 and 3.5), we will get a 1 ui J(a ) da Pi 0 (3.6) The flexibility influence co-efficient Cij will be, by definition u i 2 1 J(a ) da Pj Pi Pj 0 a C ij National Institute of Technology Rourkela (3.7) 23 Chapter 3 Theoretical Vibration Analysis To find out the final flexibility matrix we have to integrate over the breadth „B‟ u i 2 B / 2 a1 C ij J(a ) da dz Pj Pi Pj B / 2 0 (3.8) Taking the value of strain energy release rate from the Equation (3.1a), the equation (3.8) can be modifies as B 2 1 (K l1 K l 2 ) 2 da E Pi Pj 0 a C ij Putting ξ = (a/w), d (3.9) da W We get da = W dξ and when a = 0, ξ = 0; a = a1, ξ = a1/W = ξ1 Putting the above condition in Equation (3.9) we will get C ij BW 2 1 (K l1 K l 2 ) 2 d E Pi Pj 0 (3.10) From the Equation (3.10), calculating C11, C12 (=C21) and C22 we get, BW C11 E 1 C12 C 21 C 22 a 2 1 2 2 ( F ( )) d (F1 ()) 2 d 1 0 B2 W 2 BE 0 12 1 F1 ()F2 () d E BW 0 72 EBW 2 (3.11) (3.12) 1 F ()F () d 2 2 (3.13) 0 Converting the influence co-efficient into dimensionless form, we will get the final expression as C11 C11 E BW 2 BE EBW C12 C12 C 21 ; C 22 C 22 72 2 12 (3.14) The local stiffness matrix can be obtained by taking the inversion of compliance matrix. i.e. National Institute of Technology Rourkela 24 Chapter 3 Theoretical Vibration Analysis K 12 C11 C12 K K 11 K 21 K 22 C 21 C 22 1 (3.15) 3.3. Analysis of Vibration Characteristics of a Cracked Cantilever Beam 3.3.1. Analysis of Free Vibration A cantilever beam of length „L‟ width „B‟ and depth „W‟, with a crack of depth „a1‟ at a distance „L1‟ from the fixed end is considered (Fig. 3.1). Taking u1(x,t) and u2(x,t) as the amplitudes of longitudinal vibration for the sections before and after the crack and y1(x,t), y2(x,t) are the amplitudes of bending vibration for the same sections (Fig. 3.2). U2 U1 L1 L Y1 Y2 Figure: 3.2 Beam Model The normal function for the system can be defined as u1 (x) A1 cos (K u x) A 2 sin(K u x) (3.16a) u 2 (x) A 3 cos (K u x) A 4 sin(K u x) (3.16b) y1 (x) A 5 cosh (K y x) A 6 sinh(K y x) A 7 cos (K y x) A 8 sin(K y x) (3.16c) y 2 (x) A 9 cosh (K y x) A10 sinh(K y x) A11 cos (K y x) A12 sin(K y x) (3.16d) Where x L x u y ,u , y , 1 L L L L E L , C u Ku Cu 1/ 2 L2 , Ky C y 1/ 2 EI , C y National Institute of Technology Rourkela 1/ 2 , = A 25 Chapter 3 Theoretical Vibration Analysis Ai, (i=1, 12) Constants are to be determined, from boundary conditions. The boundary conditions of the cantilever beam in consideration are: u1 (0) 0 ; (3.17a) y1 (0) 0 ; (3.17b) y1 (0) 0 ; (3.17c) u 2 (1) 0 ; (3.17d) y"2 (1) 0 ; (3.17e) y2(1) 0 ; (3.17f) At the cracked section: u '1 () u ' 2 () ; (3.18a) y1 () y 2 () ; (3.18b) y1() y2 () ; (3.18c) y1() y2() (3.18d) Also at the cracked section (due to the discontinuity of axial deformation to the left and right of the crack), we have: AE du 1 (L1 ) dy (L ) dy (L ) K11 (u 2 (L1 ) u 1 (L1 )) K12 2 1 1 1 dx dx dx Multiplying both sides of the above equation by (3.19) AE we will get; LK 11K 12 M1M 2 u () M 2 (u 2 () u1 ()) M1 ( y2 () y1 ()) (3.20) Similarly at the crack section (due to the discontinuity of slope to the left and right of the crack) EI d 2 y 1 ( L1 ) dy (L ) dy (L ) K 21 (u 2 (L1 ) u 1 (L1 )) K 22 2 1 1 1 2 dx dx dx National Institute of Technology Rourkela (3.21) 26 Chapter 3 Theoretical Vibration Analysis Multiplying both sides of the above equation by EI we will get, L K 22 K 21 2 M 3 M 4 y1() M 3 (u 2 () u1 ()) M 4 ( y2 () y1 ()) Where, M1 (3.22) AE AE EI EI , M2 , M3 , M4 2 LK 22 LK 11 K 12 L K 21 The normal functions as stated in Equation (3.16) along with the boundary conditions as mentioned above equations (3.17) and (3.18) yield the characteristic equation of the system as: Q 0 (3.23) This determinant is a function of natural circular frequency (ω), the relative location of the crack (β) and the local stiffness matrix (K) which in turn is a function of the relative crack depth (a1/W). 3.3.2. Analysis of Forced Analysis If the cantilever beam with transverse crack is excited at its free end by a harmonic excitation (Y = Y0 sin(ωt) ), the non-dimensional amplitude at the free end may be expressed as y 2 (1) y0 y 0 . Therefore the boundary conditions for the beam remain same L as before except the boundary condition y2(1) 0 which modified as y 2 (1) y 0 . The constants Ai, i=1, to 12 are then computed from the algebraic condition Q1D=B1 (3.24) Q1 is the (12 x 12) matrix obtained from boundary conditions as mentioned above, D is a column matrix obtained from the constants, B1 is a column matrix, transpose of which is given by, B1T 0 0 0 y 0 0 0 0 0 0 0 0 0 (3.25) 3.4. Finite Element Formulation 3.4.1. Theory The beam with a transverse edge crack is clamped at left end, free at right end and has uniform structure with a constant rectangular cross-section of 800 mm X 38 mm X 6 mm. The EulerBernoulli beam model is assumed for the finite element formulation. The crack in this particular National Institute of Technology Rourkela 27 Chapter 3 Theoretical Vibration Analysis case is assumed to be an open crack and the damping is not being considered in this theory. Both single and double edged crack are considered for the formulation. 3.4.2. Governing Equations The free bending vibration of an Euler-Bernoulli beam of a constant rectangular cross section is given by the following differential equation as given in: d4y EI m2 y 0 i 4 dx (3.26) Where „m‟ is the mass of the beam per unit length (kg/m), „ωi‟ is the natural frequency of the ith mode (rad/sec), E is the modulus of elasticity (N/m2) and I is the moment of inertia m2 4 i equation is rearranged as a fourth-order differential equation (m ). By defining EI 4 as follows: d4y 4 y 0 dx 4 (3.27) The general solution to equation is y A cos i x B sin i x C cosh i x D sinh i x (3.28) Where A, B, C, D are constants and „λi‟ is a frequency parameter. Adopting Hermitian shape functions, the stiffness matrix of the two-noded beam element without a crack is obtained using the standard integration based on the variation in flexural rigidity. The element stiffness matrix of the un cracked beam is given as Ke [B(x)]T EI[B(x)]dx (3.29) [B(x)] {H1(x)H 2 (x)H3 (x)H 4 (x)} (3.30) Where [H1(x), H 2 (x), H3 (x), H 4 (x)] is the Hermitian shape functions defined as 3x 2 2x 3 2x 2 x 3 , H 2 (x) x H1 ( x ) 1 l l2 l3 l2 H3 (x) 3x 2 2x 3 x 2 x3 , H 4 (x) l l2 l3 l2 (3.31) Assuming the beam rigidity EI is constant and is given by EI0 within the element, and then the element stiffness is National Institute of Technology Rourkela 28 Chapter 3 Theoretical Vibration Analysis 6l 12 6l 12 6l 4l 2 6l 2l 2 EI [K e ] 0 l 3 12 6l 12 6l 2 6l 4l 2 6l 2l (3.32) [K ec ] [K e ] [K c ] (3.33) Here, K ec Stiffness matrix of the cracked element, K e Element stiffness matrix, K c Reduction in stiffness matrix due to the crack. According to (Peng et al. [50]), the matrix [Kc] is K12 K11 K K 22 [K c ] 12 K11 K12 K 24 K14 K11 K12 K11 K14 K14 K 24 K14 K 44 (3.34) 2 12E(I 0 I c ) 2l3c 2L1 Where, K11 3l c 1 2 L2 L4 L 2 2 3 6 L 12E(I 0 I c ) l c 5L K14 lc 2 1 1 2 L L3 L2 L 2 2 6 L 12E(I 0 I c ) l 3c 7 L K12 lc 2 1 1 2 L L3 L2 L 2 2 9 L 12E(I0 Ic ) 3l3c 9 L K 24 2lc 2 1 1 2 L L2 L2 L 2 12E(I 0 I c ) 3l3c 3L1 K 22 2l c 2 , L L2 L3 12E(I 0 I c ) 3l3c 3L K 44 2l c 1 1 L2 L L2 Here, lc=1.5W, L=Total length of the beam, L1=Distance between the left node and crack BW 3 B( W a ) 3 I0 Moment of inertia of the beam cross section, Ic Moment of 12 12 inertia of the beam with crack. National Institute of Technology Rourkela 29 Chapter 3 Theoretical Vibration Analysis It is supposed that the crack does not affect the mass distribution of the beam. Therefore, the consistent mass matrix of the beam element can be formulated directly as Me 1 A[H(x)]T H(x)dx (3.35) 0 22l 156 22l 4l 2 Al e M 13l 420 54 2 13l 3l 13l 13l 3l 2 156 22l 22l 4l 2 54 (3.36) The natural frequency then can be calculated from the relation. [2 M K]{q} 0 (3.37) Where, q=displacement vector of the beam 3.4.3. Process of Crack Detection Detection of crack in a cantilever beam has been performed in two methods. First the finite element model of the cracked cantilever beam is developed and the beam is discretized into a number of elements, and the crack position is assumed to be in each of the elements. Next, for each position of the crack in each element, depth of the crack is varied. Modal analysis for each position and depth is then performed to determine the natural frequencies of the beam. Y- Axis Z- Axis P2 B W P1 X- Axis L1 L Figure: 3.3 Representation of a Single Crack Cantilever Beam National Institute of Technology Rourkela 30 ANALYSIS OF FUZZY INFERENCE SYSTEM FOR IDENTIFICATION OF CRACK 4.1. Introduction 4.2. Fuzzy sets and membership functions 4.3. Fuzzy inference system 4.3.1. Fuzzy linguistic variables 4.3.2. Fuzzy controller/ Fuzzy If-then rule 4.3.3. Creating Fuzzy rules 4.3.4. Fuzzification 4.3.5. Defuzzification of output distribution 4.4. Fuzzy mechanism used for localization and identification of crack 4.5. Function of fuzzy controller for localization and identification of crack 4.5.1. Discussion 4.5.2. Comparison of Results 4.5.3. Summary 4.6. Why fuzzy logic is used National Institute of Technology Rourkela 31 Chapter 4 CHAPTER 4 Fuzzy Inference System Analysis of Fuzzy Inference System for Identification of Crack 4.1. Introduction In order to develop a reliable, efficient and economical approach to increase the safety and reduce the maintenance cost of elastic structures, many researchers have been developed various structural health monitoring techniques. Although improved design methodologies have significantly enhanced the reliability and safety of structures, it is still not possible to build a structure that has zero percent probability of failure. In the recent years, researchers have motivated to words the development of intelligent techniques for the fault detection. This current research presents methodologies for structural damage detection and assessment using fuzzy interface system. In this chapter an intelligent controller has been projected for localization and