Contents:
Health monitoring of helicopter rotor in forward flight using fuzzy logic. Torsional and bending vibration measurements on rotors using laser technology. Neural network modelling of oscillatory loads and fatigue damage estimation of helicopter components. A summary review of vibration based damage identification methods.
Detection of structural damage through changes in frequency: Delamination detection in composite laminates from variations of their modal characteristics. Free vibration behavior of a cracked beam and crack detection. Vibration based model dependant damage delamination identification and health monitoring for composite structures: Effect of modal curvatures on damage detection using model updating.
A frequency and curvature based experimental method for detecting damage in structures. Damage detection using frequency response function curvature method. Damage detection in structures using a few frequency response measurements. Experimental verification of intelligent fault detection in rotor blades. Rotor acoustic monitoring system RAMS: Damage identification of chordwise crack size and location in uncoupled composite rotorcraft flexbeams. Detecting chordwise cracks and delamination in uncoupled composite rotorcraft flexbeams under rotation.
Computed tomography as a non-destructive test method for fiber main rotor blades in development, series and maintenance. Damage detection testing on a helicopter flexbeam. Structural health monitoring techniques for aircraft structures. Adaptive estimation methodology for helicopter blade structural damage detection. Diagnostic reasoning based on a genetic algorithm operating in a Bayesian belief network.
Natural computing for mechanical systems research: Neural Networks and Learning Machines, 3rd edn. Pearson Education, Upper Saddle River An Introduction to Neural Networks. Taylor and Francis, London Fundamentals of Neural Networks: Architectures, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs Estimation of degraded laminate composite properties using acoustic wave propagation model and a reduction—prediction network. Estimation of composite damage model parameters using spectral finite element and neural network.
Artificial neural network based delamination prediction in laminated composites. Neural network application in fatigue damage analysis under multiaxial random loadings. Fatigue 28, — On-line fan blade damage detection using neural networks. Neural network method based on a new damage signature for structural health monitoring. Damage detection in vibrating bodies using genetic algorithms. Identification of delamination in composite beams using spectral estimation and a genetic algorithm.
Damage detection for composite plates using lamb waves and projection genetic algorithm. An hybrid real genetic algorithm to detect structural damage using modal properties. Structural damage detection and identification using fuzzy logic. Integrated oil debris and vibration gear damage detection technologies using fuzzy logic. Calibration of piezo-impedance transducers for strength prediction and damage assessment of concrete. A fuzzy system for concrete bridge damage diagnosis. A review of uncertainty in flight vehicle structural damage, monitoring, diagnosis and control: A historical review of evolutionary learning methods for Mamdani-type fuzzy rule based systems: Two-stage structural damage detection using fuzzy neural networks and data fusion.
Damage assessment of composite structures using fuzzy logic integrated neural-network approach. Neuro-fuzzy approach to gear system monitoring. NN-based structural damage diagnosis using measured vibration data. A neuro-fuzzy technique for fault diagnosis and its application to rotating frequency. The input information is typically obtained from sensors placed on the structure or embedded in the structure.
Sensor measurements are often subjected to some processing before being used as damage indicators. Though considerable effort is made to find damage indicators which show a high degree of sensitivity to the damage size and location, the problems of ambiguity, imprecision, and noise present in the measured data are likely to propagate into the damage indicators.
In fact, in some cases, the signal processing performed on the sensor measurements may increase the noise level in the damage indicators. As an example, differentiation of data to get a damage indicator can amplify the noise contamination. Based on this information from sensors, a damage detection system should provide definite outputs to help the maintenance engineers. For example, the following questions need to be answered. Is there damage in the system?
Where is the damage? What is the damage size? What needs to be done? How much longer can the structure be used? The answers to these questions should be as accurate as possible. Answers expressed in words instead of in numbers are often more useful for maintenance engineers. Among the several soft computing methods, fuzzy logic is the one which maps numerical inputs into linguistic outputs. However, fuzzy logic lacks the capability of learning from the given data, and the rules which govern the fuzzy system must be developed by human experts. This process of developing the fuzzy rule base is difficult and can become very complicated if the number of inputs and outputs increases.
The hybridization of fuzzy logic with a genetic algorithm gives an advanced soft computing algorithm called the genetic fuzzy system GFS , which generates the fuzzy rules automatically from the data. In this chapter, the terms used in the formulation of a GFS for a damage detection problem are explained. The first step in developing the GFS is to understand the concepts underlying fuzzy logic. These concepts could be numerically quantified using exact weight bounds and exercise bounds.
However, human reasoning does not operate using such numbers and still often reaches surprisingly accurate conclusions using fuzzy rules. Another important fact in fuzzy logic is the degree of membership.
