EXECUTIVE SUMMARY
INTRODUCTION/BACKGROUND The objective of the thesis is to predict and optimize the mechanical properties of Aircraft fuselage aluminium (AA5083). Firstly, data-driven modelling techniques such as Artificial Neural – Fuzzy networks and regressive analysis are used and by making the effective use of experimental data, FIS membership function parameters are trained. At the core, mathematical model that functionally relates tool rotational speed and forward movement per revolution to that of Yield strength, Ultimate strength and Weld quality are obtained. Also, simulations are performed, and the actual values are compared with the predicted values. Finally, multi-objective optimization of mechanical properties fuselage aluminium was undertaken using Genetic Algorithm to improve the performance of the tools industrially.
AIMS AND OBJECTIVES Objectives of the dissertation include Understanding the basic principles of operation of Friction Stir Welding (FSW). Gaining experience in modelling and regressive analysis. Gaining expertise in MATLAB programming. Identifying the best strategy to achieve the yield strength, Ultimate Tensile strength and Weld quality of Friction Stir Welding. Performing optimization of mechanical properties of FSW using Genetic Algorithm.
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To draw conclusions on prediction of mechanical properties of FSW optimization of aircraft fuselage aluminium.
ACHIEVEMENTS The basic principles of friction welding of the welding operations are well studied and understood. The theoretical concepts of modelling techniques are familiarised. Gained expertise in MATLAB programming. The best model that can predict the mechanical properties of friction stir welded aircraft fuselage aluminium has been developed using ANFISRegressive modelling. Performed optimization of mechanical properties friction stir welded fuselage aluminium using Genetic algorithm. CONCLUSIONS / RECOMMENDATIONS Making use of experimental data available, data are trained using ANFIS, and the MATLAB program was carried out to develop a mathematical model to predict the tensile strength, yield strength, weld quality of the friction stir welded Aircraft fuselage aluminium alloy AA508, of 6mm thickness. ANFIS is usually reliable as the experimental results are much closer to the predicted data when compared to the regressive analysis. The developed model can be effectively used to predict - the tensile strength, yield strength and weld quality of friction stir welded joints. Genetic algorithm program written in MATLAB was utilised to optimize the mechanical properties of FSW. Additional work can be done to improve the ANFIS model, to reduce errors and improve the mathematical model. The Genetic algorithm can be further improved to achieve better predicted values of the mechanical properties of friction stir welded fuselage aluminium.
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TABLE OF CONTENTS
Chapter 1 - Introduction ......................................................................................... 1 1.1. 1.2. 1.3. Research paradigm .................................................................................. 1 Aims and objectives.................................................................................. 1 Overview of the thesis .............................................................................. 2
Chapter 2 - Literature Review ................................................................................ 3 2.1. 2.2. Introduction and scope............................................................................. 3 System Identification ................................................................................ 3 Least-squares estimation: ................................................................. 3
2.2.1. 2.3. 2.4.
Regressive analysis ................................................................................... 5 Neural Networks ...................................................................................... 6 Model of an artificial neuron ........................................................... 7 Single layer feed forward network ................................................... 8 Multi layer feed forward network.................................................... 9
Previous experimental work .................................................................. 17 Summary ................................................................................................ 21
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Chapter 4 - Prediction of Mechanical properties of Friction Stir welded Aircraft fuselage aluminium (AA5083) .................................................... 22 4.1. 4.2. Introduction ............................................................................................ 22 Multiple linear regression using least squares ...................................... 22 Developing a mathematical model using MATLAB ...................... 23 Mathematical model to predict Tensile strength ........................... 25 Mathematical model to predict Yield strength .............................. 26 Mathematical model to predict Weld quality ................................ 27
4.2.1. 4.2.2. 4.2.3. 4.2.4. 4.3.
ANFIS modelling .................................................................................... 29 ANFIS modelling for Yield strength .............................................. 30 ANFIS modelling for Tensile Strength .......................................... 36 ANFIS modelling for Weld Quality ............................................... 39
4.3.1. 4.3.2. 4.3.3.
Chapter 5 - OPTIMIZATION OF MECHANICAL PROPERTIES OF FSW ... 43 5.1. 5.2. 5.3. Introduction ............................................................................................ 43 Genetic algorithm ................................................................................... 43 Results obtained from optimisation ....................................................... 44
List of Figures
Figure 2-1 Simple model of an artificial neuron [10] .................................................. 7 Figure 2-2 Single layer feed forward network [10] ..................................................... 8 Figure 2-3 Multi layer feed forward network (m-l_1-l_2–n configuration) [11].......... 9 Figure 2-4 Rules showing graphically [14] ............................................................. 11 Figure 2-5 ANFIS architecture (first order Sugeno fuzzy Model) [14] .................... 11 Figure 3-1 schematic diagram of FSW process [18] ................................................ 13 Figure 3-2 schematic diagram of a typical FSW showing four distinct zones [17] .. 14 Figure 3-3 stress-strain curve [21] ........................................................................... 16 Figure 3-4 FSW tensile test specimens, fracture surface [23] ................................. 17 Figure 3-5 GrC-NF modelling flowchart [23] ........................................................ 20 Figure 4-1 Plot showing Predicted, actual tensile strength with experimental data.. 26 Figure 4-2 Plot showing Predicted, actual Yield strength with experimental data .... 27 Figure 4-3 Plot showing Predicted, actual Weld quality with experimental data ..... 29 Figure 4-4 ANFIS editor [24] ................................................................................. 31 Figure 4-5 FIS structure ......................................................................................... 32 Figure 4-6 Fuzzy rules created from ANFIS ............................................................ 33 Figure 4-7 Fuzzy membership function plots ......................................................... 33 Figure 4-8 The Yield strength model...................................................................... 34