identification of crack. Fuzzy sets originated in the year 1965 and this concept was proposed by Lofti A. Zadeh. Since then it has grown and is found in several application areas. According to Zadeh, The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects of the framework used in the case of ordinary sets, but is more general than the latter and potentially, may prove to have a much wider scope of applicability, specifically in the fields of pattern classification and information processing.” Fuzzy logics are multi-valued logics that form a suitable basis for logical systems reasoning under uncertainty or vagueness that allows intermediate values to be defined between conventional evaluations like true/false, yes/no, high/low, etc. These evaluations can be formulated mathematically and processed by computers, in order to apply a more human-like way of thinking in the programming of computers. Fuzzy logic provides an inference morphology that enables approximate human reasoning capabilities to be applied to knowledge-based systems. The theory of fuzzy logic provides a mathematical strength to capture the uncertainties associated with human cognitive processes, such as thinking and reasoning. Fuzzy systems are suitable for uncertain or approximate reasoning, especially for the system with a mathematical model that is difficult to derive. Fuzzy logic allows decision making with estimated values under incomplete or uncertain information. National Institute of Technology Rourkela 32 Chapter 4 Fuzzy Inference System 4.2. Fuzzy Sets and Membership Functions The concept of a fuzzy set is an extension of the concept of a crisp set. Similar to a crisp set a universe set U is defined by its membership function from U to [0, 1]. Consider U to be a non-empty set also known as the universal set or universe of discourse or domain. A fuzzy set on U is defined as A (x) : U [0, 1] . Here A is known as the membership function, and A ( x ) is known as the membership grade of x. Membership function is the degree of truth or degree of compatibility. The membership function is the crucial component of a fuzzy set. Therefore all the operations on fuzzy sets are defined based on their membership functions. The membership function is a graphical representation of the magnitude of participation of each input. It associates a weighting with each of the inputs that are processed, define functional overlap between inputs, and ultimately determines an output response. The rules use the input membership values as weighting factors to determine their influence on the fuzzy output sets of the final output conclusion. Once the functions are inferred, scaled, and combined, they are defuzzified into a crisp output which drives the system. There are different membership functions associated with each input and output response. Reasonable functions are often piecewise linear function, such as triangular or trapezoidal functions. The value for the membership function can be taken in the interval [0, 1]. When the functions are nonlinear the Gaussian membership function will be taken for the smooth operation. 4.3. Fuzzy Inference System A fuzzy inference system is the process of formulating the mapping from a given input to an output using fuzzy logic. To compute the output of this FIS for the given the inputs, we must go through six steps: Determining a set of fuzzy rules. Fuzzifying the inputs using the input membership functions. Combining the fuzzified inputs according to the fuzzy rules to establish rule strength. Finding the consequence of the rule by combining the rule strength and the output membership function. Combining the consequences to get an output distribution, and Defuzzifying the output distribution. National Institute of Technology Rourkela 33 Chapter 4 Fuzzy Inference System Fuzzy inference systems have been successfully applied in fields such as automatic control, fault diagnosis, data classification, decision analysis, expert systems, and computer vision. If Then Rule Strength And And Input Distributions x0 Output Distributions y0 Figure: 4.1 Schematic Diagram of Operation of a Fuzzy Interface System 4.3.1.Fuzzy Linguistic Variables Just like an algebraic variable takes numbers as values, a linguistic variable takes words or sentences as values. The set of values that it can take is called its term set. Each value in the term set is a fuzzy variable defined over a base variable. The base variable defines the universe of discourse for all the fuzzy variables in short. In short the hierarchy is as follows: Linguistic variable → Fuzzy variable → Base variable. In 1973, Professor Lotfi Zadeh [51] proposed the concept of linguistic or "fuzzy" variables. Think of them as linguistic objects or words, rather than numbers. Suppose that X = "age." Then we can define fuzzy sets "young," "middle aged," and "old" that are characterized by MFs young( x) , middleaged (x) , and old ( x ) , respectively. Just as a National Institute of Technology Rourkela 34 Chapter 4 Fuzzy Inference System variable can assume various values, a linguistic variable "Age” can assume different linguistic values, such as "young," "middle aged “and” old" in this case. If "age" assumes the value of "young," then we have the expression "age is young," and so forth for the other values. 4.3.2. Fuzzy Controller/ Fuzzy If-Then Rule Fuzzy logic controllers are based on the combination of Fuzzy set theory and fuzzy logic. Systems are controlled by fuzzy logic controllers based on rules instead of equations. This collection of rules is known as the rule base usually in the form of IF-THEN-ELSE statements. Here the IF part is known as the antecedent and the THEN part is the consequent. The antecedents are connected with simple Boolean functions like AND, OR, NOT etc., Figure 4.2 outlines a simple architecture for a fuzzy logic controller [52]. The outputs from a system are converted into a suitable form by the fuzzification block. Once all the rules have been defined based on the application, the control process starts with the computation of the rule consequences. The computation of the rule consequences takes place within the computational unit. Finally, the fuzzy set is defuzzified into one crisp control action using the defuzzification module. A fuzzy if-then rule (also known as fuzzy rule, fuzzy implication or fuzzy conditional statement) assumes the form “if x is A then y is B”. Where A and B are linguistic values defined by fuzzy sets on universes of discourse x and y respectively. Often “x is A” is called the antecedent or premise, while “y is B” is called the consequence or conclusion. (Some of the linguistic terms used are shown in Table: 4.1). 4.3.3. Creating Fuzzy Rules Fuzzy rules are a collection of linguistic statements that describe how the FIS should make a decision regarding classifying an input or controlling an output. Fuzzy rules are always written in the following form: if (input1 is membership function1) and/or (input2 is membership function2) and/or …. then (outputn is output membership functionn). For example: one could make up a rule that says: if temperature is high and humidity is high then room is hot. There would have to be membership functions that define what we mean by high temperature (input1), high humidity (input2) and a hot room (output1). This process of taking an input such as temperature and processing it through a membership function to National Institute of Technology Rourkela 35 Chapter 4 Fuzzy Inference System determine what we mean by "high" temperature is called fuzzification. Also, we must define what we mean by "and" / "or" in the fuzzy rule. This is called fuzzy combination. Rule Base Input Output Computational Unit Fuzzification Defuzzification Figure: 4.2 Fuzzy Controller Architecture 4.3.4. Fuzzification The purpose of fuzzification is to map the inputs from a set of devices (for example sensors or features of those sensors such as amplitude or spectrum) to values from 0 to 1 using a set of input membership functions. In the schematic diagram shown in Figure 4.1, there are two inputs, x0 and y0 shown at the lower left corner. These inputs are mapped into fuzzy numbers by drawing a line up from the inputs to the input membership functions above and marking the intersection point. These input membership functions, can represent fuzzy concepts such as "large" or "small", "old" or "young", "hot" or "cold", etc. The membership functions could then represent "large" amounts of tension coming from a muscle or "small" amounts of tension. When choosing the input membership functions, the definition of what we mean by "large" and "small" may be different for each input. 