A fuzzy rule could then be stated as: Thus, fuzzy logic can address a variety of situations using the concept of degree of membership in a fuzzy set. This simple example clearly shows how humans constantly assign fuzzy memberships to concepts to perform decision making. The output in this case clearly represents a diagnosis of health. We see that a key feature of fuzzy logic is the use of words for computing. Other soft computing methods such as neural networks and genetic algorithms typically work with numerical data only. In other words, they convert a set of numbers to another set of numbers.
However, fuzzy logic converts a set of words into another set of words. Expert systems can also work with words but use crisp logic in the sense that the concepts can only be either 1 or 0. So the rules of expert systems are similar to if-then-else rules in computer programs. Expert systems tend to deteriorate very rapidly in the presence of noise in the data.
Despite these differences, fuzzy systems can be interpreted as generalizations of rule-based expert systems with the binary logic framework being replaced by fuzzy logic. A fuzzy logic system is a nonlinear mapping of an input feature vector into a scalar output [1]. Fuzzy set theory and fuzzy logic provide the framework for the nonlinear mapping. Fuzzy logic systems have been widely used in engineering applications, because of the flexibility they offer designers and their ability to handle uncertainty.
A fuzzy logic system can be expressed as a linear combination of fuzzy basis functions and is a universal function approximator. A schematic diagram of fuzzy logic is shown in Fig. A very condensed introduction to fuzzy logic is provided below. Further information on fuzzy logic systems is available from textbooks [1—3]. Though fuzzy logic has become a vast topic, its application for structural health monitoring SHM requires the knowledge of only a few key concepts, which are outlined next. Here, R m refers to the space of m real numbers and assumes that m measurement deltas are present as inputs to the fuzzy system.
The notation R refers to the space of real numbers and refers to the output of the fuzzy system. Thus the fuzzy system performs the mapping F: A typical fuzzy logic system maps crisp inputs to crisp outputs using four basic components: Once the rules driving the fuzzy logic system have been fixed, the fuzzy logic system can be expressed as a mapping of inputs to outputs. Rules can come from experts or can be obtained from numerical data. When the rules come from experts, they can be directly represented as words. For example, an expert may suggest that when oil temperature measured by a sensor is high and vibration level at a particular accelerometer A is high, then there is damage at a location B in the structure.
A process of interviewing of experts is often the best way to develop a fuzzy rule base directly from human knowledge which encodes the expert information often used by maintenance engineers. However, for many engineering problems, expert knowledge may not be available about the different conditions of the damaged system. In such cases, a mathematical model of the damaged system is invaluable for creating a rule base linking seeded damages to the damaged indicators. Therefore, the rules for a fuzzy logic system can come either from experts or from a mathematical model, depending on the system under consideration.
Various examples and case studies considered in this book will show some typical input-output sets. To physically understand the fuzzy rules, a damage detection problem for a cantilever beam can be considered where the inputs are the changes in natural frequencies relative to the baseline undamaged beam and the output is the location of the damage. To formulate a fuzzy rule, we need an understanding of 1. Linguistic variables words versus numerical values of a variable e.
Quantifying linguistic variables e. Logical connections between linguistic variables e. We also need to understand how to combine more than one rule. Once the inputs of the fuzzy logic system are identified, a set of linguistic variables must be associated with each input. This process of moving from the number space to the word space is accomplished by the fuzzification process.
The fuzzifier performs the fuzzification and maps numbers into fuzzy sets. This number-to-word transformation is very important in fuzzy logic, as all further operations such as rules are performed on the words. Thus, the fuzzifier is needed to activate rules that are expressed in terms of linguistic variables. An inference engine of the fuzzy logic system maps fuzzy sets to fuzzy sets and determines the way in which the fuzzy sets are combined.
The inference engine therefore performs the operations for the fuzzy rules by converting the inputs expressed in words to the output expressed in words. The application of the rules on the words rather than on the numbers is the main source of the strength of the fuzzy logic system.
The words are relatively insensitive to small changes in numbers and therefore are robust to uncertainty in the inputs. In several applications, numbers are needed as an output of the fuzzy logic system. In those cases, a defuzzifier is used to calculate crisp values from fuzzy values. Thus, the defuzzifier converts the words back to numbers. The internal architecture of the fuzzy system thus operates with words, but the interface to the outside world can be through numbers because of the fuzzifier and the defuzzifier.
We will now formally define the various fuzzy logic terms. A fuzzy set generalizes the concept of an ordinary set whose membership function only takes two values, zero and unity. Thus, an element must either belong to an ordinary set or not belong to it. However, an element can belong to two or more fuzzy 2. Though the concept of height of a person is associated with a clear numerical value, human decisions which are made with height as an input typically use height as a fuzzy variable.