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Figure 4-9 FIS editor .............................................................................................. 35 Figure 4-10 Plot showing Predicted, Actual Yield strength with experimental data obtained from ANFIS .............................................................................................. 36 Figure 4-11 Fuzzy rules created from ANFIS (Tensile strength as output) ............... 37 Figure 4-12 The Tensile Strength model ................................................................ 38 Figure 4-13 Plot showing Predicted, Actual Tensile strength with experimental data obtained from ANFIS .............................................................................................. 39 Figure 4-14 Fuzzy rules created from ANFIS (Weld Quality as output) ................. 41 Figure 4-15 The Weld Quality model ...................................................................... 41 Figure 4-16 Plot showing Predicted, Actual Weld Quality with experimental data obtained from ANFIS .............................................................................................. 42 Figure 5-1 Prediction and Optimisation of FSW represented in a Flow Chart ........... 47
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LIST OF TABLES
Table 3-1 augmented Taguchi series for the FSW experiments [23] ....................... 18 Table 4-1 FSW AA5083 data .................................................................................. 24 Table 5-1 Optimisation of Mechanical properties of FSW ........................................ 45
Chapter 1 - Introduction
1.1. Research paradigm AA5083 aluminium alloy (4.6% Mg, 0.6%Mn, and 0.3% Si) is commonly used in aircraft applications, also in sea water applications, as AA5083 is with strong resistance to corrosion and is non-heat- tractable Al-Mg alloy. Friction stir welding is a technique used for joining aluminium alloys where the characteristics of the aluminium remain unchanged. In this process, heating is created between the rotating tool and tool shoulder by friction. The heat generated softens the material without reaching the melting point, and finally solid state joint is produced [1]. Data-driven modelling techniques are used to predict the nonlinear behaviour of mechanical properties of aircraft fuselage aluminium. In general, data-driven modelling approach needs large data, but only 25 trials have been made on AA5083 aluminium alloy, which in turn affects the accuracy of predicting a model. Mathematical model is developed by training the experimental data. The ultimate tensile strength, yield strength, weld quality of friction stir welding of aircraft fuselage aluminium alloy AA5083 is predicted by the mathematical model obtained. 1.2. Aims and objectives Objectives of the dissertation include Understanding the basic principles of operation of Friction Stir Welding (FSW). Gaining experience in modelling and regressive analysis. Gaining expertise in MATLAB programming. Identifying the best technique to obtain the yield strength, Ultimate Tensile strength and Weld quality of Friction Stir Welding. Performing optimization of mechanical properties of FSW using Genetic Algorithm. 1
To draw conclusions on prediction of mechanical properties of FSW optimization of aircraft fuselage aluminium.
1.3. Overview of the thesis
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Chapter 2 - Literature Review
2.1. Introduction and scope In this chapter, an overview about system identification, neural-fuzzy networks, regressive analysis, and statistical tools in MATLAB will be discussed. 2.2. System Identification System identification is a process of building mathematical models of dynamic system from the experimental data, in order to control a system. Accurate model that gives the exact behaviour of the system is required to control a system. The accurate model is further used to develop a controller, which provides stability and desired performance [2]. System identification is an effective approach to construct accurate models of complex systems from noisy data. It involves three phases, which are interrelated. The first phase is to design the experiment, second phase involves in, constructing a mathematical model from the experimental data, and in the final phase model parameters are estimated from the measurements [3]. 2.2.1. Least-squares estimation: The term describes least-squares method often used for solving over determined or inaccurately specified system of an equation in a direction approximate. Least-squares, minimize the sum of squared residuals, and play a key role in the interference of parameters in nonlinear regression model [4]. Least-squares model was developed by Karl Friedrich Gauss at the end of the eighteenth century. According to Gauss, the unknown parameters are selected in such a way that “The sum of the squares of the differences between the actually observed and the computed values, multiplied by numbers that measure the degree of precision, is a minimum [5]”. 3
The mathematical model is of the form () () () () () Equation 2.1 are the unknown () ( ) are known functions. are the
Where, y is the examined variable, parameters of the model, and () ()
The model in Equation 2.1 is called regressive model, and the variables regressive variables [5].
The coefficients are determined so that the output calculated by the mathematical model in Equation 2.1 agrees as much as possible in accordance with the observed variables ( ) in the least-square sense [5]. The parameter has to be
chosen to minimize the least-square function ( Let, Y(t) = ( ) ( ) ) ∑ ( () ( ) ( ) , and ( ) () ) Equation 2.2
( ) ( ) ( ) ( ) ( ) ( )
The residuals are defined by () () ̂( ) () () Equation 2.3
From the above equation the cost function can be written as ( ) ∑ () ̅ ‖ ‖ Equation 2.4
Where, E can be written as
By solving the loss function to find the minimum we get ( ) ( ( ( ( ) ) ) ) ( ( ) ) Equation 2.5
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From the Equation 2.5, the minimum of the cost function is obtained for ̅ 2.3. Regressive analysis Regressive analysis is a method for studying functional relationship between variables [6]. Regressive analysis helps to understand how dependent variable varies when one of the independent variable is varied and the rest are held constant, by determining the functional relationship between a dependent and one or more independent variable [7]. Regressive parameters are estimated on the basis of collecting data using the estimation method. Least-squares method is the most commonly used valuation method for evaluating the parameters [6]. The simple form of a linear regression model has the form ( )
(
Where,
)
Equation 2.1
Is the input variable Is the response The equation involving linear combination of fixed nonlinear function of the input variable is of the form
(
Where,
)
∑
( )
Equation 2.2
( ) is the basis function and M represents the total number of ( ) , the
parameters from the index I by M-1. By assuming the Basis function equation can be written as
(
)
∑
( )
( )
Equation 2.3
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Where,
, ( )
The second order mathematical equivalent, regressive polynomial equation of the surface response ‘Y’ is given by [8] ∑ ∑ ∑ Equation 2.4 are the regressive coefficients in the model. Regressive modelling involves the following steps: Collect the test data. Use the test data to determine unknown parameters of the model. Check the effectiveness of the model. The satisfied model is used for prediction, estimation. [9] 2.4. Neural Networks The Seminal works of Mcculloh and Pitts (1943) were a pile of development in the neural network architectures. Mcculloh and Pitts proposed the unification of mathematical logic neurophysiology, which made his way to yield meaningful results in research of neural networks. Neural networks are simplified mathematical model or computational approach of the biological nervous system inspired from the computation of the human brain. In neural networks, the networks are distributed in parallel with highly interconnected network of a large number of processing elements called neurons in the architecture inspired by the human brain [10]. Artificial neural networks are nonlinear statistical tools for data modelling, usually used to model the functional relationship between inputs and outputs.