4.3.5. Defuzzification of Output Distribution In many situations, for a system whose output is fuzzy, it is easier to take a crisp decision if the output is represented as a single scalar quantity. This conversion of a fuzzy set to single crisp value is called defuzzification and is the reverse process of fuzzification. There are two common methods for defuzzification generally followed: Centroid Method: Also known as the centre of gravity or the centre of area method, it obtains the centre of area (x*) occupied by the fuzzy set. It is given by the expression; x* ( x ) x dx (For a continuous membership function) ( x ) dx National Institute of Technology Rourkela (4.1) 36 Chapter 4 Fuzzy Inference System n x i ( x i ) x * i 1 (For a discrete membership function) n ( x i ) i 1 (4.2) Here, „n‟ represents the numbers of element in the sample, xi‟s are the elements, and ( x i ) is its membership function. Mean of maxima (MOM) defuzzification method: One simple way of defuzzifying the output is to take the crisp value with the highest degree of membership. In case with more than one element having the maximum value, the mean value of the maxima is taken. The equation of the defuzzified value x* is given by; xi x M x* i M (4.3) Where M= x i (x i ) is equal to the height of fuzzyset M is the cardinality of the set M. In the continuous case, M could be defined as M= x i [-c, c] (x i ) is equal to the height of the fuzzyset In such a case, the mean of maxima is the arithmetic average of mean values of all intervals contained in M including zero length intervals. The height of a fuzzy set A, i.e. h(A) is the largest membership grade obtained by any element in that set. 4.4. Fuzzy Mechanism used for localization and identification of crack The fuzzy controller has been developed (as shown in Figure: 4.3) where there are 3 inputs and 2 outputs parameter. The natural linguistic representations for the input are as follows Relative first natural frequency = “FNF” Relative second natural frequency = “SNF” Relative third natural frequency = “TNF” The natural linguistic term used for the outputs are Relative crack depth = “RCD” National Institute of Technology Rourkela 37 Chapter 4 Fuzzy Inference System Relative crack length= “RCL” Based on the above fuzzy subset the fuzzy rules are defined in a general form as follows: If (FNF is FNFi and SNF is SNFj and TNF is TNFk) then (CD is CDijk and CL is CLijk) Where i= 1to 9, j=1 to 9, k=1 to 9 (4.4) Because of “FNF”, “SNF”, “TNF” have 9 membership functions each. From the above expression (4.4), two set of rules can be written If (FNF is FNFi and SNF is SNFj and TNF is TNFk) then CD is CDijk (4.5a) If (FNF is FNFi and SNF is SNFj and TNF is TNFk) then CL is CLijk (4.5b) According to the usual Fuzzy logic control method (Das and Parhi [1]), a factor Wijk is defined for the rules as follows: Wijk=μfnfi (freqi) Λ μsnfj (freqj) Λ μtnfi (freqk) Where freqi, freqj and freqk are the first, second and third natural frequency of the cantilever beam with crack respectively ; by Appling composition rule of interference (Das and Parhi [1]) the membership values of the relative crack location and relative crack depth (location)CL. μrclijk (location) = Wijk Λ μrclijk (location) length CL As; μrclijk (depth) = Wijk Λ μrclijk (depth) depth CD The overall conclusion by combining the output of all the fuzzy can be written as follows: μrclijk (location) = μrcl111 (location) V.….V μrclijk (location) V.V μrcl9 9 9 (location) (4.6a) μrclijk (location) = μrcl111 (depth) V.…..V μrclijk (depth) V….V μrcl9 9 9 (depth) (4.6b) The crisp values of relative crack location and relative crack depth are computed using the center of gravity method (Das and Parhi [1]) as: National Institute of Technology Rourkela 38 Chapter 4 Fuzzy Inference System Relative crack location=rcl= location. rcl (location ).d(location ) rcl (location ).d(location ) (4.7a) Relative crack depth=rcd= depth. rcd (depth).d(depth) rcd (depth).d(depth) (4.8b) Membership Functions Fuzzy (if-then) rules, Linguistic variables Crisp Values Crisp Values fnf snf I N P U T Fuzzifier Fuzzy Controller Defuzzifier O U T P U T rcl rcd tnf Figure: 4.3 Schematic diagram of Fuzzy Inference System 4.5. Function of Fuzzy Controller for Localization and Identification of Crack The inputs to the fuzzy controller are relative first natural frequency; relative second natural frequency; relative third natural frequency. The outputs from the fuzzy controller are relative crack depth and relative crack location. Several hundred fuzzy rules are outlined to train the fuzzy controller. Twenty four numbers of the fuzzy rules out of several hundred fuzzy rules are being listed in Table: 4.2. The output data has been generated from the input data and the rule base. Relative Natural Frequency = Natural frequency of uncracked beam Natural frequency of cracked beam National Institute of Technology Rourkela 39 Chapter 4 Fuzzy Inference System L1F1 L1F2 L1F3 L1F4 M1F1 M1F2 M1F3 H1F1 H1F2 H1F3 H1F4 0.96 0.98 1 0.5 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 1 Figure: 4.4(a) Triangular Membership functions for relative natural frequency for 1st mode of vibration L2F1 L2F2 L2F3 L2F4 M2F1 M2F2 M2F3 H2F1 H2F2 H2F3 H2F4 0.98 0.99 1 0.5 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 1 Figure: 4.4(b) Triangular Membership functions for relative natural frequency for 2nd mode of vibration L3F1 L3F2 L3F3 L3F4 M3F1 M3F2 M3F3 H3F1 H3F2 H3F3 H3F4 0.98 0.99 1 0.5 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 1 Figure: 4.4(c) Triangular Membership functions for relative natural frequency for 3rd mode of vibration National Institute of Technology Rourkela 40 Chapter 4 Fuzzy Inference System SL1 SL2 SL3 SL4 ML1 ML2 ML3 BL1 BL2 BL3 BL4 1 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure: 4.4(d) Triangular Membership functions for relative crack location SD1 SD2 SD3 SD4 MD1 0.15 0.2 MD2 MD3 LD1 LD2 LD3 LD4 1 0.5 0 0.05 0.1 0.25 0.3 0.35 0.4 0.45 0.5 Figure: 4.4(e) Triangular Membership functions for relative crack depth L1F1 L1F2 L1F3 L1F4 M1F1 M1F2 M1F3 0.88 0.9 0.92 H1F1 H1F2 H1F3 H1F4 1 0.5 0.8 0.82 0.84 0.86 0.94 0.96 0.9 8 1 Figure: 4.5(a) Trapezoidal Membership functions for relative natural frequency for 1st mode of vibration National Institute of Technology Rourkela 41 Chapter 4 Fuzzy Inference System L2F1 L2F2 L2F3 L2F4 M2F1 M2F2 M2F3 H2F1 H2F2 H2F3 H2F4 1 0.5 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.9 9 1 Figure: 4.5(b) Trapezoidal Membership functions for relative natural frequency for 2nd mode of vibration L3F1 L3F2 L3F3 L3F4 M3F1 M3F2 M3F3 H3F1 H3F2 H3F3 H3F4 1 0.5 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 Figure: 4.5(c) Trapezoidal Membership functions for relative natural frequency for 3rd mode of vibration SL1 SL2 SL3 SL4 ML1 ML2 ML3 0.2 0.25 0.3 BL1 BL2 BL3 BL4 1 0.5 0 0.05 0.1 0.15 0.35 0.4 0.45 0.5 Figure: 4.5(d) Trapezoidal Membership functions for relative crack location SD1 SD2 SD3 SD4 MD1 MD2 MD3 LD1 LD2 LD3 LD4 1 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure: 4.5(e) Trapezoidal Membership functions for relative crack depth National Institute of Technology Rourkela 42 Chapter 4 Fuzzy Inference System L1F1 L1F2 L1F3 L1F4 M1F1 M1F2 M1F3 H1F1 H1F2 H1F3 H1F4 1 0.5 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 Figure: 4.6(a) Gaussian Membership functions for relative natural frequency for 1st mode of vibration L2F1 L2F2 L2F3 L2F4 M2F1 M2F2 M2F3 H2F1 H2F2 H2F3 H2F4 1 0.5 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 Figure: 4.6(b) Gaussian Membership functions for relative natural frequency for 2nd mode of vibration L3F1 L3F2 L3F3 L3F4 M3F1 M3F2 M3F3 H3F1 H3F2 H3F3 H3F4 1 0.5 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 Figure: 4.6(c) Gaussian Membership functions for relative natural frequency for 3rd mode of vibration SL1 SL2 0.05 0.1 SL3 SL4 ML1 ML2 ML3 BL1 BL2 BL3 BL4 1 0.5 0 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure: 4.6(d) Gaussian Membership functions for relative crack location National Institute of Technology Rourkela 43 Chapter 4 Fuzzy Inference System SD1 SD2 0.05 0.1 SD3 SD4 MD1 MD2 MD3 LD1 LD2 LD3 LD4 1 0.5 0 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure: 4.6(e) Gaussian Membership functions for relative crack depth L1F1 L1F2 L1F3 L1F4 M1F1 M1F2 M1F3 H1F1 H1F2 H1F3 H1F4 1 0.5 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 Figure: 4.7(a) Hybrid Membership functions for relative natural frequency for 1st mode of vibration L2F1 L2F2 L2F3 L2F4 M2F1 M2F2 M2F3 H2F1 H2F2 H2F3 H2F4 1 0.5 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 Figure: 4.7(b) Hybrid Membership functions for relative natural frequency for 2nd mode of vibration L3F1 L3F2 L3F3 L3F4 M3F1 M3F2 M3F3 H3F1 H3F2 H3F3 H3F4 1 0.5 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 Figure: 4.7(c) Hybrid Membership functions for relative natural frequency for 3rd mode of vibration National Institute of Technology Rourkela 44 Chapter 4 Fuzzy Inference System SL1 SL2 SL3 SL4 ML1 ML2 ML3 BL1 BL2 BL3 BL4 1 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure: 4.7(d) Hybrid Membership functions for relative crack location SD1 SD2 SD3 SD4 MD1 MD2 MD3 LD1 0.3 0.35 LD2 LD3 LD4 1 0.5 0 0.05 0.1 0.15 0.2 0.25 0.4 0.45 0.5 Figure: 4.7(e) Hybrid Membership functions for relative crack depth 4.5.1. Discussion In this current research the fuzzy controller has been designed using four types of membership functions, i.e. triangular (Figure: 4.4), trapezoidal (Figure: 4.5), Gaussian (Figure: 4.6) and hybrid membership function (Figure: 4.7) which combines trapezoidal, Gaussian as well as triangular membership functions. By taking the consideration of triangular membership functions the Figure: 4.4(a) shows the various linguistic terms used for the Relative 1st mode natural frequency. It has total 11 number of membership functions. Similarly Figure: 4.4(b) and 4.4(c) show the membership functions and the respective linguistic terms used for 2nd and 3rd mode relative natural frequencies. And both are having 11 membership functions each. Figure: 4.4(d) and 4.4(e) represents the membership functions and respective linguistic terms used for output of fuzzy controller, i.e. the linguistic terms used for relative crack location and relative crack depth. The similar linguistic terms are also used for other membership functions are as shown in the figures (Figure: 4.5 to 4.7). The working principle for the fuzzy inference system has been depicted in Figure: 4.3. The linguistic terms used in the fuzzy membership function has been specified in Table: 4.1. The fuzzy rules being used for the fuzzy inference system are National Institute of Technology Rourkela 45 Chapter 4 Fuzzy Inference System specified in the Table: 4.2. Out of several hundreds of fuzzy rules only twenty four fuzzy rules has been indicated in the table. Figure: 4.8 to Figure: 4.11 shows the operation of fuzzy inference system to exhibits the fuzzy results after defuzzification when rule 3 and 12 of the Table: 4.2 are activated for triangular, trapezoidal, Gaussian and hybrid membership functions respectively. The comparison of the results obtained from theoretical and the fuzzy controller with triangular membership function, fuzzy controller with trapezoidal membership function, fuzzy controller with Gaussian membership function and fuzzy controller with hybrid membership