Linguistic Variables A linguistic variable u is used to represent the numerical value x, where x is an element of U. A linguistic variable is usually decomposed into a set of terms T u , which cover its universe of discourse. The designer selects the type of membership function used. There is no theoretical requirement that membership functions overlap. However, one of the major strengths of fuzzy logic is that membership functions can overlap. Fuzzy logic systems are robust because decisions are distributed over more than one input class.
For convenience, membership functions are normalized to one so they take values between 0 and 1, and thus define the fuzzy set. One advantage of the Gaussian membership functions is that they do not suddenly go to zero. This helps in a progressive degradation of the behavior of the fuzzy logic system. Inference Engine Rules for the fuzzy system can be expressed as Ri: This sort of rule covers many applications.
The algebraic product is one of the most widely used operators in applications and leads to product implication. Underlying all this mathematics is a very simple concept that the degrees of memberships of the different parts of the rule are multiplied to get the degree to which a rule has fired. The Gaussian fuzzy sets are useful here, as they ensure that each rule is fired to some degree as none of the membership functions will become exactly zero. In this book, product implication will be used for the fuzzy systems. While centroid defuzzification is widely used for fuzzy control problems where a numerical output is needed, maximum matching is often used for pattern matching problems where we need to know the output class.
For example, a fuzzy control application may require an output in terms of an angle degree or radian for a given actuator. In these applications, the defuzzifier plays an important role. The output in words given by the fuzzy system is also useful for maintenance engineers and can be easily fed directly into the graphical user output. Let Dpi be the measurements of how the pth pattern matched the antecedent conditions IF part of the ith rule, which is given by the product of membership grades of the pattern in the regions which the ith rule occupies: There are several applications of fuzzy logic systems in SHM as well as in the broader area of engineering.
Despite their considerable success, fuzzy systems are limited to problems with a small number of input variables. In addition, the process of developing the fuzzy system requires a lot of judgement and experience on the part of the designer. Two aspects in the design of the fuzzy system are particularly difficult: The rules and the membership functions must accurately capture the relationship between the independent and dependent variables. Unfortunately, the tasks of tuning the membership function and generating rules are not independent. The task of selecting membership functions and rule values is difficult since the information has to be obtained from numerical data of the system 2.
Another problem is selecting an appropriate number of fuzzy sets. Most studies use experience to come up with this number. Often, symmetric fuzzy sets are assumed. However, assuming symmetry in the fuzzy sets also implies assuming symmetry in the system being modeled [4]. The results of successful fuzzy logic systems which one reads in papers and which have been implemented in many practical systems have come after much trial and error on the part of the designer. Typically, the designer selects a level of discretization for the measurement, then assigns the membership function for each fuzzy set, creates the rules, and checks for performance.
If the performance of the fuzzy system is not good, the level of discretization, membership functions, and rules are manually tuned until a reasonably good level of performance is obtained. However, the danger of this approach is that it is ad hoc in nature and the fuzzy systems developed using this method are not optimal. To use the power of fuzzy logic for realistic health monitoring problems, it is necessary to automate the process of fuzzy rule creation.
For SHM problems, a clear metric for maximization is the success rate of the fuzzy system when confronted with test data. The design of the best fuzzy system for SHM is essentially an optimization problem which involves maximization of the success rate. As discussed in the previous chapter, the process of designing the best fuzzy logic system can be accelerated by using genetic algorithms.
We therefore discuss genetic algorithms in the next section. The GA was one of the early examples of bio-inspiration in engineering and has paved the way for many other such methods and concepts. The GA searches the design space for an optimal design point to maximize a fitness function value. Generally, a simple GA contains three basic operations: A typical GA cycle is shown in Fig. The initial population is the potential solution set comprising a reasonably large number of points in the design space and is generated randomly or heuristically.
The general method used at this stage is a random construction of individuals. It is also possible to use a combination of uniform sampling of the design space and random sampling in order to ensure that each part of the design space is represented in the initial population. The next step of the pre-evolution phase is to evaluate the initial individuals.
This step is needed to determine the next generation that will constitute the subpopulation. For a typical problem, the fitness function value needs to be calculated at each of the points in the population. There is a very small but finite probability that the initial population may possibly include the solution. After this pre-evolution stage, the evolution phase loops until 1 a solution is found or 2 the generation number reaches the predetermined maximum generation 32 2 Genetic Fuzzy System Fig.