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2.4.1. Model of an artificial neuron The behaviour of the basic artificial neuron model is shown in Figure 2-1. Each model directly bears resemblance to the real components of the biological neuron and, therefore, is defined as an artificial neuron.
Figure 2-1 Simple model of an artificial neuron [10] Here, inputs and are the synaptic weights connected to the ‘n’ are the input signals.
A biological neuron receives inputs via dendrites, sums them and produces an output in the cell body through the synapses that can accelerate or delay the arrival of the signal. The acceleration or retardation of artificial neural network is modelled by the weights. In an artificial neural network, the synapses that transmit strong signals, the greater the weight, and weak signals are lighter. So the weights are the multiplier effects of global factors in the strength of synapses [10]. Hence, the overall input received by artificial neuron is given by: Equation 2.5
∑
Equation 2.6 called
The overall input obtained is passed onto the nonlinear filter activation function to obtain the final output ‘Y’. 7
( )
Equation 2.7
Threshold function is the commonly used activation function. In this sum, obtained is compared with a threshold value . If the value of ‘I’ is greater than the output is 1, else it is 0 [10]. For this type of activation function, we have ( ) ( ) Where, is the step function known as Heaviside function. ,
2.4.2. Single layer feed forward network Single layer feed forward network has only two layers, the input layer and the output layer. The input layer neurons receive input signals, and the output layer neurons receive output signals. The synaptic weight connects all the input neurons to the output neurons, but not vice versa. Hence, this network is termed as a single layer though it has two layers. The input layer simply sends the signals only to the level of output. Hence, it is a single layer feed forward network [10].
Figure 2-2 Single layer feed forward network [10]
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In Figure 2-2, it can be seen a general Single layer feed forward network model. Where, represents the input neurons and represents the output neuron.
2.4.3. Multi layer feed forward network The name indicates the network is made of multilayer. Multi layer feed forward network has input layer, output layer, hidden layer. The hidden neurons perform intermediary computations before directing the input layer to the output layer. The weights of the input layer linked to the hidden layer are referred as inputhidden layer weights. The weights of the hidden layer-neurons linked to the output layer neurons are referred as hidden-output layer weights [10]. A Multi layer feed forward network with m neurons in the input layer, neurons in the first hidden layer, neurons in the second hidden layer, and n neurons in the output layer are written as m- - –n. Figure 2-3 illustrates the multi layer feed forward network with m- - –n configuration.
Figure 2-3 Multi layer feed forward network (m-l_1-l_2–n configuration) [11].
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2.5. Neural Fuzzy network Neural-fuzzy network is used to find the parameters of a fuzzy system by exploiting techniques of approximation of neural networks. They are also used to solve a problem if there is no mathematical model of the problem [12]. Fuzzy membership functions are created using neural networks. The set of input-output data values divided into training and checking data set. The training data set trains the neural network, and a nonlinear relation between inputs, output data sets are obtained. The checking data is used to verify how well the neural network can simulate nonlinear relationship. Neural-fuzzy networks are used for the system where the relationship between input and output is highly nonlinear [13]. 2.5.1. Adaptive Neural-Fuzzy Interface System ANFIS is an adaptive network, functionally, equal to the fuzzy interface system, and is used to tune the rules of fuzzy systems [14]. ANFIS uses multi input and a single-output system where the data is trained by using a combination of leastsquares and back propagation gradient descent method. It represents the Sugeno type fuzzy interface system and uses hybrid learning algorithm to identify the membership function parameters of a single output [15-16]. The input, output train data, is a matrix with N+ 1 column where the N columns contains data input and the last column contains single vector of output data [16]. Consider and are the two inputs and one output function f, expressing
the rules in first order Sugeno model. Rule 1: if Rule 2: if is is and and is is Then Then
Graphically the rules are shown in the Figure 2-4.
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Figure 2-4 Rules showing graphically [14] From the Figure 2-4, we can see that the output parameters are the linear combination of output and input. The ANFIS equivalent of the above graphical representation is shown in Figure 2-5.
From the Figure 2-5, the model has five layers. Layer 1 is called the membership function layer. In this layer, the output of any node gives membership degree of the input. The second layer is called multiplication layer. In this layer, each node by multiplying the membership degree of the input resulting in the firing strength or the extent to which the corresponding rule is triggered. Third layer is just a layer of normalisation. It is the ratio of a particular level of activation of a specific rule to the sum of all rules. In the fourth layer, Sugeno processing rule is applied and, the output is calculated. In the final layer, the overall performance is calculated as the sum of all incoming signals [14]. 2.6. Summary In this chapter the underlying theory is presented, so that the reader understands the basic concepts in detail about system identification, regressive analysis, neural networks, and neural-fuzzy networks. Detail explanation is given on the data-driven modelling techniques, which enables the reader to understand the best technique that can be employed for the purpose of the project.
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Chapter 3 - Friction Stir Welding (FSW)
3.1. Overview Friction stir welding is a welding technique in solid -state invented by Wayne Thomas in 1991, at the welding Institute (TWI). Initially used for joining the aluminium alloys. Friction stir welding is used in aerospace, automotive, ship building, railway rolling stock industries [17]. 3.2. Friction Stir welding process In FSW, a non-consumable, rotating tool with a shoulder, and a specially designed nib is inserted into the butting surface of the sheets or plates to be joined. The nib is rotated at constant speed, at a constant traverse rate, along the joint line between the two pieces as shown in Figure 3-1. This results in heating and softening of surrounding material. Sheets or plates are rigidly clamped in order to prevent abutting [8, 17].
