A crack in a structural member introduces local flexibility that would affect the vibration response of the structure. Finally, the details of each chapter of the dissertation for the current study have been explained in the third part of this chapter.
Framework and agenda
The final outputs of the MANFIS controller are the relative crack depth and the relative crack location. The two output parameters of the fuzzy inference system are the relative crack depth and the relative crack location. The finite element analysis considers the cracked cantilever beam with a V-shaped single crack.
The depth and location of the crack affect the natural frequencies of the vibrating structures. This section shows the steps used to implement the genetic algorithm in the research work. A comparison of the results from the Fuzzy controller and the Genetic Algorithm controller is given in Table .7.4.1.
Purpose of the research
Layout of the work
The free vibration analysis was carried out to calculate the vibration characteristics of the cracked cantilever beam. The following article describes some of the techniques used by various researchers in detecting damage to structural components.
Introduction
This property can be used to detect the presence of a crack, along with its location and depth in the structural member. Some of these techniques used in the current research work are given in the next section.
Damage detection using finite element method
A finite element model is used to estimate the undamped natural frequencies of the raw spindle. In this part of the chapter, the results for the relative crack depth and relative crack location were derived from the Takagi-Sugeno FIS using the MATLAB toolbox [42]. The strings are created randomly, that is, based on the values of the parameters they represent.
Some essential expressions of Genetic Algorithm used in this research work are given below. An analysis of the dynamic response of a cracked beam with a transverse single crack was performed. Finite element analysis of the cracked cantilever beam was performed to obtain the first three natural frequencies for different depth positions and crack locations.
Fuzzylogic System for crack detection in a beam
Genetic Algorithm for crack detection in a beam
Below are some pattern search terms used later in this section. In the chapter on finite element analysis, the effects of the single crack on the vibration signature of the cantilever beam have been elucidated.
Dynamic characteristics of beam with single transverse crack
Theoretical vibration analysis
A cantilever beam is subjected to an axial force (P1) and a bending moment (P2), shown in Figure 3.1.1, creating coupling with the longitudinal and transverse motion. The rate at which the strain energy is released at the fractured portion can be written as (Tada et al.1973).
Beam model
Take 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 shown in Figure 3.1.2. From the governing equations of free mode of vibration of the cracked beam The normal function for the system can be defined as.
Introduction
Here Fuzzy Logic is used for training data table datasets for predicting relative crack depth and relative crack location. In the defuzzification process, the inputs are standardized and transformed into the domain of membership function values. The inputs to the genetic controller are the relative values of the first three natural frequencies and the outputs are the relative crack depth and the relative crack location.
The GA was initialized with three inputs, corresponding to the crack characteristics to predict the relative crack depth and relative crack location. The grid point of the previous iteration becomes the current point in the current iteration after each successful poll. Different language terms (Table 5.5.1) are used to represent different ranges of membership functions.
Methods used in finite element analysis
Finite element formulation
Governing equation of free vibration
Where 'm' is the beam mass per unit length (kg/m), '?i' is the i-th natural frequency (rad/sec), E is the modulus of elasticity (N/m2) and I is the moment of inertia (m4). It is assumed that the crack does not affect the mass distribution of the beam.
Applications of Finite element method
Introduction
Analysis of fuzzy logic system for crack detection
- Fzzy set
- Membership function
- Fuzzy operations
- Fuzzy linguistic variables
The degree of membership µA(x )0 of the membership function µA(x) describes, for the special element x x= 0, to which degree it belongs to the fuzzy set A. Zadeh proposed the minimal operator for the intersection and the maximal operator for the union of two fuzzy sets.
Steps used in Mamdani FIS
The change in local flexibility due to the presence of a crack is used to calculate the change in natural frequencies of the cantilever beam. Vibration-based methods use the fact that due to the presence of a crack there is a change in flexibility that affects. In this chapter, mainly triangular, Gaussian, trapezoidal and hybridized membership functions are used for the input and output variables of the controller.
It is found that the results (Table .6.9.1) of the Pattern Search Algorithm are closer to the data taken and require less computation time. But in the case of Takagi-Sugeno FIS, we can directly get the sharp values of the outputs. The ALGOR V 19.3 SP 2 Finite Element Program [109] was used for vibration analysis of the uncracked and cracked cantilever beam.
Fuzzy mechanism for crack detection
Results and discussionof fuzzy controller using Mamdani FIS
Takagi-Sugeno fuzzy model
When f(x1, x2) is a first-order polynomial, the resulting fuzzy inference system is called a first-order Sugen fuzzy model. When 'f' is a constant, we have a zero-order Sugen fuzzy model, which can be considered a special case of Mamdani's fuzzy inference system, in which each consequence of the rule is specified by a fuzzy unit (or pre-defuzzified consequence).