Details about GAs are available in textbooks [5, 6], and their use in genetic fuzzy systems is described in [7]. GAs are theoretically and empirically proven to provide a robust search in complex spaces [8]. A GA operates on a population of randomly generated points P. Each point is sometimes called a chromosome and is often represented by binary strings. There exist both binary and real coded versions of GAs; however, in this book we will use the binary GA, as this works quite well when a low level of discretization in terms of the numerics are needed.
In a binary GA, any numbers 2. The binary form resulting after the operations can then again be converted to real number form. However, the binary form requires a specification of the number of bits used for representing each design variable. These bits can be kept to a low number for SHM applications in genetic fuzzy systems as they typically represent the characteristics of the fuzzy set. The GA is an optimization algorithm, and its advantage relative to traditional gradient-based algorithms lies in its ability to locate the global minimum and also operate with discrete or integer design variables.
Several terms are widely used in the GA literature, and they are discussed next. A chromosome is generally a sequence of variables of a problem placed in an organized manner. Every variable sequenced to construct the chromosome is called a gene. These definitions come from the biologically inspired nature of the GA. However, as the GA moved away from its biological roots, genes were replaced by bits and chromosomes by strings. For instance, a design variable x can be represented by the string If there are two such design variables, they can be put side by side and result in a string of double the size.
This process of moving from the real number space to the binary space is called encoding. Fitness Evaluation GAs mimic the survival-of-the-fittest principle of nature to perform a search process. Therefore, GAs are naturally suitable for solving maximization problems where a fitness function is maximized. Minimization problems are usually transformed to maximization problems by some suitable transformation. In general, a fitness function F x is first derived from the objective function f x and used in successive genetic operations.
For maximization problems, the fitness function can be considered to be the same as the objective function, i. For minimization problems, the fitness function is an equivalent maximization problem chosen such that the optimum point remains unchanged. A number of such transformations are possible. The following fitness function is often used: GA operators typically require the function value to remain positive.
Therefore, the process of changing the sign of the fitness function which is popular in gradientbased optimization is not used in GAs. If there is some chance that the fitness function may become negative in the design space, a large positive number can be added to it to ensure that the fitness stays positive. Thereafter, each string is evaluated to find the fitness value.
The population is then operated by three main operators, reproduction, crossover, and mutation, to create a new population of points. These operators are described below.
Reproduction is the first operator applied on a population. Reproduction selects good strings in a population and forms a mating pool. In doing so, it mimics the courtship phase of natural selection. There exist a number of reproduction operators in the GA literature, but the essential idea in all of them is that above-average strings are picked from the current population and their multiple copies are inserted in the mating pool in a probabilistic manner. The commonly used reproduction operator is proportionate reproduction, where a string is selected for the mating pool with a probability proportional to its fitness.
This approach is also known as roulette wheel selection. Thus, good strings in a population are probabilistically assigned a larger number of copies and a mating pool is formed. Another approach to reproduction occurs in tournament selection. Here tournaments are arranged between any two random strings, and the winners are selected for mating. It is important to note that no new strings are formed in the reproduction phase.
At the end of the reproduction process, the mating pairs are selected. The idea of exchange of genetic information which occurs between the male and female is now mimicked during the crossover process. Recall that each individual is a point or a binary string. Thus, information exchange involves the swapping of some bits between the male and female strings.
The process of information exchange between the individuals is called crossover and is a basic property of GAs. The crossover procedure creates new chromosomes or strings from the two parents. Crossover is performed after selection of a subpopulation of individuals according to their fitness values and collection of the selected individuals into a gene pool.
Crossover is achieved in three stages. The first stage is matching. Matching is the selection of two individuals in the gene pool randomly. In the second stage, a crossover point is determined in each of the individuals. In the final stage, two parts of the individuals are replaced with each other. Typically, the crossover performs an operation where two parents lead to two children. An example of crossover is shown below: In such a case, the parent strings between two sites are swapped to get the 2. For such long strings, single-point crossover is biased toward the right of the strings and changes these design variables much more often than those at the left of the string.
Mutation means a random change in the information of a chromosome or string. In other words, mutation is an operation that defines the variation in a chromosome. This variation may be local or global. A probability test determines whether a mutation will be carried out or not. For example, if the average fitness of the new generation is smaller than the average fitness of the previous generation, bit y of the chromosome x can be changed.
A bit mutation applied to a chromosome is shown below: Since the initial population is a subset of all possible solutions, an important bit of all the chromosomes may be 0 while it must be 1 to be optimal. Crossover may not solve this problem and mutation is indispensable for the solution. In general, mutation leads to small local moves in the design space, while crossover leads to larger global moves.
