A STUDY ON CO2 LASER CUTTING OF MILD STEEL AND OPTIMIZATION OF PROCESS VARIABLES

Submitted by

K.J. MURALIDHARA For the award of the degree of

DOCTOR OF PHILOSOPHY

In

Physics/Mechanical Engineering Interdisciplinary Dr. MGR Educational and Research Institute University (Declared U/S 3 of the UGC act, 1956)

Chennai – 600 095 OCTOBER 2008

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BONAFIDE CERTIFICATE

I certify that the thesis entitled A STUDY ON CO2 LASER CUTTING OF MILD STEEL

AND OPTIMIZATION

OF PROCESS VARIABLES

submitted for the degree Doctor of Philosophy by Mr K.J.Muralidhara is the record of research work carried out by him during the period from 2004 to 2008 under my guidance and supervision, and that this work has not formed the basis for the award of any degree, diploma, associate-ship, fellowship, titles in this or any other university or other similar institution of higher learning.

Dr B.J Ranganath M.Tech, Ph.D(IIT,Chennai),FIE. Head,Mechanical Engineering Department VVIET, Mysore Signature of the Supervisor with designation

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DECLARATION

I declare that the thesis titled A STUDY ON CO2 LASER CUTTING OF MILD STEEL AND OPTIMIZATION

OF PROCESS VARIABLES

submitted by me for the degree of Doctor of Philosophy is the record of work carried out by me during the period from 2004 to 2008 under the guidance of Dr B.J Ranganath and has not formed the basis for the award of any degree, diploma, associate-ship, fellowship, titles in this or any other university or other similar institution of higher learning.

K.J Muralidhara Prof & Head Department of Physics Sarada Vilas College Mysore Signature of the candidate

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ABSTRACT A laser can produce a coherent, convergent and monochromatic beam of electromagnetic radiation with wavelength ranging from ultra-violet to infrared. Laser can deliver extremely high focused power (1–100kW) with a precise spot size on to any kind of material through any medium. Laser material processing has become popular due to several unique advantages of laser namely, high productivity, ease of automation, non-contact processing, improved product quality, greater material utilization and minimum heat affected zone.CO2 laser is a four level gas laser emitting light of wavelength in the range 9-11 μm, of which laser light at 10.6 μm is widely used in metal cutting. CO2 laser delivers very high output power and can be easily interfaced with necessary computer controlled systems to meet the commercial requirements. Metal cutting is realized through melting and the molten material is removed by using an assist gas such as Oxygen or Nitrogen. When oxygen assist gas is used exothermal reaction takes place during the formation of FeO and other oxides to generate extremely high temperatures and speeds up the cutting process. The objective of the research work is to study the different aspects of metal cutting using carbon dioxide lasers and establish the relationship between the kerf width, Ra value and the process variables like laser power, assist gas pressure and cutting speed. The ANN is used for modeling because it can accommodate non-linearity, interactions, and multiple variables. ANN is found to be highly effective in identifying the functional relationship between inputs and outputs. Three different neural network learning algorithms are used to build the ANN model. The training algorithm that is best suited for the present work is found to be Levenberg-Marquardt algorithm. As the experimental results form the basis of this model, it is highly useful for optimization of process variables. An adaptive artificial neuro-fuzzy

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inference system has been developed to predict the surface finish of the cut surface. The results are compared with the conventional regression model. Heat affected zone during the laser machining is also studied using SEM photographs. Micro metallurgical changes are studied using EDX analysis.

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ACKNOWLEDGEMENT

I would like to take this opportunity to thank those people who helped to make this work possible. My deepest gratitude goes to my advisor, Dr B.J.Ranganath, without his constant inspiration, guidance and support, this thesis would not have been possible. I would also like to thank my research committee members, Dr. P. Aravindan, Dean, Research, Dr. S.Sendhilvelan and Prof Ganesan for their inspiring advice. I am also grateful to Ms GTTC, Mysore for helping me to use the CO2 laser facility. I would also like to thank the management of Sarada Vilas Educational Institutions, Dr A.S Ashok Kumar and colleagues at Sarada Vilas College, for their support in carrying out this research. My sincere thanks go to my family members, especially my wife Rama for sparing me to pursue this work. I also would like to thank Dr M.G.R. Educational and Research Institute (Deemed University) for permitting me to do this work. Last but not least my fond remembrance to two friends and motivators Prof B.J.Subbaramaiah and Shri H.S. Gurunath.

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TABLE OF CONTENTS

Abstract ……………………………………………………………… . ..……..…....iii List of tables…….…………..…………………………………………….........x List if figures…………………………………………………………………..xi List of abbreviations …………………………………………………………..xv Chapter1 Introduction and objective of the work ………………………………………...1 Chapter 2 Literature survey ……………………………………………………………….5 Chapter 3 Lasers 3.1.1 Basic principles………………………………………………………….20 3.1.2 Stimulated emission……………………………………………………..21 3.1.3 Spontaneous emission …………………………………………………..22 3.2