Figure 3-1 schematic diagram of FSW process [18]
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Strong bond is produced by utilizing the frictional heat and forging pressure. The major advantage of FSW is, it softens the surrounding medium without reaching the melting point, and transforms from solid state into a plastic like state. In this state, the two materials to be butted together are stirred mechanically under pressure to create a welding joint [19]. 3.3. Microstructure classification of Friction Stir Welding (FSW) Typical Friction stir welding section consists of four areas. Figure 3-2 shows the four distinct zones in a friction stir weld process a) Unaffected material b) Heat- affected zone (HAZ) c) Thermo- mechanical affected zone (TMAZ) d) Nugget (stirred) zone
Figure 3-2 schematic diagram of a typical FSW showing four distinct zones [17] Unaffected material: This is the material away from the weld, which has not been distorted, and although it may have undergone a thermal weld cycle, but is not affected by heat in terms of microstructure and mechanical properties. Heat-affected zone (HAZ): The heat affected zone is similar to that of traditional welds, while the peak maximum temperature is considerably lower than the solidus temperature, and heat is highly fragmented. This can lead to more or less distinct micro-structures compared to fusion welding process [19]. 14
Nugget (stirred) zone: The central nugget contains "onion ring" resemblance, is the one which encounters the most severe strain, and is a consequence of how a threading tool material deposits from the front to the back of the weld [19]. Thermo-mechanical affected zone (TMAZ): The thermo-mechanical affected zone lies between nuggets and heat-affected zone. The micro-granules of the original structure are preserved in the thermo-mechanical affected zone, but usually in a crooked state [19]. The FSW parameters such as tool rotational speed, forward movement per revolution play a vital role in determining the strength of the welding joint. 3.4. Mechanical properties of FSW joint Obtaining the mechanical properties of materials is essential to determine the usefulness of a product. 3.4.1. Yield strength According to engineering and material science, Yield strength is defined as the stress applied to the material at which the material starts to deform plastically. The material deforms elastically and regains it shape when the applied stress is removed [20]. 3.4.2. Ultimate Tensile Strength Ultimate tensile strength or Tensile strength is defined as the maximum stress that can be applied to the material, so that the material can withstand without breaking. It is the maximum stress, on the stress-strain curve, and is measured in Mega Pascal (MPa) units [20]. The stress-strain curve is shown in Figure 3-3.
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Figure 3-3 stress-strain curve [21] 3.4.3. Elongation The material extends to a certain amount before breach, when it is tested for tensile strength. When the two pieces are put together, and the expansion of the measure is against the marks made before the test, and is expressed as a percentage of the original length [22].
The typical strength of Aluminium AA5083 varies between 230 and 570 MPa. Data-driven modelling techniques are employed in order to predict the tensile strength, yield strength and weld quality of FSW welded aluminium alloys AA5083.
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3.5. Previous experimental work In previous studies, research conducted in three phases: experimental work, computational intelligence and process optimization of friction stir welded AA5083-0 aluminium alloy, developed by George Panoutsos, Mahdi Mahfouf, Kathryn Beamish, Ian Norris. Granular computing (GrC) and Neural-Fuzzy (NF) modelling are used to obtain the functional relationship with a minimum of experimental data available [23]. AA5083-0 aluminium alloy with a thickness of 5.8mm and Tri-flute MX tools was used for experimentation. The range of tool rotational speed and welding speed are selected based on the expert knowledge, and the augmented L25 Taguchi series was designed as shown in Table 3-1[23]. The target was to carryout welds of both excellent standard and poor standard, In order to capture much system information. The welded plates were subjected to tensile strength. This test is a pass-fail indicator, in terms of weld quality. Therefore, a notched prototype model was used, forcing the tensile test to occur in the weld region. The Figure 3-4 shows the standard weld quality associated with poor weld quality [23].
Figure 3-4 FSW tensile test specimens, fracture surface [23]
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Table 3-1 augmented Taguchi series for the FSW experiments [23] Granular computing was used to obtain information from data sets, and granular data set is released through fuzzy membership functions which are the basic rules of the system model. Granulation of the data was performed through a two-step iterative process that involves the following steps:
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1)
Find the two most compatible granular data and integrate them together as a new data granule containing both original granules.
2)
Repeat the process of finding the two granules until a sufficient level of abstraction of the data is reached [23].
A mathematical representation of the compatibility specification is given in equation 3.1. Compatibility Where, Distance Density ∑ ( ) ( ) Equation 3.1
∏
are the weights to balance the distance/density requirements and k is the data space dimensionality[23]. The information gathered during granulation is stored in the algorithm. The captured data define the basic structure of the fuzzy rule-base, number of rules, starting position and size of the membership function. A singleton fuzzy output can be expressed by the following equation [23]. ∑ [∑
∏ ∏ ( ) ( )
]
Equation 3.2
Where, equation
( ) is the Gaussian membership function, and can be calculated using the
( )
[
(
)
]
Equation 3.3
Neural-fuzzy learning strategy starts by initializing the centres and widths of each fuzzy weight. This can be achieved by taking the corresponding values from
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granular computing rule-base structure. The neural-fuzzy system is then trained. The GrC-NF modelling flowchart is shown in Figure 3-5.
Figure 3-5 GrC-NF modelling flowchart [23] The development of process models based on independent and dependent process variables have been described. The simulation of these process models forms
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the basis of a procedure for multi-objective optimization to achieve a successful methodology based on specific targets [23].
3.6. Summary This chapter gives brief information about FSW technology. The major factors of the FSW are explored to facilitate the terminology used for research in the next chapter. Finally, some key information was provided on previous experimental work from which this project began, to predict the yield strength, tensile strength and weld quality of the friction stir welded Aircraft fuselage aluminium alloy AA5083.