Mutation may often lower the average fitness of the population, but it also allows the GA to escape from a local minimum by adding diversity. There are two proposed methods for allowing a subpopulation to replace its ancestors. One of the methods is generational replacement. In this method, a population of size n entirely replaces the new generation. The other method is steady-state reproduction, which replaces only a few individuals in a generation. A small number of strings with high fitness values are sometimes shielded from the crossover and mutation operations.
This is known as elitism. Increasing the population means a longer computation time. On the other hand, if the population size is decreased, the accuracy of the solution also decreases because of reduced variation of chromosomes. In GA design, there must be a balance between the generation numbers and population size. Population size has another effect in the GA; it 36 2 Genetic Fuzzy System reduces the effect of the highest fitness valued chromosomes.
For example, in a population of 10 chromosomes, if one of the chromosomes has a fitness value of 9 while the others have a fitness value of 1, half of the parents will be chosen from among the relatively low fitness valued chromosomes, although the best fitness valued chromosome is nine times better. Evaluation of chromosomes and fitness calculations are the most time-consuming parts of the GA.
If the evaluation operation is reduced, the GA process will work faster, and this can be achieved by reducing the population size and number of generations needed to reach a solution. These two soft computing tools can be combined to form the genetic fuzzy system [9].
The GA provides good global search capability. Fuzzy logic presents robust and flexible inference methods in problems subject to imprecision and uncertainty. The linguistic representation of knowledge permits a person to interact with a fuzzy system in an easy manner. The hybridization of the GA and fuzzy logic gives an advanced soft computing method called the genetic fuzzy system GFS , in which a GA is used to evolve a fuzzy system by tuning fuzzy membership functions and learning fuzzy rules.
A key objective of this section is to use GAs to automate the design of fuzzy systems. The generalized GFS algorithm is explained for a damage detection problem. Input and Output Suppose inputs to the fuzzy systems are represented by z and outputs are represented by x. The objective is to find the mapping between z and x. Each measurement delta has uncertainty. Fuzzification The structure can be divided into various locations. To get a degree of resolution of the extent of damage [10], each of these damage locations is allowed several levels of damage and split into linguistic variables.
The other structural damage variables are fuzzified in a similar manner. Fuzzy sets with Gaussian membership functions are used to define these input 2. These fuzzy sets can be defined using the following equation: We can see that the Gaussian fuzzy sets depend on the appropriate choice of the midpoint and the standard deviation. The midpoint is a measure of the point of maximum likelihood of a fuzzy set, while the standard deviation represents the scatter and accounts for the uncertainty.
Rule Generation Rules for the fuzzy system can be obtained by fuzzification of the numerical values obtained by numerical analysis of the model using the following procedure. The standard deviation of each set is initially fixed randomly within a prescribed range. For each measurement delta corresponding to given fault, the degree of membership in the fuzzy set is calculated. Each measurement delta is assigned to the fuzzy set with the maximum degree of membership.
One rule is obtained for each damage type by relating the measurement deltas. Suppose for rule i we are giving d inputs. Then the fuzzy system will generate d membership functions A taking the change in measurements obtained by numerical analysis of the model as the midpoint. The standard deviation is obtained by optimization as discussed later. A1d level1 Damage at location1 A21 A A2d level1 Damage at location A3d level1 Damage at locationk.. In the preceding rules, the membership values of membership function A will change from structure to structure.
These rules are symbolically tabulated in Table 2. For calculation of the uncertainty associated with variables, i.
As we have already discussed, there will be uncertainty and some noise in the measurement deltas. By generating noisy deltas and testing the fuzzy system for a known damage, we can define a success rate. The success rate is calculated using the results obtained after defuzzification.
Fuzzy logic involves computing with words instead of numbers and is robust to the presence of uncertainty in the inputs. However, fuzzy logic systems are difficult to design due to the interaction between the choice of appropriate membership functions, the rule base obtained, and the performance of the fuzzy system. The process of designing fuzzy systems can be automated by using an optimization procedure for tuning the membership functions and rule base. Genetic algorithms provide an excellent approach for finding the global minimum and can be used to design the fuzzy systems.
The genetic fuzzy system combines the uncertainty representation characteristics of fuzzy logic with the learning ability of the genetic algorithm. Using the changes in measurement deltas, a fuzzy system is generated, and the rule base and membership functions are optimized by the genetic algorithm. The generalized formulation of the genetic fuzzy system is explained using a damage detection problem. The genetic fuzzy system will be used for solving damage detection problems using modal data in the next two chapters.