Population inversion…………………………………………………….24

3.3

Pumping…………………………………………………………………25

3.3.1 Three level pumping scheme……………………………………………25 3.3.2 Four level pumping scheme……………………………………………..26 3.4

Laser pumping sources…………………………………………………..27

3.4.1 Electron pumping………………………………………………………..27 3.4.2 Optical pumping…………………………………………………………28 3.5

Amplification and gain…………………………………………………..29

3.6

Threshold condition……………………………………………………..29

3.7

Growth of beam and saturation…………………………………………30

3.8

Critical population inversion……………………………………………31

3.8.1 Laser rate equations and four level lasers……………………………….32 3.9

Laser beam properties…………………………………………………...35

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3.10 Shape of gain medium…………………………………………………..36 3.11 Longitudinal cavity modes………………………………………………37 3.12 Transverse modes………………………………………………………..39 3.13 Collimation………………………………………………………………41 3.14 Monochromaticity……………………………………………………….42 3.15 Coherence………………………………………………………………..43 3.16 Intensity………………………………………………………………….44 3.17 Radiance…………………………………………………………………44 3.18 Focusability………………………………………………………………45 3.19 Laser types……………………………………………………………….46 3.19.1 Carbon-di-oxide laser…………………………………………………..46 3.19.2 Axial flow CO2 laser…………………………………………………...52 3.20 Attributes of laser energy field…………………………………………..53 3.21 CO2 laser as a cutting tool……………………………………………….55 3.22 Laser –material interaction………………………………………………58 3.23 The role of assist gas in laser cutting……………………………………63 Chapter 4 Artificial neural network and ANFIS 4.1 Artificial neural networks…………………………………………………65 4.2 Training algorithm………………………………………………………...68 4.3 ANFIS…………………………………………………………………….73 4.3.1 Takagi-Sugeno model…………………………………………………...75 Chapter 5 Experimentation 5.1 Laser cutting system………………………………………………………78 5.1.1 Beam delivery…………………………………………………………...80 5.1.2 Lenses – meniscus lens………………………………………………….81 5.1.3 Plano convex lens………………………………………………………82 5.1.4 Circular polarizer……………………………………………………….84

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5.1.5 Gas impurities.…………………………………………………………..85 5.1.6 Nozzles.………………………………………………………………….87 5.2

Melt removal mechanism………………………………………………..88

5.3

Kerf flow conditions…………………………………………………….88

5.4

Surface finish measurement……………………………………………..90

5.5 Scanning electron microscope and EDAX studies……………………….91 Chapter 6 Results and discussion 6.1 Effect of laser power on surface finish…………………………………...93 6.2 Effect of cutting speed on surface finish………………………………….94 6.3 Effect of assist gas pressure on surface finish…………………………….95 6.4 Effect of laser power on kerf width……………………………………….96 6.5 Effect of assist gas pressure on kerf width………………………………..98 6.6 Effect of cutting speed on surface kerf width…………………………….99 6.7 Key hole technique………………………………………………………100 6.8 White layer………………………………………………………………105 6.9 Heat affected zone……………………………………………………….107 Chapter 7 Process modeling 7.1 ANN model……………………………………………………………...110 7.2 Comparison of results obtained from ANN with experimental values….119 7.3 Comparison of ANN predicted and experimental values of kerf width at different laser powers……………………………………………………120 7.4 Comparison of ANN predicted and experimental Ra values at different laser powers……………………………………………………………..121 7.5 Comparison of ANN predicted and experimental Ra values at different assist gas pressures………………………………………………………122 7.6

Comparison of ANN predicted and experimental Ra values at Different cutting speeds………………………………………………123

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7.7

Comparison of the kerf width obtained from the ANN and regression Methods………………………………………………………………127

7.8

Comparison of Ra value obtained from the ANN and regression Methods……………………………………………………………….129

7.9

Adaptive neuro-fuzzy inference system ( ANFIS )…………………..134

7.9.1 ANFIS structure………………………………………………………135 7.9.2 Input membership functions………………………………………….136 7.9.3 Fuzzy rules……………………………………………………………136 7.9.4 Comparison of ANFIS predicted Ra values with experimental values at different laser powers...……………………………………………141 7.9.5 Comparison of ANFIS predicted Ra values with experimental values at different assist gas pressures……………………………………….142 7.10

EDX analysis…………………………………………………………143

Chapter 8 Conclusions 8.1

Summary and conclusions…………………………………………...145

8.2

Scope for future work………………………………………………..149

Appendix 1 – Learning algorithms of ANN…………………………………150 Appendix 2 – Tables of experimental and ANN predicted values…………..158 Appendix 3 – Regression result for kerf width……………………………....168 Appendix 4 – Regression result for Ra value………………………………...171 Appendix 5 – ANN weights for L-M algorithm……………………………..174 Appendix 6 – ANFIS details…………………………………………………175 References……………………………………………………………………181 Vitae……………………………………………………………………….....187 List of papers published on this thesis……………………………………….188

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LIST OF TABLES

Table 6.1 - The kerf width at different laser powers …………………….…97 Table 6.2 - variation of kerf width assist gas pressure …………………….98 Table 7.1 Comparison of ANN predicted values of kerf width using L-M algorithm, regression model and experimental values …………..127 Table 7.2 Data statistics - kerf width …………………………………………….129

Table 7.3 Comparison of ANN predicted Ra values using L-M algorithm, regression model and experimental values ……………………..130 Table 7.4 Data statistics – Ra value ……………………………………….132 Table 7.5 Data statistics – Ra value from ANFIS ………………………………...142 Table 7.6 EDAX analysis- composition before and after cutting………………...143 Table A2.1 Comparison of experimental and ANN (with L-M algorithm) predicted values ………………………………………………………………….164 Table A2.2 Comparison of experimental and ANN (with CG algorithm) predicted values ………………………………………………………………….168