Steps for Diagnosing Crack using Takagi-Sugeno FIS
Third, the parts of the strings to the left and right of the randomly selected location in the two strings are swapped. Finite element analysis was performed based on the above theoretical analysis to study the influence of crack depth and crack location on the dynamic response of the cracked beam. The first two steps, that is, fuzzification and rule formulation, are the same as those of the Mamdani FIS.
Douka E., Hadjileontiadis L.J., Time-frequency analysis of the free vibration response of a beam with a breathing crack, NDT & E International pp.3-10.
Difference between Mamdani and Takagi-Sugeno FIS
Results and discussions of fuzzy controller using Takagi-Sugeno FIS
The results are listed in Table 5.9.1, and the input and output variables of the Takagi-Sugeno FIS and the Takagi-Sugeno FIS rule viewer are given in Figures 5.9.1 and 5.9.2. The difference between Mamdani FIS and Takagi-Sugeno FIS is that in the first type we get the output in the form of membership functions as input, while in the case of Takagi-Sugeno FIS the output is linear or constant.
Summary
Introduction
Methodology used in Genetic Algorithm
Crossover is the process in which strings are able to mix and match their desirable qualities randomly. First, two new strings are selected whose fitness value is close to that of the field variables (natural frequencies).
Genetic Algorithm terms
- Encoding of chromosome
- Initial population
- Selection
- Reproduction
- Crossover
- Mutation
- Iterating the Algorithm and stopping criterion
In this study, single point crossover is used in the design of a genetic algorithm controller. With the mutation operator the entire chromosome is changed or each bit can be reversed.
Crack identification with Genetic Algorithm deployment
Procedure of genetic algorithm anlysis
Results and discussions
In this chapter, a genetic algorithm based method has been used to find out the relative crack depth and relative crack location. After finding the results after several runs of GA, it is ruled out that GA can be used to get accurate results for damage detection in a structure and which can be observed from Table 6.5.1.
Pattern search Algorithm for crack detection
Terminologies used in Pattern searchAlgorithm
Patterns
Meshes
Polling
Expanding and contracting
Methods used in Pattern searchAlgorithm
The total procedure is repeated until the algorithm reaches a convergent result or the stopping criteria are met. The following criteria can be used to stop the algorithm: i) maximum iteration which is the maximum number of iterations the algorithm performs; (ii) Evaluations of the maximum function which is the maximum number of evaluations of the objective functions and constraints; (iii) timeout which is the maximum time in seconds that the pattern search algorithm runs before stopping; and finally (iv) function tolerance which is the termination tolerance for the objective function value.
Results and discussions
In this step, first multiply the current mesh size by ηexp then multiply by the pattern vector and add it to the current point; to update the current point and finally build a new mesh on this current point Mj⇔ xjstroom + ∆x , jj =1..N and go to Step 2.1.
Summary
Behzad M., Meghdari A., A new approach for vibration analysis of a cracked beam, International Journal of Engineering pp.319. Nahvi H., Jabbari M., Crack detection in beams using experimental modal data and finite element model, International Journal of Mechanical Sciences pp.482-490. Lee Jinhee., Identification of multiple cracks in a beam using natural frequencies, Journal of Sound and Vibration pp.482-490.
Parhi D.R.K., Dash A.K., Fault detection by finite element analysis of a multi-crack beam using vibration signatures, International Journal of Vehicle Noise and Vibration pp.40-54.
Description of instruments used in experimental analysis
Experimental set-up
The vibration amplitude of the uncracked and cracked cantilever beam is taken by the accelerometer (vibration receiver) and fed into the vibration analyzer, which is connected to the vibration indicator. The vibration indicator is a laptop which is loaded with PULSE LabShop Version 12 vibration analysis software.
Experimental procedure
Experimental results and discussions
Introduction
Discussions of results
In this analysis, the first two steps (fuzzification and rule formulation) are the same as Mamdana FIS, but the defuzzification process is different. The stochastic nature of GA can be considered a weakness as it can lead to a suboptimal solution.
Contributions
Conclusion
Comparisons of the Fuzzy Controller results with the theoretical, finite element and experimental results demonstrate the effectiveness of the proposed method in identifying the location and extent of damage. Further choices and adjustments are made in the context of a genetic algorithm search (selection, crossover points, mutation rate, etc.), which strongly influence both the efficiency of the results and the computational time required.
Applications
Scope for Futurework
Parhi D.R.K., Dash A.K, Analysis of methodologies applied for fault diagnosis in vibrating structures, International Journal of Vehicle Noise and Vibration pp.271-286. Zang S., Imregun M., Structural damage detection using artificial neural networks and measured frf data reduced via principal component projection, Journal of Sound and Vibration pp.813-827.