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Chapter 4 - Prediction of Mechanical properties of Friction Stir welded Aircraft fuselage aluminium (AA5083)
4.1. Introduction A mathematical model needs to be developed, in order to predict the Yield strength, Tensile strength and weld quality of friction stir welded aircraft fuselage aluminium alloy (AA5083). Statistical tools were used in MATLAB program to evaluate the functions. The functions evaluated in MATLAB [24] program, were used predict the yield strength, tensile strength, and weld quality. 4.2. Multiple linear regression using least squares Regression analysis is the process of evaluating a mathematical equation that best fits the data. General form of the probabilistic model of the regressive analysis is given by [9]: ( ) Where is the dependent variable ( ) is the expected value of is the error component Linear regression analysis is to use the model to estimate E (y), the average value of y, or predict a future value of y for a given value of x. The aim of regression is to develop an optimal model. We need to know the reliability of prediction, to predict for given values of . This is often called as regression
model [9]. The second order mathematical equivalent, regressive polynomial equation of the surface response ( ) is given by
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( )
∑
∑
Equation 4.1
Therefore, the probabilistic model of the regressive analysis is given by ∑ ∑ Equation 4.2
4.2.1. Developing a mathematical model using MATLAB MATLAB uses least-squares in multiple linear regressions, in evaluating the mathematical model that best fits the data. REGRESS and REGSTATS are the two statistical functions used in MATLAB for computing a mathematical model [26]. Syntax for evaluating the coefficients in MATLAB is given by: ( )
Where, X is a model input n-by-p matrix with rows corresponding to observations and columns for the predictor variables. Y is an N-by-1 vector response to the results. B = regress (Y, X) returns a vector B, the regression coefficients for the linear model Y = X * B [26]. The second order mathematical equivalent, regressive polynomial in Equation 4.1 was used to predict the functional relationship of Yield strength (YS), Tensile strength (UTS) and Weld quality (WQ) in relation to the Tool rotational speed (N) and Welding speed (S). The FSW data are shown in the Table 4-1. The main objective was to demonstrate the effectiveness of the regression model.
4.2.2. Mathematical model to predict Tensile strength The Tool rotational speed (N), Welding speed (S) are taken as inputs, to predict the mechanical properties of Tensile strength (UTS). Tensile strength of the joint is the function of tool rotational speed (N), welding speed (S) and it can be expressed as ( ) Equation 4.3
The second order regressive polynomial could be expressed as ( ) ( ) ( ) ( ) ( ) Equation 4.4
The tool rotational speed and welding speed are taken as inputs and the ultimate tensile strength being the dependent variable, the regressive coefficients of the linear model UTS = B*X was found. The final mathematical model to predict Tensile strength of FSW welded joints of aircraft fuselage aluminium AA5083 is given below: ( ) ( ) ( ) ( ) ( )
Equation 4.5
Figure 4-1 shows the plot between the measured Tensile strength and predicted tensile strength produced from MATLAB program. The m-file of the program is attached in Appendix I.
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Figure 4-1 Plot showing Predicted, actual tensile strength with experimental data
4.2.3. Mathematical model to predict Yield strength The Tool rotational speed (N), Welding speed (S) are taken as inputs, to predict the mechanical properties of Yield strength (YS). Yield strength of the joint is expressed as the function of tool rotational speed (N), welding speed (S) and it can be expressed as ( 4.6 The second order regressive polynomial could be expressed as ( ) ( ) ( ) ( ) ( ) Equation 4.7 ) Equation
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The tool rotational speed and welding speed are taken as inputs, and the Yield strength being the dependent variable, the regressive coefficients of the linear model YS = B*X was found. The final mathematical model to predict Tensile strength of FSW welded joints of aircraft fuselage aluminium AA5083 is given below: ( ) ( ( )) ( ( ) ( )) ( ) Equation 4.8
Figure 4-2 Plot showing Predicted, actual Yield strength with experimental data
Figure 4-2 shows the plot between the measured Yield strength and predicted Yield strength produced from MATLAB program. The m-file of the program is attached in Appendix II. 4.2.4. Mathematical model to predict Weld quality The mathematical model to predict the Weld quality follows the same procedure that was mentioned in obtaining the mathematical model for yield strength and Tensile strength. The Tool rotational speed (N), Welding speed (S) are taken as 27
inputs, to predict the mechanical properties of Weld Quality (WQ). Weld Quality of the joint is expressed as the function of tool rotational speed (N), welding speed (S) and it can be expressed as ( 4.9 The second order regressive polynomial could be expressed as ( ) ( ) ( ) ( ) ( ) Equation 4.10 ) Equation
The tool rotational speed and welding speed are taken as inputs and the Weld Quality being the dependent variable, the regressive coefficients of the linear model WQ = B*X was found. The final mathematical model to predict Tensile strength of FSW welded joints of aircraft fuselage aluminium AA5083 is given below: ( ) ( ( )) ( ( ) ( ( )) ( ))
Equation 4.11
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Figure 4-3 Plot showing Predicted, actual Weld quality with experimental data Figure 4-3 shows the plot between the measured Weld Quality and predicted Weld Quality produced from MATLAB program. The m-file of the program is attached in Appendix III. Figures 4-1, 4-2 and 4-3, clearly indicates that the mathematical model obtained is generally not an appropriate model, as predicted output values are not optimized with respect to the value of real output. The mathematical model must be developed that can effectively predict the mechanical properties of FSW joints. Thus, a mathematical model was developed using the ANFIS, as ANFIS is more reliable than the regressive model. 4.3. ANFIS modelling ANFIS toolbox is used to tune the rules of fuzzy interface system. ANFIS builds a fuzzy interface system using a given input-output data set. It uses multiple inputs and a single output system where the data is trained by using a combination of 29
least squares and back propagation gradient descent method (hybrid optimization) [14-15]. The set of input-output data values are divided into training and checking dataset. The training data set of the system to be modelled is loaded to train the FIS. Hybrid optimization method is selected in optim.method. 4.3.1. ANFIS modelling for Yield strength The Tool rotational speed (N), Welding speed (S) are taken as inputs and Yield strength as output. The set of input-output data values are divided into training and checking dataset. In the Load data portion of GUI The training and checking data set are loaded from workspace, as shown in Figure 14. After loading the data, FIS has to be initialized using ANFIS. Choose Grid partition has been selected as partitioning method. A Gaussian membership function with different number of MF’s to each input is to be assigned. The next step is to train FIS. After training the FIS has to be checked against checking data. The process has to be repeated by randomly selecting the training and checking data with different number of MF’s and epochs, until a minimum checking error is obtained. As the experimental data values are vey less. By making the efficient use of experiment data available FIS membership function parameters are trained. After many trials the minimum checking error is reached with the number of training epochs to 3 and 3MF’s to each input. The minimum checking error obtained is 2.9002. The plot showing checking data and FIS output is shown in Figure 4-4.