Table A2.3 Comparison of experimental and ANN (with quasi Newton algorithm) predicted values ……………………………………………………….171

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LIST OF FIGURES Fig.3.1 Three level pumping scheme………………………………………….25 Fig 3.2 Four level pumping scheme…………………………………………...26 Fig 3.3 Four level laser……………………………………………………………..32 Fig 3.4 Basic components of a laser…………………………………………..36 Fig 3.5 Longitudinal modes occurring within the gain bandwidth of a typical gas laser………………………………………………………38 Fig 3.6 Two longitudinal modes occurring within a laser cavity……………...38 Fig 3.7 Two transverse modes occurring simultaneously within a laser Cavity…………………………………………………………………39 Fig 3.8 Collimation of Gaussian beam………………………………………...42 Fig 3.9 Modes of vibration of CO2 molecule………………………………….47 Fig 3.10 CO2 laser energy level diagram……………………………………...49 Fig 3.11 Rotational mode of CO2 molecule…………………………………...50 Fig 3.12 Transition between vibrational rotational states……………………..51 Fig 3.13 Axial flow CO2 laser…………………………………………………52 Fig 3.14 Laser cutting front……………………………………………………58 Fig 4.1Input neuron with bias…………………………………………………65 Fig 4.2 Neural network interconnections……………………………………...66 Fig 4.3Transfer functions……………………………………………………...67 Fig 4.4 Flow chart for training algorithm……………………………………...69 Fig 5.1 Block diagram of laser cutting system………………………………...78 Fig 5.2 Rofin sinar CO2 laser cutting machine used in the present work……..79 Fig 5.3 Laser cutting head with beam delivery system………………………..80 Fig 5.4 Meniscus lens………………………………………………………….82 Fig 5.5 Plano-convex lens……………………………………………………..83 Fig 5.6 Circular polarizer assembly used in the cutting machine…………….85

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Fig 5.7Different types of laser cutting nozzles.……………………………….87 Fig 5.8 Mitutoya surf test meter used for Ra value measurement.……………90 Fig 5.9 Scanning electron microscope used in the present study.…………….91 Fig 6.1 Variation of surface finish with laser power at different assist gas Pressures.……………………………………………………………..94 Fig 6.2 Variation of Ra value with cutting speed..…………………………………...95

Fig 6.3Variation of Ra with assist gas pressure.………………………………96 Fig 6.4 Kerf width variation at different laser powers..……………………….97 Fig 6.5 Variation of kerf width with assist gas pressure..…………………….98 Fig.6.6Variation of kerf width with cutting speed.……………………………99 Fig 6.7.1 SEM photograph of the front side of the key hole…………………100 Fig 6.72 SEM photograph of the back of the key hole……………………...101 Fig 6.8 SEM photographs showing periodic striations………………………102 Fig 6.9 Enlarged view of the striation showing lateral cracks………………102 Fig 6.10.1 SEM photograph of the kerf showing transverse Cracks.……………………………………………………………103 Fig 6.10.2 SEM photograph of the kerf showing surface Damage. ..……………………………………………………….. 103 Fig 6.10.3 SEM photograph of 5mm thick mild steel cut using laser……….104 Fig 6.10.4 SEM photograph of the work etched across the thickness……….104 Fig 6.11.1 SEM photographs showing white layer when the laser power is 1200watts…………………………………………………………..106 Fig 6.11.2 SEM photographs showing white layer when the laser power is 800watts……………………………………………………………106 Fig 6.12 Micrograph near the surface of the work exposed to the laser Beam……………………………………………………………….107 Fig 6.13Micrograph away from the surface of the work exposed to the laser Beam………………………………………………………………..108 Fig 6.14 Digitized image of the cut kerf showing lateral cracks and surface

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Damage…………………………………………………………….108 Fig 7.1 The neural network model scheme………………………………….111 Fig 7.2 Multilayer perceptron with 3-9-7-1 architecture for kerf width……..112 Fig 7.3 Multilayer perceptron with 3-9-7-1 architecture for Ra value..……..113 Fig 7.4 MLP with 3-12-2 architecture………………………………………114 Fig 7.5 Mean square error versus number of epochs for Levenberg – Marquardt algorithm……………………………………………………………...........116 Fig 7.6 Mean square error versus number of epochs for Conjugate Gradient algorithm…………………………………………………….117 Fig 7.7 Mean square error versus number of epochs for Quasi Newton Algorithm…………………………………………………………….118 Fig 7.8 Comparison of results for kerf obtained from the three algorithms…119 Fig 7.9 Comparison of results for Ra value obtained from the three Algorithms ………………………………………………………….119 Fig 7.10 ANN predicted values of kerf width with laser power ……………………120

Fig 7.11 Variation of Ra value with laser power at a constant Cutting speed of 3000mm/minute……………………………………………………121 Fig.7.12 Variation of Ra value with assist gas pressure……………………..122 Fig.7.13 Variation of kerf width cutting speed………………………………123 Fig 7.14 Surface plot-Variation Ra value with cutting speed and laser power……………………………………………………………………..124 Fig 7.15 Contour plot of kerf width with laser power and cutting speed…………..125 Fig 7.16 Surface plot -Variation of Ra value with laser power and assist

gas pressure …………………………………………………………………………………126 Fig 7.17 Comparison of ANN predicted values of kerf width using L-M algorithm, regression model and experimental values…………….128