These applications will make the process of developing genetic fuzzy systems for structural health monitoring very clear. Prentice-Hall, Englewood Cliffs 2. Engineering Applications of Fuzzy Logic. Wiley, New York 3. Springer, Heidelberg 4. Optimization of hydrocyclone operation using a geno-fuzzy algorithm. Wiley Interscience, New York 7. Generating the knowledge base of a fuzzy rule-based system by the genetic learning of the data base.
Theoretical and numerical constraint handling techniques used with evolutionary algorithms: World Scientific, Singapore A fuzzy logic system for ground based structural health monitoring of a helicopter rotor using modal data.
Neuro-Fuzzy Methods and Their Comparison. Springer, London Chapter 3 Structural Health Monitoring of Beams Structural damage detection is an inverse problem of structural engineering having four main parts: In this chapter, a genetic fuzzy system GFS is used to find the location and extent of damage in a beam. The beam is a fundamental structural element used to model systems such as helicopter rotor blades, airplane wings, columns, bridges, and buildings.
In fact, the beam is probably the most ubiquitous structural member. The GFS automatically generates the rules for a fuzzy system using a genetic algorithm GA for application to structural damage detection. The GFS is demonstrated for damage detection in a beamtype structure modeled using the finite element method. The finite element method is a numerical method for solving governing differential equations and has proved to be particularly successful for problems in solid mechanics where nonuniformity of the domain and boundary conditions do not permit analytical solutions.
We first consider a cantilever beam for illustrating the GFS for the damage detection problem. Cantilever beam-type structures include, e. These structures are fixed at one end and free at the other end. We also discuss results for the BO hingeless helicopter rotor blade at the end of this chapter. In the present work, the fuzzy rules are automatically generated. The GA is used to solve the optimization problem. It is based on roulette wheel selection, fixedpoint crossover, and bitwise mutation. The population size, crossover probability, mutation probability, and maximum number of generations are 20, 0.
These values are determined by numerical experimentation. It should be noted that the computational expense of the GA here depends on the underlying finite element analysis. For the beam-type structures considered in this chapter, the number of degrees of freedom is small, and hence the computer time for solving the eigenvalue problem is much less. All numerical results in this book are obtained on PCs, reflecting the growing power of computers. In this chapter, a GFS is developed and demonstrated for damage detection in two isotropic beam-type structures using modal-based measurement deltas.
A cantilever beam and a hingeless helicopter blade with only flapwise out-of-plane bending are considered. To obtain the natural frequencies, harmonic excitation is assumed: This ODE has an exact solution for natural frequencies for a uniform beam which has constant flexural stiffness and mass per unit length. However, structural damage causes a localized reduction in the flexural stiffness at the point where the damage occurs. Therefore, modeling of damaged beams requires analysis of a nonuniform beam. For a nonuniform beam, an approximate method such as the finite element method is needed to calculate the natural frequencies.
In finite element analysis, the structure is divided into many small finite elements and a local displacement field is assumed within the element with degrees of freedom present at the element nodes. The elements are then assembled together and appropriate boundary conditions are applied. Basically, finite element analysis transforms a differential equation into a matrix equation and gives a discrete system approximation of the continuous system.
Details of the finite element method are given in standard textbooks [1—4]. The beam element used here has four degrees of freedom, two at each node. The nodal degrees of freedom are the displacement and slope at the two ends of the element. Between the finite elements, there is continuity of the displacement and slope degrees of freedom.
The element matrices for the entire beam are assembled and boundary conditions are enforced to give the global stiffness matrix Kg and the global mass matrix Mg. For a cantilever beam which is hinged at the root, the displacement and slope degree of freedom at the root node is set to zero. These are known as geometric boundary conditions for the cantilever beam.
The change in frequency is considered as the measurement delta and is calculated by the finite element method for a combination of five different locations and four different levels of damages: Undamaged, Slight Damage, Moderate Damage, and 3. Thus both the damage locations and the damage size are expressed in terms of words to make the problem suitable for fuzzy logic.
Since modal properties such as frequencies are global properties in contrast to strains, which are local properties, it is best to use the modal methods for a broad classification of the damage and then to subject the structure to more detailed local inspection using nondestructive testing methods. The difference between the frequency of the damaged and undamaged beam is used as a system indicator for damage and referred to as 44 3 Structural Health Monitoring of Beams Fig.
The measurement delta is expressed as a percentage change: Note that in this problem, the database used to create rules is obtained from a mathematical model. It is also possible to obtain such a database from experiments. However, the costs of performing experiments on the large number of damaged beams can be very high.