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Figure 4-4 ANFIS editor [24] Figure 4-5, shows the FIS structure with two inputs, 3MF’s to each input providing 9rules. The 9 rules presented below 1. If (Tool-Rotational-Speed(N) is N-MF1) and (Welding-Speed(S) is SMF1) then (Yield-Strength is YS-MF1) 2. If (Tool-Rotational-Speed(N) is N-MF1) and (Welding-Speed(S) is SMF2) then (Yield-Strength is YS-MF2) 3. If (Tool-Rotational-Speed(N) is N-MF 1) and (Welding-Speed(S) is SMF3) then (Yield-Strength is YS-MF3) 4. If (Tool-Rotational-Speed(N) is N-MF2) and (Welding-Speed(S) is SMF1) then (Yield-Strength is YS-MF4) 5. If (Tool-Rotational-Speed(N) is N-MF2) and (Welding-Speed(S) is SMF2) then (Yield-Strength is YS-MF5) 6. If (Tool-Rotational-Speed(N) is N-MF2) and ((Welding-Speed(S) is SMF3) then (Yield-Strength is YS-MF6)
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7. If (Tool-Rotational-Speed(N) is N-MF3) and (Welding-Speed(S) is SMF1) then (Yield-Strength is YS-MF7) 8. If (Tool-Rotational-Speed(N) is N-MF3) and (Welding-Speed(S) is SMF2) then (Yield-Strength is YS-MF8) 9. If (Tool-Rotational-Speed(N) is N-MF3) and (Welding-Speed(S) is SMF3) then (Yield-Strength is YS-MF9)
Figure 4-5 FIS structure
The colour coding of the branches of the graph, characterize the rules and the logical operations involved. The fuzzy rules created from the ANFIS are shown in Figure 4-6.
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Figure 4-6 Fuzzy rules created from ANFIS
Figure 4-7 Fuzzy membership function plots
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The fuzzy membership functions obtained are shown in Figure 4-7. In fuzzy membership function, the degree of membership is from 0-1, and the type of the membership function is Gaussian membership function.
Figure 4-8 The Yield strength model The 3-D surface plot obtained after training the FIS is shown in the Figure 4-8. The response of the yield strength is clearly observed with respect to the tool rotational speed, and welding speed. In general, the ANFIS needs a large number of data to give an accurate response. The checking error is 2.9002, and distortions in the surface are due to the insufficient data values. The FIS editor is shown in Figure 4-9. The FIS obtained is exported into the workspace and, the mathematical model that relates the yield strength to the tool rotational speed and welding speed is obtained using the MATLAB program mentioned in APPENDIX IV.
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Figure 4-9 FIS editor In ANFIS the functional model obtained is of the form Equation 4.12 This satisfies only a specific input. As we have 20 experimental data sets, we get 20 functional outputs. The aim is to find a mathematical model that can effectively predict the mechanical properties. Therefore, the values obtained from Equation4.12 are considered as an input-output data set, and regressive model is used in ANFIS, to obtain a final model that can effectively predict the tensile strength for different values of inputs. Figure 4-10 shows the plot between the measured Yield strength and predicted Yield strength produced from MATLAB program. The mathematical model that can effectively predict the Yield strength is given by ( ) ( ) ( ) ( ) ( ) ( ( ) ) ( ) ( )
Equation 4.13
Where N is the Tool rotational speed, S is the welding speed and k is the constant value that is obtained from the Equation 4.12.
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Figure 4-10
Plot showing Predicted, Actual Yield strength with experimental data obtained from ANFIS
4.3.2. ANFIS modelling for Tensile Strength The Tool rotational speed (N), Welding speed (S) are taken as the inputs and Tensile strength as output. The set of input-output data values are divided into training and checking dataset. The ANFIS modelling of Tensile strength is same as that of the modelling of Yield strength. The FIS with two inputs and three Membership functions providing nine rules, and the rules are mentioned below: 1. If (Tool-Rotational-Speed(N) is N-MF1) and (Welding-Speed(S) is S-MF1) then (Tensile-Strength is UTS-MF1) 2. If (Tool-Rotational-Speed(N) is N-MF1) and (Welding-Speed(S) is S-MF2) then (Tensile-Strength is UTS-MF2) 3. If (Tool-Rotational-Speed(N) is N-MF1) and (Welding-Speed(S) is S-MF3) then (Tensile-Strength is UTS-MF3)
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4. If (Tool-Rotational-Speed(N) is N-MF2) and (Welding-Speed(S) is S-MF1) then (Tensile-Strength is UTS-MF4) 5. If (Tool-Rotational-Speed(N) is N-MF2) and (Welding-Speed(S) is S-MF2) then (Tensile-Strength is UTS-MF5) 6. If (Tool-Rotational-Speed(N) is N-MF2) and (Welding-Speed(S) is S-MF3) then (Tensile-Strength is UTS-MF6) 7. If (Tool-Rotational-Speed(N) is N-MF3) and (Welding-Speed(S) is S-MF1) then (Tensile-Strength is UTS-MF7) 8. If (Tool-Rotational-Speed(N) is N-MF3) and (Welding-Speed(S) is S-MF2) then (Tensile-Strength is UTS-MF8) 9. If (Tool-Rotational-Speed(N) is N-MF3) and (Welding-Speed(S) is S-MF3) then (Tensile-Strength is UTS-MF9)
The Figure 4-11 shows the Fuzzy rules created from ANFIS
Figure 4-11 Fuzzy rules created from ANFIS (Tensile strength as output) 37
The 3-D surface plot obtained after training the FIS is shown in the Figure 412. The response of the Tensile strength is clearly observed with respect to the tool rotational speed, and welding speed. The distortions in the graph at some points, is due to the insufficient data values to train the FIS.
Figure 4-12 The Tensile Strength model The FIS obtained is exported into the workspace and, the mathematical model that relates Tensile strength to the tool rotational speed and welding speed is obtained using the MATLAB program mentioned in APPENDIX V. Figure 4-13 shows the plot between the measured Tensile strength and predicted Tensile strength produced from MATLAB program.