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Fig 7.18 The comparison of Ra values obtained from experiment, regression analysis and ANN with L-M algorithm…………………………….131 Fig 7.19 Relative error between the experimental and predicted values…….132 Fig.7.20 Post regression analysis result……………………………………...133 Fig 7.21 Block diagram of the ANFIS system……………………………….134 Fig 7.22 ANFIS structure…………………………………………………….135 Fig 7.23.1 Member ship functions of input variable laser power…………...136 Fig 7.23.2 Member ship functions of input variable pressure….…………...137 Fig 7.23.3 Member ship functions of input variable cutting speed…………137 Fig 7.24 3D Surface plot of laser power, assist gas pressure versus Ra value………………………………………………………….139 Fig 7.25 3D Surface plot of laser power, cutting speed versus Ra value………………………………………………..………..140 Fig 7.26 Comparison of ANFIS predicted Ra values at different laser powers with experimental values……………………………………………141 Fig 7.27 Comparison of ANFIS predicted Ra values at different pressures with experimental values………………………………………………...142 Fig.7.28 EDX spectrum of cut kerf of mild steel…………………………….144

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LIST OF ABBREVIATIONS ANN – Artificial neural network MLP -

Multilayer perceptron

BPNN – Backpropagation neural network L-M –

Levenberg –Marquardt

CG –

Conjugate gradient

BFG -

Quasi Newton with Broyden, Fletcher, Goldfarb and Shanno algorithm

ANFIS- Adaptive neuro-fuzzy inference system. Ra –

Centre line average(Surface roughness)

K–

Boltzmann constant

Nd -YAG - Neodymium – Yttrium Aluminum Garnet

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CHAPTER I

INTRODUCTION AND OBJECTIVE OF THE WORK

Metal cutting is one of the most significant manufacturing processes in the area of material removal application. Metal cutting is defined as the removal of metal from a workpiece in order to obtain a finished product with desired attributes of size, shape, and surface roughness.. In high speed cutting operations such as laser cutting, dimensional accuracy, kerf width and quality of surface finish are three factors that manufacturers must be able to control Among various process variables, surface finish is central to determining the quality of a workpiece . Surface roughness is harder to attain and track than physical dimensions are, because relatively many factors affect surface finish. Some of these factors can be controlled and some cannot. Key controllable process parameters in laser metal cutting are cutting speed, assist gas pressure and laser power. The parameter used to evaluate surface finish in this study is the Roughness Average, Ra value. This parameter is also known as the arithmetic mean roughness value, arithmetic average (AA), or centerline average (CLA). Ra is recognized universally as the commonest international parameter of surface roughness, as defined by the following equation

L

Ra

1 Y ( x) dx L0

where L is the sampling length, and Y is the ordinate of the curve of the profile, the arithmetic mean of the departure of the roughness profile from the mean

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line. One of the aims of the present work is to develop a mechanism using which online surface finish assessment can be made. The other parameter of interest is the kerf width. In laser machining the quality of the cut is assessed by the narrowness of the kerf. This will result in minimal loss of the material being processed. Metal cutting industry in this regard has derived substantial benefit from the innovations in the field of physics. One such device which has changed the perception of cutting tool is a laser. Over the years new category of lasers with compact size and

higher energy delivering ability has been

developed. The relative ease with which they can be integrated into automated manufacturing process has increased their application. Carbon-di-oxide laser which is a molecular laser generates energy in the far infrared region of electromagnetic spectrum. The energy is localized to a very narrow region which is of the order of a fraction of a millimeter. The energy deposition process from a pulsed/continuous wave laser beam into the near-surface regions of a solid involves electronic excitation and de-excitation within an extremely short period of time. The laser–matter interaction within the near-surface region achieves extreme heating and cooling rates of 103–1010 K/s. The total deposited energy which is of the order of 0.1–10 J/cm2 is insufficient to affect the temperature of the bulk material. This allows the near-surface region to be processed under extreme conditions with little effect on the bulk properties. The deposited energy of laser irradiation is converted into heat on a time scale shorter than the pulse duration or laser interaction time. The resulting temperature profile depends on the deposited energy profile and thermal diffusion rate during laser irradiation. Depending on the temperature profile, the irradiated material may undergo only heating, melting or vaporization. The increasing demand of laser in material processing can be attributed to several unique advantages of laser namely, high productivity, ease of automation, noncontact processing, elimination of finishing operation, reduced processing cost, improved product quality, greater material utilization and minimum heat

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affected zone. The major advantage of laser cutting over other non traditional and traditional techniques speed, amenability to a wide variety of materials narrow kerf. Several physical models of laser material interaction have been developed. Some are material dependent and some have parameters which are difficult to measure. The laser material interaction is yet to be fully understood. The soft computing which has made rapid progress during last decade can be used in non linear process modeling. This technique is basically driven by the experimental database and it is extremely useful for online optimization of process parameters. The knowledge derived from soft computing can be easily backward integrated into machine to improve the quality of the process. This work is focused on the study of kerf width and surface finish with the variation of laser power, assist gas pressure and cutting speed in oxygen assisted laser cutting. The real physical processes in laser machining are very complex. Modeling of the Process can be divided according to the area of interests, some are focused on the target material, some are on gas dynamics and some are on the plasma phenomena. The governing equations are usually Energy equation and momentum equations. Most of the models have parameters which are very difficult to measure. Such models, though explains a domain of laser machining are found wanting when it comes to finding the optimal process parameters. Laser cutting is widely used in aerospace, armament, defence, sheet metal industry, automobile and shipping industry. The current research work is focused on developing an interactive model using artificial neural network(ANN) based on the information derived through the experimental work. The ANN is used for modeling because it can accommodate nonlinearity, interactions, and multiple variables. They are also tolerant of noisy data and can operate very quickly in software, and in real time in hardware. Apart from being a useful tool for optimization of process variables, ANN model unlike analytical model, is devoid of complex mathematical equations