functions are presented in Table: 4.3 at the end of this Chapter and also a comparison of result shown between different fuzzy controllers and the kohonen network technique in Chapter 7. Table: 4.1 Linguistic Terms used for Fuzzy Membership Functions Name of the Membership functions L1F1,L1F2,L1F3,L1F4 Linguistic terms fnf1to4 M1F1,M1F2,M1F3 fnf5to7 H1F1,H1F2,H1F3,H1F4 fnf8to11 L2F1,L2F2,L2F3,L2F4 snf1to4 M2F1,M2F2,M2F3 snf5to7 H2F1,H2F2,H2F3,H2F4 snf8to11 L3F1,L3F2,L3F3,L3F4 tnf1to4 M3F1,M3F2,M3F3 tnf5to7 H3F1,H3F2,H3F3,H3F4 tnf8to11 SD1,SD2,SD3,SD4 rcd1to4 National Institute of Technology Rourkela Description and range of the linguistic terms Low ranges of relative natural frequency for first mode of vibration in ascending order respectively. Medium ranges of relative natural frequency for first mode of vibration in ascending order respectively. Higher ranges of relative natural frequency for first mode of vibration in ascending order respectively Low ranges of relative natural frequency for second mode of vibration in ascending order respectively. Medium ranges of relative natural frequency for second mode of vibration in ascending order respectively. Higher ranges of relative natural frequency for first mode of vibration in ascending order respectively Low ranges of relative natural frequency for second mode of vibration in ascending order respectively Medium ranges of relative natural frequency for second mode of vibration in ascending order respectively Higher ranges of relative natural frequency for first mode of vibration in ascending order respectively Small ranges of relative crack depth in ascending order respectively. 46 Chapter 4 Fuzzy Inference System MD1,MD2,MD3 rcd5to7 LD1,LD2,LD3,LD4 rcd8to11 SL1,SL2,SL3,SL4 rcl1to4 ML1,ML2,ML3 rcl5to7 BL1,BL2,BL3,BL4 rcl8to11 Medium ranges of relative crack depth ascending order respectively Larger ranges of relative crack depth ascending order respectively. Small ranges of relative crack location ascending order respectively. Medium ranges of relative crack location ascending order respectively. Bigger ranges of relative crack location ascending order. in in in in in Table: 4.2 Fuzzy Rules for Fuzzy Inference System Sl. No Some Examples of Fuzzy rule used in the Fuzzy Controller 1 If fnf is L1F1, snf is L2F1, tnf is L3F1 then rcd is SD1 and rcl is SL1 2 If fnf is L1F1, snf is L2F2, tnf is L3F2 then rcd is SD2 and rcl is SL2 3 If fnf is L1F1, snf is L2F2, tnf is L3F3 then rcd is SD1 and rcl is SL2 4 If fnf is M1F1, snf is M2F1, tnf is M3F1 then rcd is MD1 and rcl is ML1 5 If fnf is M1F1, snf is M2F2, tnf is M3F2 then rcd is MD2 and rcl is ML2 6 If fnf is M1F1, snf is M2F2, tnf is M3F3 then rcd is MD1 and rcl is ML2 7 If fnf is M1F2, snf is M2F1, tnf is M3F1 then rcd is MD2 and rcl is ML1 8 If fnf is M1F2, snf is M2F2, tnf is M3F2 then rcd is MD2 and rcl is ML3 9 If fnf is M1F3, snf is M2F1, tnf is M3F2 then rcd is MD3 and rcl is ML1 10 If fnf is M1F2, snf is M2F3, tnf is M3F2 then rcd is MD1 and rcl is ML3 11 If fnf is L1F2, snf is L2F1, tnf is L3F1 then rcd is SD2 and rcl is SL1 12 If fnf is L1F2, snf is L2F3, tnf is L3F3 then rcd is SD2 and rcl is SL3 13 If fnf is L1F3, snf is L2F1, tnf is L3F2 then rcd is SD3 and rcl is SL1 14 If fnf is L1F2, snf is L2F3, tnf is L3F2 then rcd is SD1 and rcl is SL3 15 If fnf is L1F3, snf is L2F3, tnf is L3F3 then rcd is SD3 and rcl is SL3 16 If fnf is M1F3, snf is M2F3, tnf is M3F3 then rcd is MD3 and rcl is ML3 17 If fnf is H1F1, snf is H2F1, tnf is H3F1 then rcd is LD1 and rcl is BL1 18 If fnf is H1F1, snf is H2F2, tnf is H3F2 then rcd is LD2 and rcl is BL2 19 If fnf is H1F1, snf is H2F3, tnf is H3F3 then rcd is LD1 and rcl is BL2 20 If fnf is H1F2, snf is H2F1, tnf is H3F1 then rcd is LD2 and rcl is BL1 21 If fnf is H1F2, snf is H2F2, tnf is H3F2 then rcd is LD2 and rcl is BL3 22 If fnf is H1F3, snf is H2F1, tnf is H3F2 then rcd is LD3 and rcl is BL1 23 If fnf is H1F2, snf is H2F3, tnf is H3F2 then rcd is LD1 and rcl is BL3 24 If fnf is H1F3, snf is H2F3, tnf is H3F3 then rcd is LD3 and rcl is BL3 National Institute of Technology Rourkela 47 Chapter 4 Fuzzy Inference System Inputs for triangular membership function Rule No. 3 of Table: 4.2 is activated Rule No.12 of Table: 4.2 is activated Outputs obtain from triangular membership function 0.169 Relative Crack Depth 0.255 Relative Crack Location Figure: 4.8 Resultant values of relative crack depth and relative crack location of triangular membership function when Rules 3 and 12 of Table: 4.2 are activated National Institute of Technology Rourkela 48 Chapter 4 Fuzzy Inference System Inputs for trapezoidal membership function Rule No. 3 of Table: 4.2 is activated Rule No.12 of Table: 4.2 is activated Outputs obtain from trapezoidal membership function 0.170 Relative Crack Depth 0.260 Relative Crack Location Figure: 4.9 Resultant values of relative crack depth and relative crack location of trapezoidal membership function when Rules 3 and 12 of Table: 4.2 are activated National Institute of Technology Rourkela 49 Chapter 4 Fuzzy Inference System Inputs for Gaussian membership function Rule No. 3 of Table: 4.2 is activated Rule No.12 of Table: 4.2 is activated Outputs obtain from Gaussian membership function 0.165 Relative Crack Depth 0.245 Relative Crack Location Figure: 4.10 Resultant values of relative crack depth and relative crack location of Gaussian membership function when Rules 3 and 12 of Table: 4.2 are activated National Institute of Technology Rourkela 50 Chapter 4 Fuzzy Inference System Inputs for Hybrid membership function Rule No. 3 of Table: 4.2 is activated Rule No.12 of Table: 4.2 is activated Outputs obtain from Hybrid membership function 0.161 Relative Crack Depth 0.262 Relative Crack Location Figure: 4.11 Resultant values of relative crack depth and relative crack location of hybrid membership function when Rules 3 and 12 of Table: 4.2 are activated National Institute of Technology Rourkela 51 Chapter 4 Fuzzy Inference System 4.5.2. Comparison of Results Table: 4.3 Comparison of Results between Theoretical Analysis and different Fuzzy Controller Analysis Relative First natural frequency fnf Relative Second natural frequency snf Relative Third natural frequency tnf 0.8142 0.9537 0.8635 Theoretical Triangular Fuzzy Controller rcd rcl Trapezoidal Fuzzy Controller rcd rcl Gaussian Fuzzy Controller rcd rcl Hybrid Fuzzy Controller rcd rcl Relative crack depth rcd Relative crack location rcl 0.9266 0.3167 0.125 0.314 0.124 0.315 0.126 0.316 0.125 0.316 0.128 0.9737 0.9335 0.3 0.1875 0.296 0.189 0.297 0.190 0.298 0.190 0.305 0.186 0.9013 0.9813 0.9470 0.2834 0.25 0.280 0.255 0.285 0.260 0.281 0.245 0.287 0.262 0.9315 0.9867 0.9523 0.2667 0.3125 0.270 0.320 0.272 0.323 0.268 0.313 0.261 0.312 0.9544 0.9888 0.9664 0.25 0.375 0.252 0.379 0.252 0.382 0.245 0.372 0.245 0.379 0.9692 0.9905 0.9757 0.2334 0.4375 0.240 0.442 0.239 0.445 0.233 0.440 0.224 0.425 0.9839 0.9917 0.9845 0.2167 0.5 0.224 0.505 0.227 0.495 0.214 0.512 0.215 0.498 0.9908 0.9946 0.9855 0.2 0.5625 0.213 0.571 0.212 0.565 0.21 0.561 0.23 0.567 0.9964 0.9967 0.9993 0.1834 0.625 0.185 0.635 0.179 0.637 0.182 0.629 0.182 0.632 0.9986 0.9980 0.9994 0.1667 0.6875 0.169 0.69 0.170 0.72 0.165 0.687 0.161 0.681 4.5.3. Summary A fuzzy controller has been designed, which uses three natural frequencies as inputs where as the crack depth and crack location as output. It has been observed that the natural frequency of the beam is changing with Crack depth and crack location. The predicted results from fuzzy controllers for crack location and crack depth are compared with the theoretical results. It is observed from the Table: 4.3 that the results obtained from Gaussian membership function fuzzy controller predict more accurate result in comparison to other three controllers. 4.6. Why Fuzzy Logic is Used Provides an easy to use interface for applying modern fuzzy logic techniques. Does not require mathematical formulation. Powerful tool for dealing with imprecision, uncertainty. Precision is traded for tractability, robustness and low cost solution. National Institute of Technology Rourkela 52 Chapter 4 Fuzzy Inference System Provides the ability to use fuzzy logic when appropriate with other control techniques. It may be used for real world model like behavior of a human being. Supplies a fuzzy inference engine that can execute the fuzzy system as a stand-alone application. National Institute of Technology Rourkela 53 ANALYSIS OF KOHONEN NETWORK FOR IDENTIFICATION OF CRACK 5.1. Introduction 5.2. Essential processes of kohonen network/ the SOM training Algorithm 5.2.1. Initialization 5.2.2. Competitive Process 5.2.3. Co-operative Process 5.2.4. Synaptic adaptation Process 5.3. Mechanism 5.3.1. Competition Mechanism 5.3.2. Co-operative Mechanism 5.3.3. Adaptive Mechanism 5.4. Flow chart of Kohonen network 5.5. Comparison of Results 5.5.1. Discussion 5.5.2. Summary National Institute of Technology Rourkela 54 Chapter 5 CHAPTER 5 Kohonen Network Analysis of Kohonen Network for Identification of Crack 5.1. Introduction The Self-Organizing Map (SOM), commonly also known as Kohonen network (Kohonen 1982, Kohonen 2001) is a computational method for the visualization and analysis of highdimensional data, especially experimentally acquired information [53]. The self-organizing map (SOM) network was originally designed for solving problems that involve tasks such as clustering, visualization, and abstraction. While Kohonen‟s SOM networks have been successfully applied as a classification tool to various problem domains, their potential as a robust substitute for clustering and visualization analysis remains relatively un researched. The self-organizing map (SOM) network is a special type of neural network that can learn from complex, multi-dimensional data and transform them into visually decipherable clusters. The Kohonen network (Kohonen, 1982, 1984) can be seen as an extension to the competitive learning network, although this is chronologically incorrect. Also, the Kohonen network has a different