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Figure 4-13 Plot showing Predicted, Actual Tensile strength with experimental data obtained from ANFIS The model that can effectively predict the Tensile strength is given by ( ) ( ( ) ) ( ( ) ) ( ) ( ) ( ) ( )
Equation 4.14
4.3.3. ANFIS modelling for Weld Quality The Tool rotational speed (N), Welding speed (S) are taken as the inputs and Weld quality as output. The set of input-output data values are divided into training and checking dataset. The ANFIS modelling of Weld quality is same as that of the process carried out in modelling Yield strength, Tensile strength. The FIS with two inputs and three Membership functions providing nine rules, and the rules are mentioned below: 1. If (Tool-Rotational-Speed(N) is N-MF1) and (Welding-Speed(S) is S-MF1) then (Weld-Quality is WQ-MF1) 2. If (Tool-Rotational-Speed(N) is N-MF1) and (Welding-Speed(S) is S-MF2) then (Weld-Quality is WQ-MF2)
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3. If (Tool-Rotational-Speed(N) is N-MF1) and (Welding-Speed(S) is S-MF3) then (Weld-Quality is WQ-MF3) 4. If (Tool-Rotational-Speed(N) is N-MF2) and (Welding-Speed(S) is S-MF1) then (Weld-Quality is WQ-MF4) 5. If (Tool-Rotational-Speed(N) is N-MF2) and (Welding-Speed(S) is S-MF2) then (Weld-Quality is WQ-MF5) 6. If (Tool-Rotational-Speed(N) is N-MF2) and (Welding-Speed(S) is S-MF3) then (Weld-Quality is WQ-MF6) 7. If (Tool-Rotational-Speed(N) is N-MF3) and (Welding-Speed(S) is S-MF1) then (Weld-Quality is WQ-MF7) 8. If (Tool-Rotational-Speed(N) is N-MF3) and (Welding-Speed(S) is S-MF2) then (Weld-Quality is WQ-MF8) 9. If (Tool-Rotational-Speed(N) is N-MF3) and (Welding-Speed(S) is S-MF3) then (Weld-Quality is WQ-MF9)
The Figure 4-14 shows the Fuzzy rules created from ANFIS. The 3-D surface plot obtained after training the FIS is shown in the Figure 4-15. The response of the Tensile strength is clearly observed with respect to the tool rotational speed, and welding speed. The checking error obtained is 1.2166. The small distortion in the graph is due to the insufficient data values, in training the FIS.
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Figure 4-14 Fuzzy rules created from ANFIS (Weld Quality as output)
Figure 4-15 The Weld Quality model 41
The mathematical model relating Weld Quality to that of the input parameters is obtained using MATLAB program mentioned in APPENDIX VI. The plot between the actual weld quality and the predicted weld quality is shown in the Figure 4-16.
Figure 4-16 Plot showing Predicted, Actual Weld Quality with experimental data obtained from ANFIS
The mathematical model that can effectively predict the weld quality is given by ( ) ( ( ) ) ( ( ) ) ( ( ) ( ) ) Equation 4.15 ( )
It is clearly seen that the ANFIS is more reliable when compared to regressive analysis. The predicted mathematical models using ANFIS is used in Genetic algorithm, to optimize the mechanical properties of friction stir welded aircraft fuselage aluminium AA5083. The optimization of mechanical properties of FSW is presented in Chapter 5.
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Chapter 5 - OPTIMIZATION OF MECHANICAL PROPERTIES OF FSW
5.1. Introduction ANFIS models in the earlier phase were used to predict the mechanical properties of friction stir welded aluminium AA5083. Predictions of mathematical models using ANFIS are used in the genetic algorithm, to optimize the mechanical properties of AA5083 aluminium aircraft fuselage. The main objective of the optimization process is to improve the performance of industrial tools. To optimize the mechanical properties of FSW aluminium structures Genetic algorithm is carried out using MATLAB program. 5.2. Genetic algorithm Genetic algorithm is a stochastic global search algorithm, proposed by John Holland in 1975. Customizing the search is entirely based on the principles of natural genetics and natural selection. Genetic algorithm is designed to simulate the process, especially those who follow the principles outlined by Charles Darwin’s “survival of the fittest” [26]. Genetic operators: i. ii. iii. Selection Recombination Mutation
Selection: Genetic algorithm works with the estimates of the current population. The Individuals are initially first drawn from which improvement is sought. Fitness of each individual is evaluated according to the objective function that describes the problem to be solved. At the stage of reproduction, highly fit individuals have a greater probability of being selected to take part in the next stage. Those
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selected individuals are then modified by applying genetic operators to obtain the next generation [26]. Recombination: Recombination leads to couples or individuals with high genetic exchange of information with each other. At recombination, by combining information from parents, new individuals are produced. It is applied, with a high probability for reproduction, and is considered as the most significant genetic operator. This is a case of cross-site parts of chromosomes exchange with their parents in this matter and to produce offspring of elements from both parents [26]. Mutation: The mutation changes the genetic cause of individual performance, according to a probabilistic rule. The mutation is generally regarded as the operator of the substance applied with a low probability to the offspring. Mutation randomly selects or interrupts a new gene value, and can use the high mutation rate without affecting the research [26]. 5.3. Results obtained from optimisation The mathematical models that were used to optimize the mechanical properties of air craft fuselage aluminium AA5083 is shown in Equation 5.1, 5.3 and 5.3 ( ) ( ) ( ) ( ) ( ( ) ) ( ( ) ) ( ) ( ( ) ) ( ( ) ) ( ( ) ( ) ) Equation 5.3 ( ( ) ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( )
Equation 5.1
Equation 5.2 ( )
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The Equations 5.1, 5.2, 5.3 mentioned above are used in the objective function, and the objective function program written in MATLAB is shown in APPENDIX VII. The genetic algorithm tool box has been used. The tensile strength, yield strength and weld quality as the targets, the parameters have to be initialized for the genetic algorithm. After initialization, the genetic algorithm with a physical condition of the initial population of each individual evaluates the objective function that characterizes the problem to be solved. If the optimization criteria are not met, the genetic
algorithm generates a new population by performing selection, recombination and mutation. The new population generated is used to evaluate the fitness, and the process is repeated until the optimization criteria are met. Finally, the best individuals are generated. The MATLAB code for evaluating the optimal tool profile was shown in APPENDIX VIII. The results obtained for different values of yield strength, ultimate strength and weld quality are shown in the Table 5-1.