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and is more useful for online parametric adjustments. Despite the active research and progress in global optimization in recent years,

no efficient

solution procedure is available for the general nonlinear problems. The objective of this work is to develop a scientific, systematic and reliable methodology based purely on experimental knowledge to arrive at optimal cutting conditions to achieve specified process performance goals. Towards achieving this goal, the ANN model with different learning algorithms and their applicability to the present work and development of a artificial neuro –fuzzy inference system to predict the surface finish Ra is employed. This work also includes the study of surface character produced by the lasers by using scanning electron microscopy and diffused X-ray analysis. Main thrust of the research work is on the effect of laser power, assist gas pressure and cutting speed on the kerf width and surface finish. This will go a long way in online optimization of process variables in laser metal cutting applications.

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CHAPTER 2

Literature survey

A theoretical model for high velocity laser fusion cutting of metals assuming that the light absorption occurs as per the classical Fresnel formulae has been given by W Schulz et al(1987). The dependence of the cutting depth and the mean absorbed laser power on the laser intensity and the mode number has been discussed and they have obtained an optimal choice for the laser focus position and the beam divergence. Jorgensen etal.(1991 ) have studied on-line quality detection during CO2 laser cutting. Fourier Analyses and statistical analyses of the signals have been carried out and from these analyses they have estimated the surface roughness in the cut kerf, dross attachment at the backside of the work piece and the penetration of the laser beam The quality of a laser cut surface is of paramount importance in laser material processing. The mechanisms governing the laser cutting process are not fully understood. In this context P. Di Pietro and Y. L. Yao(1994) have investigated quality improvement techniques in laser cutting. They have also reviewed the present trends and future directions in the field of laser cutting.. Ivarson et al(1994) have studied the phenomena which give rise to a cyclic cutting event when a CO2 laser is used in conjunction with an oxygen jet to cut mild steel. They have postulated a oxidation cycle to explain the cut edge striation which affects the cut surface quality. Powell et al (2000) have studied the conductive losses in CO2 laser cutting. Their studies have shown that in oxygen assisted laser cutting of steel the oxidation reaction is the major contributor for cutting at higher thickness of the sample. For a particular laser power the power of oxidation exceeds that of laser. It is also observed that the power supplied to the cut zone increases

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with the material thickness. But this is offset by a greater conduction loss. The process of metals cutting with a laser beam is described by a mathematical model comprising a set of balance equations for mass, momentum and energy by A. F. H. Kaplan(1996). All the parameters of the laser beam and of the process gas flow are considered in this model. Numerical evaluations of the analytical model provide explanations of the process behavior both for inert and for oxygen gas cutting. The narrowing of the kerf width with increasing speed is explained by heat conduction. B. S. Yilbas(2004)

has given a

Statistical method based on factorial analysis to identify the influence of cutting parameters on the resulting cut quality by using International standards for thermal cutting. Contribution of exothermic oxidation reaction in oxygen assisted cutting is also considered in the analysis. First and second law efficiencies for laser cutting process are formulated. They have shown that increasing laser beam scanning speed reduces the Kerf width while Kerf width increases with increasing laser output power.

The importance of the

exothermic reaction to CO2 laser cutting was found by many previous researchers and this reaction was also investigated widely over the years. However, a systematic investigation about the effects of gas composition on high power CO2 laser cutting has not yet been performed. Shang-Liang Chen(1998) Chen have investigated the importance of the exothermic reaction to CO2 laser cutting and effects of gas composition on high power CO2 laser cutting of mild steel. Non-linear interacting factors responsible for laser cutting process performance. The effects of gas-composition variation and the gas pressure on the cut quality are investigated, with particular reference to small variations in gas composition. Yilbas, B. S.( 1997) has developed a mathematical modeling of CO2 laser cutting and has obtained a numerical solution of the heat transfer equation. The melting front velocities at different laser powers and work thicknesses are predicted. Model for the metal cutting with a gas assisted CO2 laser beam has been proposed by Chen S-L (1998)

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considering the cutting front as a surface phenomenon for reaction and absorption. The effects of oxide films, polarization, cut front shape, cutting speed and laser power has been analysed. They have shown that very small levels of impurity in the oxygen will have a significant effect on the cutting performance. The effect of assist gas pressure on the material removal rate in laser cutting has been investigated by K. Farooq and A. Kar(1998). A model for melt depth is developed on the overall energy balance, and the cut depth is obtained by considering the effect of the assist gas. The model for kerf width is also developed on the basis of modified Rosenthal solution taking into account the melting effect. Ming-Jye Tsai and Cheng-I Weng(1998) have used a perturbation method to investigate the linear stability behavior of a thin molten layer in laser cutting. They have obtained a nonlinear generalized kinematic equation for the film thickness. Linear stability analysis has shown that there exists an optimum cutting speed which decreases with increasing cutting thickness of the work piece and increases with increasing gas flow velocity. A large number of non-linear interacting factors responsible for the quality of laser cutting makes it impracticalto investigate all the factors by experimental method only. To understand further the essential phenomena of the oxidation reaction in reactive CO2 laser cutting, a mathematical model was created by Chen S-L(1998) to investigate the effects of cutting parameters on the dynamic behaviour of cutting front. By using a continuous-wave laser beam having a Gaussian intensity distribution in Oxygen assisted CO2 laser cutting of a thin metallic plate