set of applications. In the Kohonen network, the output units are ordered in some fashion, often in a two dimensional grid or array, although this is application-dependent. The ordering, which is chosen by the user1, determines which output neurons are neighbors. The main function of SOM networks is to map the input data from an n-dimensional space to a lower dimensional (usually one or two-dimensional) plot while maintaining the original topological relations. The physical location of points on the map shows the relative similarity between the points in the multi-dimensional space. Competitive learning (Kohonen, 1982) is a special case of Self organizing Map. In Self-Organizing Map, it Transform as input signal pattern of arbitrary dimension into one or two dimensional discrete map, Perform this transformation adaptively in a topological ordered fashion. Winner takes all neuron. Two possible architectures are existing in Kohonen Network, which are shown in Figure: 5.1. National Institute of Technology Rourkela 55 Chapter 5 Kohonen Network One Dimensional Output Two Dimensional Outputs Figure: 5.1 Possible Architectures in Kohonen Network 5.2. Essential Processes of Kohonen Network/ the SOM Training Algorithm The SOM network typically has two layers of nodes, the input layer and the Kohonen layer. The input layer is fully connected to a two-dimensional Kohonen layer. During the training process, input data are fed to the network through the processing elements (nodes) in the input layer. An input pattern xm (m=1, 2, 3…, m) is denoted by a vector of order m as: x m =(x1, x2… xm), where xm is the mth input signal in the pattern and m is the number of input signals in each pattern. An input pattern is simultaneously incident on the nodes of a twodimensional Kohonen layer. Associated with the N nodes in the m × l (N =m × l) Kohonen layer, is a weight vector, also of order l; denoted by: wj = (wj1, wj2… wjl), where wjl is the weight value associated with node j corresponding to the lth signal of an input vector. As the training process proceeds, the nodes adjust their weight values according to the topological relations in the input data. The node with the minimum distance is the winner and adjusts its weights to be closer to the value of the input pattern. In this study, Euclidean distance is the most common way of measuring distance between vectors, is used. The procedure with details being as follows. 5.2.1. Initialization Each nodes weight is initialized. A vector is chosen at random from the set of training data and presented to the lattice. When the input vector is presented to the map, its distance to National Institute of Technology Rourkela 56 Chapter 5 Kohonen Network the weight vector of each node is computed. The map returns the closest node which is called the Best Matching Unit (BMU). BMU Figure: 5.2 Process of Initialization in Kohonen Network 5.2.2. Competitive Process It is also called long range inhibition*. For each input pattern, the neurons in the output layer will determine the value of a function. That function will be calling as the discriminant function. Each neuron computes a discriminant function. The neuron with largest discriminant function is the winner. “A continuous input space of activation pattern is mapped onto a discrete output space of neurons by a process of competition among the neurons in the network”. Winner neuron= arg j max( w jT x ) arg i min( x w j ) (5.1) fnf snf tnf Winner Figure: 5.3 Process of Competition in Kohonen Network National Institute of Technology Rourkela 57 Chapter 5 Kohonen Network 5.2.3. Co-operative Process It is also called short range excitation*. The winning neuron locates the Centre of a topological neighborhood of co-operating neurons. A neuron that is firing tends to excite the neurons in its immediate neighborhood more than those farther away from it. Excitation is a co-operation. It strengthens the neuron which are closer to winner, where as the process of competition the neurons which are far apart, they are eliminated. d 2 j,i h ji exp 2 2 (5.2) *Neurons which are there at the output, they are act in a competitive manner, in the sense that they inhibit the responses of each other. The neurons which are close to the winning neuron, they tends to have an excitatory response; that means to say around the winning neuron an excitatory response is generally created where as an inhibitory response is created for the neurons which are there far distant apart. This short of networks will exhibit long range inhibition and short range excitation. Figure: 5.4 Process of Co-operative in Kohonen Network 5.2.4. Synaptic Adaptation Process It enables the excitation/excited neurons to increase their individual values of discriminant function in relation to the input pattern. The discriminant function values for the nonexcited will be kept unchanged. Response of winning neuron is increased. National Institute of Technology Rourkela 58 Chapter 5 Kohonen Network The output of the map is usually sent to another learning machine which will finish the process of pattern recognition. 5.3. Mechanism The training of the Kohonen Network is done by a specific algorithm. The goal is to obtain a map where two points, which are nearby in the input space are also closed in the map. The algorithm of Kohonen network is processed through various mechanisms as explained below. 5.3.1. Competition Mechanism [ x] x1, x 2 ,..., x m T Where, x is the input vector, m= Dimensional Input. [ w j ] w j1, w j2 ,..., w j T , j=1, 2, 3...l, l= Number of output neurons. (5.3) (5.4) w j = Weight Vector Every output is connected to all input. There will be m l number of arrays. We have to determine the best match between x and w j . Compute w jT x for j=1, 2 …l. Winning neuron= arg maximize ( w jT x ) , we have to minimize the Euclidian distance ( xwj ) Winner neuron= arg j max( w jT x ) arg i min( x w j ) (5.5) 5.3.2. Co-operative Mechanism In addition to winning neuron all the neighborhood neuron should adjust their weights. Winning neuron: i Topological neighborhood: h j,i : Topological neighborhood centered on i; encompassing neuron j. d j,i : Lateral distance between the winning neuron „i‟ and excited neuron j. Satisfying two properties: National Institute of Technology Rourkela 59 Chapter 5 Kohonen Network Symmetric about dj,i→0 and considering monotonically decaying function, we can get d 2 j,i , h j,i ( x ) exp 2 2 (5.6) =width of Gaussian function, is not constant with time/iteration. As the iteration processes, is going to be decrease. The topological neighborhood hj,i shrinks and narrow down with time. t ( t ) 0 exp , 1 (5.7) t = number of iteration. 0 = Initial (at t=0), 1 = time constant is a function of iteration number. When t 1 , ( t ) decreases to 0.37 of its maximum value. t= 0, 1, 2 . . . d 2 j,i h j,i ( x ) exp 2 2 t (5.8) h j,i t is called the neighborhood function (the more a node is far from the BMU the smaller value is returned by this function). 5.3.3. Adaptive Mechanism In this step we have to update the weights in relation to inputs. w j t 1 w j t t h j,i t x w j t (5.9) Where t = Learning Rate and t t 0 exp 2 (5.10) Where, 2 is another time constant National Institute of Technology Rourkela 60 Chapter 5 Kohonen Network 5.4. Flow Chart of Kohonen Network Start Initialization Draw an Input Sample Winner neuron= Update End Figure: 5.5 Flow Chart showing the processes of Kohonen Network National Institute of Technology Rourkela 61 Chapter 5 Kohonen Network 5.5. Comparison of Results Table: 5.1. Comparison of Results between Theoretical Analysis and Kohonen Network Technique Relative First natural frequency fnf Relative Second natural frequency snf Relative Third natural frequency tnf 0.8142 0.9537 0.8635 Theoretical Kohonen Network Technique Relative crack depth rcd Relative crack location rcl Relative Crack depth rcd Relative crack location rcl 0.9266 0.3167 0.125 0.317 0.122 0.9737 0.9335 0.3 0.1875 0.302 0.187 0.9013 0.9813 0.9470 0.2834 0.25 0.291 0.258 0.9315 0.9867 0.9523 0.2667 0.3125 0.267 0.321 0.9544 0.9888 0.9664 0.25 0.375 0.253 0.374 0.9692 0.9905 0.9757 0.2334 0.4375 0.242 0.438 0.9839 0.9917 0.9845 0.2167 0.5 0.217 0.505 0.9908 0.9946 0.9855 0.2 0.5625 0.198 0.563 0.9964 0.9967 0.9993 0.1834 0.625 0.186 0.615 0.9986 0.9980 0.9994 0.1667 0.6875 0.167 0.689 5.5.1. Discussion Kohonen network technique has been developed for the prediction of crack depth and crack location. Kohonen network can be viewed as a clustering method so that similar data samples tend to be mapped to nearby neurons. The complete architecture of the Kohonen network, the essential processes and the mechanism of the competitive learning algorithm has been discussed in the different sections. Finally the steps involved for Kohonen network technique is presented through a flow chart. 5.5.2. Summary Kohonen network technique is nothing but a competitive learning algorithm is developed for crack detection, which uses three natural frequencies as inputs where as the crack depth and crack location as output. The predicted results from Kohonen network technique for crack location and crack depth are compared with the theoretical results and it is observed that process of Kohonen network predict the depth and location accurately as close to theoretical technique. National Institute of Technology Rourkela 62 EXPERIMENTAL SETUP FOR IDENTIFICATION OF CRACK 6.1. Introduction 6.2. Experimental Setup 6.2.1. Instruments Used 6.2.2. Description 6.3. Discussion National Institute of Technology Rourkela 63 Chapter 6 CHAPTER 6 Experimental Analysis Experimental Setup for Identification of Crack 6.1. Introduction Experimental Analysis plays a vital role in the research work. Experimental Analysis is being carried out to justify the validation of theoretical analysis and different intelligent techniques projected in the chapter 3 to 5 for localization and identification of crack. For the analysis, the experimental setup is made to measure the natural frequency and to observe the response of cantilever beam with the presence of transverse crack. The experimental setup is discussed in detail in the subsequent sections of this chapter. An aluminum beam specimen of dimension (800 x 38 x 6mm) is selected for the experimental analysis. The schematic diagram of the complete experimental setup is shown in Figure: 6.1. It is shown in the figure that the vibration exciter is driven by a function generator connected to a power amplifier. An oscilloscope is connected to observe the vibrational response of cracked cantilever beam after getting the signal from the vibration pick up. The detailed specifications of the instruments used in this analysis are given below. 