Therefore, a model-based diagnostics is very useful. These results are obtained from the finite element model of the cantilever beam by seeding the damage at different locations along the beam. However, any modeling process including the finite element method contains errors due to uncertainty in the material properties and discretization and numerical errors. Uncertainties in the model can be of two types: Besides random uncertainty, epistemic or model uncertainty may also be present.
For instance, if Euler—Bernoulli beam theory is used to model a short thick beam, the results will not be accurate, as the effects of shear deformation and rotary inertia are neglected in the Euler— Bernoulli beam theory. These effects are included in Timoshenko beam theory, and the use of such a theory for short thick beams will reduce epistemic uncertainty.
While epistemic uncertainty can be reduced by choosing good models, random uncertainty is harder to reduce as it involves manufacturing and quality issues. Discretization errors arise from not using enough elements in the finite element model. It is very important to use a sufficient number of finite elements especially at locations of sudden change in the material properties which occur in the vicinity of the damage. As a rule, it is important to perform a convergence study by increasing 3. It is better to overdiscretize and err on the side of caution than to underdiscretize.
We thus see that there is always some difference between predictions by models and test results, even if discretization errors have been minimized. This difference is called modeling uncertainty.
Structural health monitoring (SHM) has emerged as a prominent research area in recent years owing to increasing concerns about structural safety, and the. Request PDF on ResearchGate | On Jan 1, , Prashant M. Pawar and others published Structural Health Monitoring Using Genetic Fuzzy Systems.
This is called measurement uncertainty. In this work, we will assume that random noise models both forms of uncertainties present in the structure. Finally, numerical errors involved in algorithms such as eigenvalue solvers, linear system solvers, and differential equation solvers should be minimized by using robust methods and double precision computer arithmetic.
The objective of the GFS is to find a mapping between the frequency-based measurement deltas and damage at 46 3 Structural Health Monitoring of Beams the five locations. Helicopter rotor blades are designed to have a high level of damage tolerance and therefore only significant levels of damage are of interest to us here. Therefore, the use of a global method such as frequency-based damage detection is appropriate, as the damage indicator acts as a filter which prevents minor and insignificant damage from being detected.
Fuzzy sets with Gaussian membership functions are used to define these input variables. This midpoint will change for different structures and damage combinations since it is dependent on geometrical properties and material properties of the structure. As this GFS automatically adjusts the midpoints of fuzzy sets, it has the flexibility to deal with different beams. In addition, by selecting the number of fuzzy sets as equal to the number of measurements or modes, we automate the process of selecting the number of fuzzy sets.
Of course, the greater the number of measurements, the more accurate the fuzzy system. For each different beam, the GFS will automatically adjust midpoints according to the conditions of damage and location. For calculation of the uncertainty associated with the variables, i. As we have already discussed, there will be the presence of uncertainty and some noise in the measurement deltas.
The added noise in the data simulates the uncertainty present in the experimental measurements and the modeling process. The data for training and testing of the GFS is developed by using the noise model given in 2. The schematic representation of the GFS is shown in Fig. Data from the finite element model are used to obtain the midpoints of the fuzzy sets.
The outputs of the GFS are damage presence, location, and size, which are the same as those for the fuzzy system in Fig. Once we obtain the proper fuzzy rules which are generated automatically, we have a knowledge base to isolate structural damage using frequency shifts. When these rules are applied for a given measurement, we have the degree of membership for each fault. For fault isolation, we are interested in the most likely fault.
Therefore, the fault with the highest degree of membership is selected as the most likely fault. The output is left as a fuzzy set, as it is closer to the human reasoning process. The dimensions and material properties of the beam Fig. The beam is divided into 20 finite elements of equal length. The selection of the number of elements is justified in Fig.
In this figure, the ratio of the eighth mode, which is the highest mode used in the numerical results, with the first mode is shown. From the graph, it appears that 20 elements give a good level of discretization. The undamaged beam is uniform. Therefore, the frequency predictions from the finite element method FEM model of the undamaged beam are validated by comparing with exact solutions for a continuous beam.
The first eight natural frequencies of the beam are We shall assume in this study that the finite element analysis is accurate and any deviations from reality are random noise and not systematic biased modeling error. For applications to real structures, some methods such as finite element model updating can be used to match the model predictions of the undamaged structure with experimental data. Material and geometric properties of the beam are: The eight input deltas represent the first eight natural frequencies of the beam.
During the process of developing the fuzzy system, eight frequencies are used, but during testing, we will also experiment with missing measurements. In the GFS, only the beam geometry and material properties need to be specified along with the number of measurement deltas modes needed, and the rule base and success rate for the optimal system are automatically generated.
The linguistic forms will remain constant for different structures, but the numerical values of midpoints and standard deviation will change.