a numerical relationship has been

obtained.This result is used to predict the variations in the location, speed and acceleration of the cutting front edge, given various gas compositions, gas pressures, cutting speeds and beam absorptivities. B. S. Yilbas, S. J. Hyder, and M. Sunar(1998) have used Taguchi method to classify the relevant parameters such as waviness and flatness, Cutting speed, oxygen pressure, and workpiece thickness. Scanning electron

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microscope photography is used to examine the resulting cut surfaces. They have found that the cut quality is mainly affected by the oxygen gas pressure and cutting speed. Shang-Liang Chen(1999) has investigated CO2 laser cutting performance on 3 mm thick mild steel plate with assistant-gas pressures of up to 10 bar. The results have shown that for laser cutting of mild steel up to 3mm oxygen is the most suitable assist gas.S. M. Shariff, G. Sundararajan, and S. V. Joshi (1999)

have studied the influence of two major process parameters

namely laser power and cutting speed on cut surface quality attributes such as surface roughness, kerf width, heat affected zone. The oxygen-assisted cutting involving an exothermic oxidation reaction, which contributes significantly to the overall energy input to the cutting front is also studied. B S Yilbas(2001) has studied the Striation formation due to the slow drifts and disturbances in various parameters during the laser cutting process. The effects of laser power, cutting speed and energy coupling factor on the kerf size are investigated. His work has shown that increasing laser power and energy coupling factor increase the kerf width size. Also small changes in laser power, cutting speed and energy coupling factor modify the kerf width to a great extent. Dayana Espinal and Aravinda Kar(2000) have developed A simple mathematical model to relate kerf depth to laser power, laser scanning speed and kerf width during laser cutting. The model is based on a lumped parameter technique in which the overall energy balance is considered. The energy released during the chemical reaction between the material that is being cut and the assist gas is considered for oxygen-assisted cutting. Their work has shown that Oxygen increases the cut depth significantly compared to when nitrogen is used as an assist gas. Asano et al(2003) have shown that the range of processing conditions which allow cutting is determined by the energy input per unit area . The values of roughness of the cutting surface on both entry and exit sides of the plates can be reduced if the cutting speed is 1000 mm/min or higher. They change little

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at small values if the heat input per unit area is within a range under 20 J/mm2. Cutting with small heat input always results in better finish of cut surface. A self-consistent model of the laser cutting process has been developed by taking into account the laser beam intensity distribution, the thermal phenomena controlling the melt parameters, cutting gas , melt hydrodynamics, and the local equilibrium of the geometry of the cutting front by Cédric Mas and Rémy Fabbro(2003).BasemF. Yousef etal(2003) have developed a neural network model using the experimentally acquired data. Using mean depth and mean diameter of the crater as inputs, and pulse energy, variance of depth and variance of diameter as output parameters. Their study has shown that the proposed neural network approach is useful in understanding the behavior of the cutting process during laser machining to a high degree of accuracy. Uslan.I(2005) has investigated the influence of laser power and cutting speed variations on the kerf width size. A lump parameter analysis is introduced when predicting the kerf width size and an experiment is conducted to measure the kerf size and its variation during the cutting process. He has found that the workpiece surface influences significantly the kerf width size. He has also shown that the variation in the power intensity results in considerable variation in the kerf size during the cutting, which is more pronounced at lower intensities.Zhang et al (2005) have proposed a synthetic evaluation method for laser cutting quality.. The

cut quality

indicators, such as kerf width, striations, dross, roughness, under different conditions have been studied by a neural network based method. G V Ermolaev(2006) et al have given a physico-mathematical model for the phenomenon of formation of periodic striations in oxygen laser cutting of mild steel sheets. Surface roughness is assumed to be caused by a cyclic reaction of iron–oxygen oxidation. The mathematical description is based on solving the adjoint problems of heat and mass transfer in the liquid phase and in the solid metal with nonlinear moving interfaces between the substances

10

and phase changes. They have described roughness as functions of the cutting velocity, purity of oxygen and thickness of the film of the iron oxide being formed by numerical simulation technique. B S Yilbas et al(2006) using a CO2 laser with variable pulse frequency and Oxygen, as assisting gas, at different pressures have investigated the cutting process. SEM and X ray diffraction are carried out to obtain micrographs and oxide compounds formed in the dross. They have found that the liquid layer thickness increases with increasing laser output power and reduces with increasing assisting gas velocity. Lee Mein Wee, and Lin Li(2005) have developed a two-dimensional, analytical model placing special stress on the effect of laser parameters such as power, scan speed and spot sizes. Power absorption over the cut front, effect of oxidation and melt film thickness are predicted in the model. Markus S Gross (2006) in his paper concentrates only on exothermic reactions of the melt with the assist gas while neglecting other important physical mechanisms for modeling the laser cutting process. Giovanni Tani et al(2004) have investigated the striation and dross formation on the basis of an analytical model by considering mass, force and energy balances. They have evaluated the 3D geometry of the cutting front, and the geometry and temperature fields of the melt film. An interpretation for the striation pattern on the basis of the evolution of the melt film is given and the effect of the assist gas pressure is predicted. The mechanics of dross formation has been studied by using the relation between kinetic energy of the melt film and the local temperature. Striation produced during the laser cutting affects the surface roughness, appearances and geometry of laser cut products. Lin Li, M. Sobih and P.L. Crouse (2007) have given a theoretical model to predict the critical cutting speed at which striation-free cutting takes place. It is also observed that at cutting speeds above the critical cutting speed, striation reappears and surface roughness increases with the cutting speed. Gabzdyl et al(1992) in their experimental results have shown that the purity of the assist