Table 5-1 Optimisation of Mechanical properties of FSW Sl.No TS (MPa) YS (MPa) WQ N Predicted S Predicted
1 2 3 4 5
305 312 258 291 320
174 174 176 184 180
0 0 3 1 0
546.2838 369.739 576.07 446.97 509.76
0.818733 0.8156 0.6006 1.023566 1.20134
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The Table 5-1 shows the predicted tool rotational speed (N) and welding speed (S) that are obtained for specific values of Tensile strength, Yield strength and Ultimate Tensile strength. The modelling technique representing, the prediction and optimization of mechanical properties of Friction Stir welded fuselage aluminium AA5083, is represented in the form of a flow chart, and is shown in Figure 5-1.
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Figure 5-1 Prediction and Optimisation of FSW represented in a Flow Chart 47
Chapter 6 - Conclusion and future recommendations
6.1. Conclusion The primary objective of the project is to develop a mathematical model using modelling techniques guided by data such as ANFIS and the regression analysis, and determine the best approach, in order to predict the mechanical properties of FSW. Statistical tools such as regressive analysis, ANFIS are used to develop a mathematical model that can predict the Tensile strength, yield strength, and Weld quality of friction stir welded Aircraft fuselage aluminium (AA5083) alloy joints. Linear regressive technique was used to develop a mathematical model that can predict the tensile strength, yield strength and weld quality. But, the predicted values are not much accurate when compared with the experimental values obtained. This is due to the lack of information in the dataset. Further, an Adaptive Neural network Fuzzy Interface System tool box in MATLAB was used to predict the mathematical model. This is because, the advantages of both fuzzy and neural networks are combined together to design the ANFIS. It is one of the best exchanges between fuzzy systems and neural systems, which provide smoothness, because the interpolation of fuzzy control and adaptive capacity due to back propagation of neural networks. The training dataset of the system to be modelled is loaded and FIS is trained. After training, the FIS is checked against the checking data. The process is repeated by randomly selecting the training and checking data, with different number of MF’s and epochs until a minimum checking error is obtained. The minimum checking error is obtained with epochs set to 3 and 3 MF’s to each input. The minimum checking error obtained is found to be 2.9002 and 1.2166 for yield strength and weld quality. In ANFIS, the input variables with MF’s give the membership degree of the input. The membership values are combined resulting in the firing strength or the 48
extent to which corresponding rule is triggered. The rules obtained in the ANFIS clearly show the rules, which triggers based on the input values. From the rules obtained, it can be concluded that poor weld quality is obtained if the welding speed and rotational speed are high or low. The optimal weld quality is obtained if the welding speed is low and the tool rotational speed is moderate or, the welding speed is high and the tool rotational speed is moderate. Rule 1 and Rule 9 gives a poor weld quality, Rule 4 and Rule 6 gives better weld quality, and the weld quality partially fails for the rest of the rules, if the inputs are triggered at these values. Similarly it is observed that the tensile strength is high if the inputs are triggered for Rule 2, Rule 3 and Rule 4, and the Tensile strength is low if the inputs are triggered for the Rule 5, Rule 7, and Rule 9. Rules’ 1, 2,3 gives high value in yield strength, Rules 4,5,6 gives moderate values, and Rules 7,8,9 gives low values of yield strength if the inputs are triggered at these values. Finally, we can conclude that optimal solution is obtained for tensile strength, weld quality and yield strength, if the Rule is triggered at 4. The output of each rule is a linear combination of the input variables plus a constant term, and the final output is the weighted output of each rule’s output. The final output obtained from ANFIS satisfies to a specific input. Therefore, 20 functional outputs are obtained which is a linear combination of input variables plus a constant term. The functional model obtained from neural-fuzzy interference is used in regressive analysis to develop a second order mathematical equivalent, polynomial equation that can predict the mechanical properties of friction stir welded air craft fuselage aluminium. The mathematical model developed using NN-Regressive is much reliable when compared to that of the mathematical model developed using only regressive. The predicted values are much more accurate than the actual values and a successful model was developed using very minimal data. Finally, Genetic algorithm program written in MATLAB is used to optimise the mechanical properties of friction stir welded air craft fuselage aluminium. The purpose of using the genetic algorithm is to improve the performance of the tools industrially. 49
The tensile strength, yield strength and weld quality as the targets, the parameters have to be initialized for the genetic algorithm. After initialization, the genetic algorithm with a physical condition of the initial population of each individual evaluates the objective function that characterizes the problem to be solved. If the optimization criteria are not met, the genetic algorithm generates a new population by performing selection, recombination and mutation. In selection the Individuals are initially first drawn from which improvement is sought. Fitness of each individual is evaluated according to the objective function that describes the problem to be solved. At the stage of reproduction, highly fit individuals have a greater probability of being selected to take part in the next stage. Those selected individuals are then modified by applying genetic operators to obtain the next generation. At recombination, by combining information from parents, new individuals are produced. This is a case of cross-site parts of chromosomes exchange with their parents in this matter and to produce offspring of elements from both parents. The mutation changes the genetic cause of individual performance, according to a probabilistic rule. The new population generated is used to evaluate the fitness, and the process is repeated until the optimization criteria are met. Finally, the best individuals are generated. The mechanical properties of FSW using genetic algorithm are optimised in such a way that, for a given value of yield strength, ultimate strength and weld quality, satisfactory tool profiles are obtained. 6.2. Future recommendations Due to limited time mathematical model to predict elongation, reduction of the area and average grain size is not developed. It is recommended to develop a mathematical model to predict the elongation, reduction of the area, and average grain size. Additional work can be done to improve the ANFIS model, so that the error is reduced, and improved mathematical model can be obtained with the reduction in error.
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It is recommended to improve the genetic code of the algorithm with the introduction of additional constraints to obtain better estimates of the mechanical properties friction stir welded fuselage aluminium.
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