Fuzzy associations between observable structural responses and damage conditions were generated by finite element simulations. Another function of HUMS is automated recording in flight exceedance, thus removing the dependency upon the crew to perform this task manually. The finite element method is a numerical method for solving governing differential equations and has proved to be particularly successful for problems in solid mechanics where nonuniformity of the domain and boundary conditions do not permit analytical solutions. Therefore, GAs are naturally suitable for solving maximization problems where a fitness function is maximized. The output faults of the fuzzy logic system are four levels of damage undamaged, slight, moderate, and severe at five locations along the blade root, inboard, center, outboard, tip. This number-to-word transformation is a key step which gives fuzzy systems the power to handle uncertainty.
We can see that a lot of information is contained in this table, and it is very difficult to find all this information manually through a trial-and-error process. As expected, the success rate improves as the data quality becomes better. Optimizing the fuzzy system for a higher noise level therefore results in good performance at lower noise levels. Furthermore, the fuzzification process of fuzzy logic ensures that the deterioration in the performance of the GFS is gradual with the increase in the noise level. Even though a success rate of These 19 cases are called false alarms, and one of the main aims of damage detection algorithms is to minimize these false alarms, as they can be expensive and also reduce the faith of the users in the health monitoring system.
To get a physical feel for the change in frequencies due to structural damage and the effect of noise level, we look at the numbers used in rule 4 of Table 3. Rule 4 of Table 3. For a given 3. It is clear that, for each mode, the change in damaged frequency due to structural damage is larger than the change due to noise.
Of course, this fact is an important requirement for successful damage detection. Even the best soft computing approach cannot perform damage detection if the change in the damage indicator due to damage is less than the noise level. Therefore, the search for good damage indicators is as important in structural health monitoring as the development of soft computing methods. Eight frequency measurement deltas were used during the training phase of the GFS. However, for a physical system, the first eight natural frequencies may not be available.
Furthermore, some of the higher mode frequencies are difficult to measure accurately, and there is a greater risk of noise contamination in these measurements. We test the automatic rule generating GFS for 4, 5, 6, 7, and 8 measurements. As expected, the addition of measurements increases the accuracy of damage detection. Therefore, for reasonably good quality measurement data,the fuzzy system performs extremely accurately. The effect of the number of measurements used for damage detection on the average success rate of the fuzzy system is shown in Fig.
These values are selected so that the fuzzy sets defined in Table 3. The results are shown in Table 3. This case can represent a scenario where there is missing measurement or bad data such as a bit error. Then the fuzzy system would calculate results while neglecting that measurement. As shown in Table 3. These cases can represent a scenario where there are sensor faults or measurement errors effecting one or two modal frequencies.
See if you have enough points for this item. Structural health monitoring SHM has emerged as a prominent research area in recent years owing to increasing concerns about structural safety, and the need to monitor and extend the lives of existing structures. The use of a genetic algorithm automates the development of the fuzzy system, and makes the method easy to use for problems involving a large number of measurements, damage locations and sizes; such problems being typical of SHM.
The ideas behind fuzzy logic, genetic algorithms and genetic fuzzy systems are also explained. The functionality of the genetic fuzzy system architecture is elucidated within a case-study framework, covering: Structural Health Monitoring Using Genetic Fuzzy Systems will be useful for aerospace, civil and mechanical engineers working with structures and structured components.
It will also be useful for computer scientists and applied mathematicians interested in the application of genetic fuzzy systems to engineering problems. Advanced Structural Damage Detection. Signal Processing for Neuroscientists. Chaos in Switching Converters for Power Management.
Mechatronics and Automation Engineering. Smart Structures and Materials. Synthetic Aperture Radar Processing. Dynamics of Coupled Structures, Volume 4. Robotic Manipulators and Vehicles. Model Predictive Vibration Control. Vehicle Dynamics of Modern Passenger Cars. Cellular Nanoscale Sensory Wave Computing.
Special Topics in Structural Dynamics, Volume 6. Fourier Optics in Image Processing. Dynamics of Civil Structures, Volume 2. Finite Element Analysis for Satellite Structures. The Finite Element Method in Engineering. Satellite Communications Payload and System. Fundamentals of Structural Engineering. Optimization and Optimal Control in Automotive Systems. Introduction to Continuum Mechanics. Nonlinear Dynamics, Volume 1. Computer-aided Nonlinear Control System Design. Structural Rehabilitation of Old Buildings. Simplified Theory of Plastic Zones. Sensors and Instrumentation, Volume 5.
Advances and Applications in Nonlinear Control Systems. Introduction to Solid Mechanics. Lighter than Air Robots. Statics of Historic Masonry Constructions.