11

gas can significantly affect cutting productivity and even cut quality. detailed experimental work allowed the authors to theoretically clarify why the process is highly sensitive to small levels of contamination of the oxygen jet. Three-dimensional analysis of gas entrainment operating during the lasercutting process was studied by O'Neill and Steen(1995), the turbulent gas flow from a 1.5 mm diameter circular orifice through a model cut was examined to determine the magnitude of impurity entrainment as a function of kerf width and cut depth. Calculated entrainment levels were given for each combination of kerf width and cut depth within the chosen range. An experimental investigation of the benefits of using an annular jet in conjunction with the standard cutting jet was made during CO2 laser cutting trials to examine the validity of the theoretical calculations. Role of oxygen purity in laser cutting of mild steel is also studied by Powell et al. (1992),Design and characteristic analysis of supersonic nozzles for high gas pressure laser cutting was studied by Man et al.(1996) dynamic characteristics of gas flow inside a laser cut kerf under high cut-assist gas pressure was also studied by Man, et al.(1999). The behaviour of the cut-assist gas jet inside a simulating laser cut kerf for a supersonic and a conical nozzle tip were studied by a shadow graphic technique under conditions of inlet stagnation pressure from 3 to 7 bar. The effects of the stand-off distance, kerf width, material thickness and the inlet stagnation pressure upon the dynamic characteristics and momentum thrust of the gas flow inside the cut kerf were investigated. They found that the sensitivity to the stand-off distance and the workpiece thickness of the supersonic nozzle are much less as compared with the conical nozzle. With the supersonic nozzle, a dross free clean cut on 5 mm stainless steel can be achieved at an inert cut-assist gas pressure as low as 5 bar instead of the normal operating range of 10 bar or above for the conical nozzle. Instead of removing the melt by using a gas jet, a different method was tried by Weingartner et al.(1996). The molten material is sucked into a

12

bore coaxial to the beam and then taken away by a fast horizontal gas jet. The vacuum in the suction bore is obtained by fast expansion of highly compressed air. This design is then used to cut 1.25 mm mild steel under the variation of several parameters. It shows very similar results to conventional cutting processes, only the characteristic appearance of the top and bottom kerf edge being changed upside down. Currently molten material in laser cutting is still removed through gas jet mainly. Ilavarasan et al(1995) have developed an off-axial gas jet that has the potential to extend the laser's effectiveness by improving the rate at which parts can be machined, producing high-quality surfaces, enhancing the cutting thickness, and adding to the range of materials that can be machined. In laser cutting, an erosion front (liquid-gas region) forms at the momentary end of the cut. Laser heating, exothermic reactions and shear force between the gas flow and the molten layer dictate the material removal rates. The principle of the offaxial gas jet is to provide straight, nonturbulent flow to the cutting erosion front, causing further oxidation reactions and transferring momentum to the molten slag and dross, thereby improving the cutting speed and the quality of cut. Hsu and Molian (1995) have used a dual gas-jet, laser-cutting technique involving coaxial and off-axial oxygen gas flows to cut 6.35-mm thick AISI 304 stainless steel plates with a 1.2-kW CO2 gas transport laser at a cutting speed of 12.7 mm/sec (30 in./min). Under identical process conditions, the single, coaxial gas jet could not cut the stainless steel although the cutting speed was reduced to 2.11 mm/sec (5 in./min). Thresholds of off-axial nozzle diameter, gas-impinging angle, oxygen pressure, and other process parameters were determined to obtain clean-cut edge quality. O'Nell et al. (1992) examined the characteristics of oxygen gas jets during CO2 laser cutting of steel. Particular emphasis is placed on the mass transfer effects that are operating

13

within the kerf. Oxygen concentration levels within a model kerf are measured for various laser cutting set-ups. The results show a substantial reduction in oxygen concentration within the kerf. A system for oxygen concentration maintenance is described. A Model has been developed for laser cutting of metals with non-reactive assist gas jet by Tong(1997).The results from continuous flow and pulsed versions of the model are in reasonable agreement with cutting data for Aluminum, Stainless Steel and Titanium. The interaction of the gas jet with the molten layer during reactive-gas assisted laser cutting is investigated by Lim (1998) and Yilbas (1998) . The erosion front is modeled as a free surface and its profile is determined from the interaction. Interaction studies between the liquid metal and gas jet during laser cutting were made and relationships between the various parameters influencing the cutting action were identified theoretically by Yilbas et al. (1992) . The effect of the jet on the process, due to jet momentum and the interface (gas-liquid) shear stress, were considered. Cutting of mild steel at the different thickness levels was carried out using an argon/oxygen gas mixture at 5/1 pressure ratio and 138 kPa total pressure. The depth of resolidification around the keyhole resulted from the cutting process and was examined. Flow visualization and pressure distribution measurement were conducted by Horisawa(1999) to characterize the flow in the laser cut kerf for various conditions of the gas jets. It is shown that the flow separation becomes significant at higher feed pressure with smaller kerf width. This tendency is considered to be undesirable for the gas dynamic forces of the jet effective on the cutting front in the cutting processes, although the separation point moves downward to about 5 mm in depth with very high feed pressure higher than about 1 MPa. The results show that the shorter nozzleworkpiece separation and the wider kerf width are more favorable for gas flow in a cutting kerf. A model to arrive at optimal value of cutting velocity and assist gas pressure at given laser power and material properties has been given by Vladimir A. Karasev Et al(2005). The optimization has been carried out by

