Damage detection in structural elements: using adaptive Mamdani model

In real life all the structural and machine elements work under dynamic or variable loading. Application of dynamic loading leads to fluctuating stress. Due to fluctuating stress fatigue cracks initiates. These fatigue cracks are the main reason of failures. So, it is very important to detect the crack and predict the crack life. There are different types of damages but crack is one of the most encountered damage. There are different conventional methods to detect the damage but these methods are time taking and requires removal from the machines. Therefore, researchers are giving more importance to the unconventional methods to find the damage. In the present work a method has been introduced to find the damage site using Fuzzy Logic System and Regression Analysis. In particular, this paper focuses on applying statistical process control methods. A data pool has been created from the dynamic analysis of the cracked cantilever beam and then the data pool is trained in the proposed methodology to find the crack location. It has been noticed that the proposed methodology gives result within the tolerable range.


Introduction
Nowadays structural health monitors have become very important aspect in both structural and Aerospace industries [1,2]. In this regard Artificial Intelligence and the soft computing methods are taken as a substitute to traditional methods in many industries. From the last two decades vibration analysis method is gaining importance for damage detection in structural and machine parts [3,4]. A very minute hairline crack can lead to a catastrophic failure. In Aerospace applications minute damage leads to dangerous human casualties. So, crack detection and localization in Aerospace is becoming a topic for constant research. Due to the above-mentioned reasons, the materials used in Aerospace applications need constant health monitoring [5,6]. There are different conventional structural health maintenance methods. But these types of SHM need time and money. But particularly Aerospace applications need quick and effective detection of damage.
In the current research work one such health monitoring method for damage detection has been proposed. Here an adaptive fuzzy logic system has been proposed. In large buildings, bridges or in aircrafts, it is really very difficult to get the damage location and predict its severity [7,8]. But a hair line crack may play a very dangerous role to initiate a very serious failure, which could be dangerous to mankind. It has been observed from a thorough literature survey that though many research works have been done, FLS has some loopholes to be applied. As most of the data from the field are given by a person where there is a chance that the data mining process may not be proper. It is known to all that due to the data mining process, there may be some unclarified data, which may create problem to get a proper solution. So, to minimize the error data, Regression Analysis has been incorporated to reduce the percentage of error [9,10,11].
In this research work, the data has been collected from Finite Element Analysis (FEA). Due to the changes in the physical parameters for the initiation of crack, the stiffness of the material is changed. The changes in the stiffness of the material lead to changes in the natural frequencies and the mode shapes. To keep the proposed methodology simple, only the natural frequencies of the first three mode shapes are taken as the input variables and the crack depth and crack location are taken as the output variables to the system. First a data pool is generated using Finite Element Analysis. Then values of all the input and output parameters are normalized by dividing with the respective values of the undamaged beam.

Application of Finite Element Analysis for data generation
The beam to be assumed in this analysis is Euler-Bernoulli type. Here a hairline crack is taken and the crack is assumed to be open. The vibration type is free vibration for a rectangular cross-section. The governing equation for the above-mentioned beam type is given below. 4 2 i 4 d y EI -mω y=0 dx (1) 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/m 2 ) and 'I' is the moment of inertia (m 4 ). By characterizing 2 4 i mω λ = EI mathematical statement is adjusted as a fourtharrange differential equation as follows: 4 4 4 d y -λ y=0 dx (2) The general solution to equation is Where A, B, C, D are constants and 'λi' is a frequency parameter. The governing differential equation for the system is given as: But it is assumed that there is no damping and there is no external force applied on the system. So the governing equation becomes The equation of motion for natural frequencies for undamped free vibration is given in equation (4.5). To solve equation (4.5) it is assumed that Where  -The Eigen vector or mode shape ω -Circular natural frequency

Fuzzy mechanism for crack detection
For several years, fuzzy control has emerged as most active areas of reserch in the application of the fuzzy set theory. This logic based theory is much closer to human thinking and natural language than the traditional logical system. Fuzzy logic provides a mean of using approximate, in exact nature of the real world problems. The important part of the FLC is a set of linguistic control rules connected by the concept of fuzzy implication and inference fuzzy associative rules. The fuzzy logic system provides a method to convert the linguistic control strategy to an expert knowledge automatic control strategy. From the experience of other researchers it can be observed that FLC provides. Superior results from those obtained by conventional control algorithms. Fuzzy logic control system becomes useful when the available sources of informations are inexact and uncertain. This logic is a step forward of conventional precise mathematical control and human like decision making.
The fuzzy controller has been developed where there are 3 inputs and 2 outputs parameter. The natural linguistic representations for the input are as follows First natural frequency in normalized form= 'nfnf' Second natural frequency in normalized form = 'nsnf' Third natural frequency in normalized form = 'ntnf' The natural linguistic term used for the outputs are Crack depth in normalized form = 'ncd' Crack length in normalized form = 'ncl' Based on the above fuzzy subset the fuzzy rules are defined in a general form asfollows: If (nfnf is nfnfi and nsnf is nsnfj and ntnf is ntnfk) then (ncd is ncdijk and ncl is nclijk) (7) Where i= 1to 9, j=1 to 9, k=1 to 9 Because of 'nfnf', 'nsnf', 'ntnf' have 9 membership functions each. From the above expression, two set of rules can be written

If (fnf is fnfi and snf is snfj and tnf is tnfk) then cd is cdijk (8) If (fnf is fnfi and snf is snfj and tnf is tnfk) then cl is clijk
According to the usual Fuzzy logic control method (Parhi, 2005), a factor Wijk is defined for the rules as follows:  

Application of Regression Analysis to make Mamdani FLS Adaptive
Regression Analysis comes under the category of supervised learning based on the statistical modeling of the problem. Regression Analysis is used to find the relationship between the two variables. This method is mainly a statistics-based method. It can be described in the form of Cause (independent variable) and Effect (dependent variable).

The fundamental equation for theRegression Analysis is
Y=a+bX+c (12) where, Y= dependent variable X=independent variable a=constant b= slope of the regression line c= error term or residual factor Before doingRegression Analysis, some assumptions must be considered. For the current Regression Analysis method following are the assumptions considered by the author. For the present analysis, Y=nfnf, nsnf, ntnf and X=ncd, ncl.
This analysis mainly related to the dynamic responses of the damaged structural element. This has been observed that the responses from any damaged element are scalar values. Due to which univariate stastical tests have been proposed to predict the possible alterations in the parameters related to a particular location.The application of Regression Analysis (RA) to increase the adaptiveness of Fuzzy Logic System (FLS) is explained in the schematic diagram given in the Figure 1.

Result and Discussions
The following tables give the comparison of results from different methods. The first three columns of the table give the dimensionless values of first natural frequency, second natural frequency and third natural frequency. Next two columns present the dimensionless values of crack depth and crack location. Then the percentage of error is calculated using the following formulae.

Conclusion
In the current research work an online damage detection method has been proposed. The method comprises of Fuzzy Logic System (FLS) and Regression Analysis (RA). As human errors are involved inFLS in generating data and making the rules. So, regression analysis is used to clear the errors in the data generation.The percentage error in case of FLS is around 4.5%. But the percentage error sharply decreases to 3.5% when the proposed method is applied for damage detection. So, from the results it can be observed that the proposed method gives a better result as compared to Fuzzy Logic System and can be used as an online tool for damage detection.