6.2. Experimental Setup The experiment has been conducted in two ways. The pictorial view of experimental setup1 and setup2 are shown in the Figures: 6.1. In the Figure: 6.1(a) a cracked cantilever beam is rigidly clamped to the concrete foundation base. The free end of the cantilever beam is excited with a vibration exciter. The vibration exciter is excited by the signal from the function generator. The signal is amplified by a power amplifier before being fed to the vibration exciter. The natural frequency is measured from the function generator at the point of resonance under the excitation. In the Figure: 6.1(b) the same cantilever beam is taken into consideration. The free end of the cantilever beam is excited freely with the help of thumb and allowed to vibrate freely. The amplitude of vibration of un cracked and cracked cantilever beam is taken by the vibration pick up and is fed to the digital storage oscilloscope. The vibration signatures are analyzed graphically in the oscilloscope and the natural frequency of the beam is calculated. National Institute of Technology Rourkela 64 Chapter 6 Experimental Analysis 2 4 5 3 Figure: 6.1(a) Pictorial View of complete assembly of Experimental Setup1 6 5 1 Figure: 6.1(b) Pictorial View of complete assembly of Experimental Setup2 6.2.1. Instruments Used The various instruments used in the experimental analysis, the specification as well as the views of the instruments are shown in the Table: 6.1 National Institute of Technology Rourkela 65 Chapter 6 Experimental Analysis Table: 6.1 Instruments used in the experimental Analysis Instruments Specifications View of the Instrument 1. Vibration Pick up (Accelerometer) electromechanical Delta Tron Accelerometer An transducer capable of converting mechanical vibration into electrical voltages. Depending upon their sensing element and Type : Make : 4513-001 Bruel & kjaer Frequency Range : 1Hz-10KHz Supply voltage output : 24volts characteristics, such pickups are Operating temperature Range referred to as accelerometer, : -500C to +1000c velocity pickups or displacement pickups. 2. Function Generator It is a moving coil device with a Model :FG200K frequency in excess of 0.2Hz to Frequency Range : 0.2Hz to 200 KHz 200 KHz. In our analysis the natural frequency is calculated from the function generator at the point of resonance. VCG IN connector for Sweep Generation Sine, Triangle, Square, TTL outputs. Output Attenuation up to 60dB. Output Level: 15Vp-p into 600 ohms Rise/Fall Time: <300nSec Make :Aplab 3. Power Amplifier An RF Power Amplifier is a Type : 2719 type of electronic amplifier used Power Amplifier: to convert a low power radio frequency signal into a larger Make : 180VA Bruel & kjaer signal of significant power. National Institute of Technology Rourkela 66 Chapter 6 Experimental Analysis 4. Vibration Exciter A vibration exciter (shaker) is Type : 4808 an electro-mechanical device Permanent which transforms electrical a. c. signals into mechanical vibrations and is used to excite vibrations in bodies or Magnetic Vibration Exciter Force rating 112N (25 lbf) sine peak (187 N (42 lbf) with cooling) Frequency Range: structures for testing purposes. 5 Hz to 10Hz During the past decade a wide First Axial Resonance: variety of vibration exciters Hz have been developed, their Maximum Bare 10 Table fields of application ranging Acceleration : 700 m/s2 (71 from fatigue testing of g) automobile, missile and aircraft Continuous 12.7 mm (0.5 in) peak-to-peak displacement components, to the calibration with over travel stops Two high-quality, 4-pin of vibration pick-ups. Neutrik® Speakon® connectors Make : 5. Specimen with Bruel & kjaer the Concrete Foundation A cantilever type cracked Dimension of the Aluminum beam specimen is specimen: (800 × 38 × 6 used for the analysis. mm) 6. Oscilloscope An oscilloscope is commonly Band Width: DC ~ 60 known as CRO (Cathode Ray MHz Oscilloscope). It is an electronic Sample Rate : 100 Sa/S test instrument that allows observation of constantly varying signal voltages, usually in a twodimensional graph of one or more electrical potential differences using the vertical or 'Y' axis, Channels: 2 Storage Memory: 16 K/Ch Vertical Sensitivity: 2 mV ~ 5V Rise Time: 3.5ns plotted as a function of time using Power Supply: 100V ~ National Institute of Technology Rourkela 67 Chapter 6 the horizontal Experimental Analysis or 'x' axis. 240V/ 50W max Oscilloscopes are commonly used Make: Metravi OS 5060 to observe the amplitude of the signal. 6.2.2. Description An Aluminum beam specimens of dimension (800 × 38 × 6mm) with a transverse crack is taken for the experimental analysis for determining the natural frequencies at different crack locations and crack depths. These specimens are allowed to vibrate under 1st, 2nd and 3rd mode of vibrations. The experimental results of corresponding amplitudes are recorded in the digital storage oscilloscope at various locations along the length of the beam and also observed through function generator at the point of resonance. 6.3. Discussion The experimental results of relative natural frequencies at different relative crack locations and relative crack depths for 1st, 2nd and 3rd vibration are presented in Chapter 7. By feeding the relative natural frequency to the fuzzy controller as well as Kohonen network the relative crack depth and relative crack location is obtained. The detail mythologies have been explained in Chapter 4 and 5. A comparison made between the results obtained from the experiment, fuzzy logic and Kohonen network. The results are shown in a tabular form in the Chapter 7. A graph has been plotted to show the comparison among all the results. National Institute of Technology Rourkela 68 RESULTS AND DISCUSSIONS 7.1. Introduction 7.2. Discussions 7.3. Comparison of Results 7.4. Characteristic Curves National Institute of Technology Rourkela 69 Chapter 7 Results and Discussions CHAPTER 7 Results and Discussions 7.1. Introduction An Aluminum beam (cantilever beam) specimen with transverse crack is used to obtain the natural frequencies. Further the natural frequencies have been used as the training data for the fuzzy inference system and Kohonen competitive learning network. The results obtained from both the techniques have been discussed and analyzed in this chapter. At the end of this chapter a comparative result has been shown between the theoretical analysis, fuzzy logic and kohonen network. 7.2. Discussions Including the introductory part and literature survey the present research work has been processed in six stages. The other stages are theoretical vibration analysis for identification of crack, analysis of fuzzy inference system for identification of crack, analysis of kohonen network for identification of crack and the experimental setup for identification of crack. The various methodologies used for identification of crack since last few decades has been stated in the literature survey. In the first phase of theoretical vibration analysis, various expressions are developed to obtain the natural frequencies under the consideration of free and forced vibration analysis of the single crack cantilever beam and in the second phase, the expressions has been developed using finite element analysis. The data obtained from the theoretical analysis has been used as the training data for the fuzzy inference system and kohonen network. The detail analysis has been explained in Chapter 3. In the Chapter 4 a fuzzy inference system has been developed for the identification of crack (crack depth and crack location) using different types of membership functions, such as triangular, trapezoidal, Gaussian and hybrid function. The developed fuzzy inference system uses three natural frequencies as inputs and the crack depth and crack location as output, the schematic diagram of the fuzzy inference system is stated in Figure: 4.3. Several linguistic terms and the fuzzy rules have been developed for the design of fuzzy inference system. Some of the linguistic terms and the fuzzy rules are stated in the Table: 4.1 & 4.2. The National Institute of Technology Rourkela 70 Chapter 7 Results and Discussions complete structure of the membership function has been presented in the Figures: 4.4 to 4.7. The defuzzification results have been obtained by activating the rule no. 3 and 12 of Table: 4.2, which are demonstrated in the Figures: 4.8 to 4.11. It is observed from the Table: 4.3 that the results obtained from Gaussian membership function fuzzy controller predict more accurate result in comparison to other three controllers and the computational time for crack detection using fuzzy inference system is considerably lower as compared to theoretical analysis. In the Chapter 5 a Kohonen network technique has been developed for the prediction of crack depth and crack location. Kohonen network can be viewed as a clustering method so that similar data samples tend to be mapped to nearby neurons. The architecture of the kohonen network is presented in the Figure: 5.1. The essential processes of kohonen network technique are presented in the section 5.2 and also explained through the Figures: 5.2 to 5.4. The kohonen network technique is an extension to competitive learning network and the training of the Kohonen Network is done by a specific algorithm. The processes of the kohonen network have been presented with the help of a flow chart, which is depicted in the Figure: 5.5. The data obtained from the theoretical analysis has been trained to the kohonen network and the predicted results of crack location and crack depth is shown in the Table: 5.1. It has been observed that the prediction of crack location and crack depth from the kohonen network technique is very close to the actual results. Chapter 6 describes the complete architecture of the experimental setup. The complete view of various instruments used with the descriptions and the specifications are presented in the Table: 6.1. The experiment has been conducted in two ways and the pictorial view of complete assembly of setup-1 and setup-2 are shown in Figure: 6.1. Experimental Analysis is being carried out to justify the validation of theoretical analysis and different intelligent techniques used in this research work (Fuzzy logic and Kohonen Network). In the last section of this chapter a comparative result shown between theoretical, experimental, fuzzy logic and kohonen network. The conclusions and scope for future work of the above analysis have been discussed in the next chapter. National Institute of Technology Rourkela 71 Chapter 7 Results and Discussions 7.3. Comparisons of the Results Table: 7.1. Comparison of Results between Theoretical Analysis, Experimental Analysis, Fuzzy Controller Analysis and Kohonen Network Technique Relative First natural frequency fnf Relative Second natural frequency snf Relative Third natural frequency tnf 0.8142 0.8635 0.9013 0.9315 0.9544 0.9692 0.9537 0.9737 0.9813 0.9867 0.9888 0.9905 0.9839 0.9908 0.9964 0.9986 0.9917 0.9946 0.9967 0.9980 Theoretical Experimental Fuzzy Controller Kohonen Network Technique rcd rcl Relative crack location rcl 0.125 0.1875 0.25 0.3125 0.375 0.4375 Rcd rcl rcd rcl 0.9266 0.9335 0.9470 0.9523 0.9664 0.9757 Relative crack depth rcd 0.3167 0.3 0.2834 0.2667 0.25 0.2334 0.315 0.299 0.282 0.267 0.248 0.234 0.124 0.185 0.260 0.315 0.377 0.438 0.316 0.298 0.281 0.268 0.245 0.233 0.125 0.190 0.245 0.313 0.372 0.440 0.317 0.302 0.291 0.267 0.253 0.242 0.122 0.187 0.258 0.321 0.374 0.438 0.9845 0.9855 0.9993 0.9994 0.2167 0.2 0.1834 0.1667 0.5 0.5625 0.625 0.6875 0.219 0.203 0.182 0.165 0.505 0.569 0.623 0.679 0.214 0.21 0.182 0.165 0.512 0.561 0.629 0.687 0.217 0.198 0.186 0.167 0.505 0.563 0.615 0.689 7.4. Characteristic Curves Relative Crack Depth 0.35 0.3 0.25 0.2 Theoretical Experimental 0.15 Fuzzy 0.1 Kohonen 0.05 0 0.8 0.820.840.860.88 0.9 0.920.940.960.98 1 Relative First Natural Frequency Figure: 7.1 Relative First Natural Frequencies versus Relative Crack Depth National Institute of Technology Rourkela 72 Chapter 7 Results and Discussions 0.8 Relative Crack Location 0.7 0.6 0.5 Theoretical 0.4 Experimental 0.3 Fuzzy 0.2 Kohonen 0.1 0 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 Relative First Natural Frequency Figure: 7.2 Relative First Natural Frequencies versus Relative Crack Location 0.35 Relative Crack Depth 0.3 0.25 0.2 Theoretical Experimental 0.15 Fuzzy 0.1 Kohonen 0.05 0 0.95 0.96 0.97 0.98 0.99 Relative Second Natural Frequency 1 Figure: 7.3 Relative Second Natural Frequencies versus Relative Crack Depth Relative Crack Location 0.8 0.7 0.6 0.5 Theoretical 0.4 Experimental 0.3 Fuzzy 0.2 Kohonen 0.1 0 0.95 0.96 0.97 0.98 0.99 1 Relative Second Natural Frequency Figure: 7.4 Relative Second Natural Frequencies versus Relative Crack Location National Institute of Technology Rourkela 73 Chapter 7 Results and Discussions 0.35 Relative Crack Depth 0.3 0.25 0.2 Theoretical Experimental 0.15 Fuzzy 0.1 Kohonen 0.05 0 0.92 0.94 0.96 0.98 Relative Third Natural Frequency 1 Figure: 7.5 Relative Third Natural Frequencies versus Relative Crack Depth Relative Crack Location 0.8 0.7 0.6 0.5 Theoretical 0.4 Experimental 0.3 Fuzzy 0.2 Kohonen 0.1 0 0.92 0.94 0.96 0.98 Relative Third Natural Frequency 1 Figure: 7.6 Relative Third Natural Frequencies versus Relative Crack Location National Institute of Technology Rourkela 74 CONCLUSIONS 8.1. Conclusions 8.2. Applications 8.3. Scope for Future Work References Publications National Institute of Technology Rourkela 75 Chapter 8 Conclusion CHAPTER 8 Conclusions 8.1. Conclusions The effects of transverse cracks on the vibrating uniform cracked cantilever beam have been presented in this thesis. The main purpose of this research work has been to develop a proficient technique for diagnosis of crack in a vibrating structure in short span of time. The vibration analysis has been done using theoretical, experimental analysis and also it has been carried through using some intelligent techniques like fuzzy logic and kohonen network. In this analysis natural frequency plays an important role for the identification of crack. Crack has been identified in terms of crack depth and crack location. Based on the results of various analyses performed on the cracked cantilever beam structure, the following conclusions are drawn: The analysis has been done on the presence of a transverse crack and it is observed that the presence of crack affects the natural frequency, as a result the natural frequency decreases with the increase in crack depth and it increases with the increase in crack location. So it is concluded that the analysis of change of natural frequencies is effective for prediction of crack in beam like structures. The results of the crack depth and crack location have been obtained from the comparison of the results of the uncracked and cracked beam during the vibration analysis. A fuzzy inference system has been developed using different membership functions for the analysis of crack detection and it is observed that the fuzzy controller predict the results of crack depth and crack location as close to the theoretical and experimental analysis. The important factor of the fuzzy inference system is that it is predicting the results with less computational time. The natural frequencies obtained from the theoretical analysis are used as the training data for fuzzy inference system. It shows a good agreement between theoretical, experimental and fuzzy analysis. National Institute of Technology Rourkela 76 Chapter 8 Conclusion The kohonen network technique is developed using a competitive learning algorithm to predict the crack depth and location by using relative values of three natural frequencies. The predicted results of kohonen network technique are reasonably adequate and in agreement with the experimental result. With the successful detection of crack in a cantilever beam, it has been observed that the new technique developed can be used as an intelligent fault detecting tool for different types of vibrating structures. The experimental analysis shows the effectiveness of the proposed methods towards the identification of location and extent of damage in vibrating structures and it is observed that the changes in the vibration signatures become more prominent as the crack grows bigger. 8.2. Applications As the techniques used for crack detection are non-destructive in nature, so these techniques can be effectively used for online condition monitoring of engineering systems. The techniques developed for crack detection can be used for prediction of crack in flow lines, turbo machinery, nuclear plants and ship structures, biomedical engineering system etc. 8.3. Scope for Future Work Analysis of Fuzzy Inference system and Kohonen Network system can be extended for localization and identification of crack in complex beam structures with multiple cracks. The complete analysis of the current research work is carried out based on the Euler beam like structure and it can be extended for Timoshenko beam like structure. Kohonen network technique and Fuzzy Logic technique can be hybridized to propose another technique for identification of crack in beam like structure. National Institute of Technology Rourkela 77 References References [1] Harish Ch. Das and Dayal R. Parhi, “Online fuzzy logic crack detection of a cantilever beam”, International Journal of Knowledge-based and Intelligent Engineering Systems, Vol. 12, Issue 2, (2008), Pages: 157-171. [2] S.Suresh, S.N.Omkar, Ranjan Ganguli and V. Mani, “Identification of crack location and depth in a cantilever beam using a modular neural network approach” IOP electronic journal, Vol. 13, Issue 4, (2004), Pages: 907-915. [3] Prashant M. Pawar, Ranjan Ganguli, “Matrix Crack Detection in Thin-walled Composite Beam using Genetic Fuzzy System” Journal of Intelligent Material Systems and Structures, Vol. 16, Issue 5, (2005) Pages: 395-409. [4] Harish Ch. Das and D. R. Parhi, “Detection of the Crack in Cantilever Structures Using Fuzzy Gaussian Inference Technique” AIAA Journal, Vol. 47, Issue 1, (2009). [5] Mohammad Taghi and Vakil Baghmisheh, “Crack detection in beam-like structures using genetic algorithms”, Applied Soft Computing ,Vol. 8, Issue 2, (2008), Pages: 1150-1160. [6] Norhisham Bakhary, Hong Hao and Andrew J. Deeks, “Damage detection using artificial neural network with consideration of uncertainties”, Engineering Structures, Vol. 29, Issue 11, (2007), Pages: 2806-2815. [7] Damodar Maity and Asish Saha, “Damage assessment in structure from changes in static parameter using neural networks”, Publisher Springer India, Issue 3, Vol. 29, (2004). [8] X. Fang, H. Luo and J. Tang, “Structural damage detection using neural network with learning rate improvement”, Computers & Structures, Vol. 83, Issues 25-26, (2005), Pages: 2150-2161. [9] K. M. Saridakis, “Applying neural networks, genetic algorithms and fuzzy logic for the identification of cracks in shafts by using coupled response measurements”, Computers and Structures, Vol. 86, Issue 11-12, (2008), Pages: 1318-1338. [10] Huijian Li, Changjun He, Jialin Ji, Hui Wang and Caizhe Hao, “Crack damage detection in beam-like structures using RBF neural networks with experimental validation”, International Journal of Innovative, Computing, Information and Control, Vol. 1, Issue 4, (2005), Pages: 625-634. National Institute of Technology Rourkela 78 References [11] Fernando S. Buezasa, Marta B. Rosalesb, and Carlos P. Filipich, “Damage detection in structural elements taking into account a crack contact model”, Mecanica Computacional, Vol. XXVII, Issue 35, Pages: 2715-2732. [12] S. K. Panigrahi, S. Chakraverty and B. K. Mishra, “Vibration based damage detection in a uniform strength beam using genetic algorithm”, Springer Publication, Vol. 44, Issue 6, (2009), Pages: 697-710. [13] Jung-Huai Chou and Jamshid Ghaboussi, “Genetic algorithm in structural damage detection”, Computers & Structures, Vol. 79, Issue 14, (2001), Pages: 1335-1353. [14] M. W. Suh, M. B. Shim and M. Y. Kim, “Crack identification using hybrid neurogenetic technique”, Journal of Sound and Vibration, Vol. 238, Issue 4, (2000), Pages: 617-635. [15] T. Horibe and N. Asano, “Crack identification in elastically supported beam by genetic algorithm”, International Symposium on Inverse Problems in Engineering Mechanics, (2001), Pages: 161-166. [16] B. Sahoo and Damodar Maity, “Damage assessment of structures using hybrid neurogenetic algorithm”, Applied Soft Computing, Vol. 7, Issue 1, (2007), Pages: 89-104. [17] S.W. Doebling, C.R. Farrar and M.B. Prime, “A summary review of vibration based damage identification methods”, Shock Vibration Digest, 30, (1998), Pages: 91–105. [18] Messina, E.J. Williams and T. Contursi, “Structural damage detection by a sensitivity and statistical-based method”, Journal of Sound Vibration, 216, (1998), Pages: 791– 808. [19] J.B. Kosmatka and J.M. Ricles, “Damage detection in structures by modal vibration characterization”, Journal of Structural Engineering, 125, (1999), Pages: 1384–1392. [20] C.P. Ratcliffe, “Frequency and curvature based experimental method for locating damage in structures”, Journal of Vibration Acoustic, 122, (2000), Pages: 324 329. [21] F. Vestroni and D. Capecchi, „Damage detection in beam structures based on frequency measurements”, Journal of Engineering Mechanics, 126, (2000), Pages: 761–768. [22] W. Gawronski and J.T. Sawicki, “Structural damage detection using modal norms”, Journal of Sound and Vibration, 229, (2000), Pages: 194–198. [23] N. Hu, X. Wang, H. Fukunaga, Z.H. Yao, H.X. Zhang and Z.S. Wu, “Damage assessment of structures using modal test data”, Int. Journal of Solids & Structure, 38, (2001), Pages:3111–3126. [24] S.S. Law, T.H. Chan and D. Wu, “Efficient numerical modal for the damage detection of large scale structure”, Engineering Structure, 23, (2001), Pages: 436–451. National Institute of Technology Rourkela 79 References [25] M. Sahin and R.A. Shenoi, “Quantification and localization of damage in beam-like structures by using artificial neural networks with experimental validation”, Engineering Structure, 25, (2003), Pages: 1785–1802. [26] Sadettin Orhan, “Analysis of free and forced vibration of a cracked cantilever beam”, Journal of Sound and Vibration, Vol. 40, (2007), Pages: 443-450. [27] C. Mei, Y. Karpenko, S. Moody and D. Allen, “Analytical approach to free and forced vibrations of axially loaded cracked Timoshenko beams”, Journal of Sound and Vibration, Vol. 291, (2006), Pages: 1041–1060. [28] E. Viola, P. Ricci and M.H. Aliabadi, “Free vibration analysis of axially loaded cracked Timoshenko beam structures using the dynamic stiffness method”, Journal of Sound and Vibration, Vol. 304, Issues 1-2, (2007), Pages: 124-153. [29] P. N. Saavedra and L. A. Cuitino, “Crack detection and vibration behavior of cracked beams”, Computers & Structures, Vol. 79, Issue 16, (2001), Pages: 1451-1459. [30] Murat Kisa and M. Arif Gurel, “Free vibration analysis of uniform and stepped cracked beams with circular cross sections”, International Journal of Engineering Science, Vol. 45, Issues 2-8, (2007), Pages: 364-380. [31] M. H. H. Shen and J. E. Taylor, “An identification problem for vibrating cracked beams”, Journal of Sound and Vibration, Vol. 150, Issue 3, (1991), Pages: 457-484. [32] Marta B. Rosales, Carlos P. Filipich and Fernando S. Buezas, “Crack detection in beam-like structures”, Engineering Structures, Vol. 31, Issue 10, (2009), Pages: 22572264. [33] G. Gounaris and A. Dimarogonas, “A finite element of a cracked prismatic beam for structural analysis”, Computers & Structures, Vol. 28, Issue 3, (1988), Pages: 309-313. [34] Francois leonard, Jacques lanteigne, Serge lalonde and Yvon turcotte, “Free-vibration behaviour of a cracked cantilever beam and crack detection”, Mechanical Systems and Signal Processing, Vol. 15, Issue 3, (2001), Pages: 529-548. [35] Harish Ch. Das and Dayal. R. Parhi, “Detection of the Crack in Cantilever Structures Using Fuzzy Gaussian Inference Technique”, Institute of Aeronautics and Astronautics, Vol. 47, Issue 1, (2009), Pages: 105-115. [36] M. Chandrashekhar and Ranjan Ganguli, “Damage assessment of structures with uncertainty by using mode- shape curvatures and fuzzy logic”, Journal of sound and vibration, Vol. 326, (2009), Pages: 939-957. National Institute of Technology Rourkela 80 References [37] Harish Ch. Das and Dayal R. Parhi, “Fuzzy-neuro controller for smart fault detection of a beam”, International Journal of Acoustics and Vibrations, Vol. 14, Issue 2, (2009), Pages: 70-80. [38] R.P.R. Hasanzadeh, S.H.H.Sadeghi, M.Ravan, A.R.Moghaddamjoo and R.Moini, “Fuzzy alignment approach to sizing surface cracks by the AC field measurement technique”, NDT&E International, Vol. 44, (2010), Pages: 75–83. [39] Karim Salahshoor, Majid Soleimani Khoshro and Mojtaba Kordestani, “Fault detection and diagnosis of an industrial steam turbine using a distributed configuration of adaptive neuro-fuzzy inference systems”, Simulation Modelling Practice and Theory, Vol.19, (2011), Pages: 1280–1293. [40] T. Kohonen and O. Simula, “Engineering Applications of the Self Organizing Map”, Proceeding of the IEEE, Vol. 84, No. 10, (1996). [41] Marie Cottrell and Patrick Letremy, “Analyzing surveys using the Kohonen algorithm”, Proceedings-European Symposium on Artificial Neural Networks Bruges (Belgium), D-side Publication, ISBN 2-930307-03-X, (2003), Pages: 85-92. [42] Juha Vesanto, Johan Himberg, Esa Alhoniemi and Juha Parhankangas, “Selforganizing map in Matlab: the SOM Toolbox”, Proceedings of the Matlab DSP Conference, Espoo, Finland, (1999), Pages: 35–40. [43] Hannu Kauppinen, Hannu Rautio and Olli Silven, “Non-segmenting defect detection and SOM based classification for surface inspection using color vision”, Conference on Polarization and Color Techniques in Industrial Inspection (SPIE 3826), Munich, Germany, (1999), Pages: 270-280. [44] R.P.R. Hasanzadeh, S.H.H. Sadeghi, M.Ravan, A.R. Moghaddamjoo and R. Moini, “A Fuzzy alignment approach to sizing surface cracks by AC field measurement technique”, NDT&E International, 44 (2011), Pages:75–83. [45] Fernando S. Buezas, Marta B. Rosales and Carlos P. Filipich, “Damage detection with genetic algorithms taking into account a crack contact model”, Engineering Fracture Mechanics, 78 (2011), Pages: 695–712. [46] M. Nobahari and S. M. Seyedpoor, “Structural damage detection using an efficient correlation-based index and a modified genetic algorithm”, Mathematical and Computer Modeling, 53 (2011), Pages: 1798–1809. [47] P. Beena and Ranjan Ganguli, “Structural damage detection using fuzzy cognitive maps and Hebbian learning”, Applied Soft Computing, 11 (2011), Pages: 1014–1020. National Institute of Technology Rourkela 81 References [48] Yi-Yan Liu, Yong-Feng Ju, Chen-Dong Duan and Xue-Feng Zhao, “Structural damage diagnosis using neural network and feature fusion”, Engineering Applications of Artificial Intelligence, 24 (2011), Pages: 87–92. [49] H. Tada, PC Paris and Irwin GR, “The stress analysis of cracks hand book”, Del Research Corp. Hellertown, Pensylvania, (1973). [50] Z. K. Peng, Z. Q. Lang, and S. A. Billings, “Crack detection using nonlinear output frequency response functions”, Journal of Sound and Vibration, Vol. 301,(2007) Pages: 777-788. [51] Eiji Mizutani, Jyh-Shing Roger Jang and Chuen-Tsai Sun, “Neuro-Fuzzy and Soft Computing”, Computational Approach to Learning and Machine Intelligence, Pearson Prentice Hall, (2007). [52] S. Sumathi and P. Surekha, “Computational Intelligence Paradigms”, Theory and Applications Using Matlab, CRC Press, Taylor & Francis Group, (2010). [53] Melody Y. Kiang, “Extending the Kohonen self-organizing map networks for clustering analysis”, Journal of Computational statistics and data analysis, Vol.38, (2001), Pages: 161-180. National Institute of Technology Rourkela 82 Publications Publications [1] Dayal R. Parhi and Sasanka Choudhury, “Intelligent Fault Detection of a Cracked Cantilever beam using Fuzzy Logic Technology with Hybrid Membership Functions”, International Journal of Artificial Intelligence and Computational Research, 3(1), 2011, Pages:13-20. [2] Dayal R. Parhi and Sasanka Choudhury, “Analysis of Smart Crack Detection Methodologies in various structures”, Journal of Engineering and Technology Research (Academic Journals). Accepted for Publication. [3] Dayal R. Parhi and Sasanka Choudhury, “Smart crack detection of a Cracked Cantilever beam Using Fuzzy Logic Technology with hybrid membership functions”, Journal of Engineering and Technology Research (Academic Journals), Accepted for Publication. [4] Sasanka Choudhury and Dayal R. Parhi, “Analysis of fault detection methodologies in various structures”, International Journal of Structural Engineering (Inderscience Publishers), Accepted for Publication. [5] Sasanka Choudhury, Sasmita Sahu, Dayal R. Parhi and Satish Choudhury, “ Intelligent Fault Detection in Beam like Structures using Fuzzy Logic Technology”, International Conference on Process automation Control and Computing, IEEE,2011, Accepted for Publication. [6] Sasmita Sahu, Sasanka Choudhury and Dayal R. Parhi, “Vibration Analysis of Cracked Beam using Genetic Controller”, International Conference on Process automation Control and Computing, IEEE, 2011, Accepted for Publication. National Institute of Technology Rourkela 83

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