14

measuring laser beam energy expended on heating the bulk of the specimen to be cut. B Tirumala Rao and A.K.Nath(2002) have correlated the cut surface quality with the melt film thickness. They have estimated the optimum pressure required for melt ejection under laminar flow regime. The thickness of melt film inside the kerf is estimated using mass balance and the shear force acting on the cutting front assuming melt flow profile as linear. The dependence of melt film thickness on gas pressure, cutting velocity and work piece thickness has been estimated. In their work they have estimated the melt film thickness using the striation wavelength. Since striation wavelength is decided by surface tension effects of the melt, oxidation front propagation speed and laser beam diameter at the metal surface, besides the cut speed and gas flow velocity, the estimated melt film thickness from the striation wavelength is expected to be on the higher side. Operating at higher gas pressure results in minimum melt film thickness .When operated at higher cutting velocity thick melt film sustains as very coarse striations on the cut surface. This study has revealed that the impact of change in assist gas pressure on the melt film thickness is not as much as that of cutting speed. In order to achieve minimum melt film thickness on the cut surface laser cutting should be performed at optimum cutting velocity and maximum assist gas pressure such that laminar gas flow is sustained in the entire kerf depth. The mild steel plates cut with higher cutting speeds had coarse striations. For larger kerf width the operating range of gas pressure is more in thin sheets than in thick sheets. The effects of a gas jet in laser cutting are examined by K. Chen, Y.L. Yao, and V. Modi (2001). They have found that the material removal capability of the gas jet, in terms of shear stress and pressure gradient, is affected by the shock structure of the impinging jet interacting with the workpiece. When an oblique shock directly interacts with the normal shock, a large reduction in mass flow rate is observed. However, when an oblique shock merges upstream of its interaction with the normal shock, the reduction of the mass flow rate is small, preserving the removal

15

capability. Measurement of cut surface characteristics such as surface roughness, dross attachment, and recast layer thickness confirms their association with the shock structure and gas jet removal capability. It is observed that the increase in cutting speed results in liquid film and hence results in the increase in the interfacial velocity. The striation wavelength thus increases with the cuting speed. However the striation depth is reduced. Most of the predictions on the striation depth and frequency are based on the assumption that the properties of both the liquid and gas phase is practically unaltered. This is not true when the cutting speed and assist gas pressure changes.Same authors in their work on numerical model for the process of oxygen-assisted laser cutting of mild steel(2004) have solved Coupled oxygen concentration and energy balance equations by a control-volume- based computational scheme, while the velocity field is obtained by analytical boundary theory. The enthalpy method is adopted to trace the free boundary of the phase change and thus determine the molten-layer thickness. The steadystate simulation results include the temperature and oxygen concentration profiles at the cut front, the effects of impurity gas on the cutting speed, reaction energy, conduction loss and heat affected zone. The striation phenomenon is studied and the temperature fluctuation range versus cutting speed and oxygen pressure. Investigation the effects of laser beam spot size and the cutting speed on the pitch of the stiation and its frequency. They have stressed the need for a thermo fluid model. According to them the Gaussian distribution of the laser beam is responsible for the dependence of cut quality on the laser power and cutting speed for a given work thickness. The increase in cutting speed shifts the cutting towards the centre of the Gaussian beam and the kerf narrows. The oxidation energy suppliments the laser energy. But the energy supplimentation is cyclic in nature and leads to striations on the cut surface. The oxidation energy depends on the oxygen mass flux which itself depends on the flow rate and purity of oxygen. J powell et al (2000) have

16

investigated the conduction losses experienced during the laser cutting, Their work has shown that as the work thickness increases the conductive losses exceed the laser power in oxygen assisted laser cutting. The oxidation energy is more dominant as the material thickness is increased. This results in diminished efficiency. J Duan et al (1996) have developed a model for laser fusion cutting process to predict the effects of various process parameters on the cut kerf quality. Though the model is developed for inert gas cutting, it can be bextended for oxygen assisted cutting where the heat of oxidation adds up to the laser generated heat flux. They have found that the flow field distribution along the cutting front is indicative of the fact that the pressure gradient is passive from the entrance point of the key hole to the point of curved shock. But when thr flow passes the curved shock, the laminar flow gets transformrd into turbulent flow. As the turbulent flow consumes more energy the quality of the cut surface decreases. If the exit pressure at the bottom of the kerf is higher than the ambient pressure, the momentum thrust is reduced due to the decrease in the assist gas velocity. This may cause some of the molten slag to cling to the sides of the kerf to form thr dross at the exit point of the gas jet. Same authors have presented a model by cosidering the effect of multiple reflections of the laser beam on the cut surface quality in case of thick specimens. Time scale effects in laser material removal has been studied by Lawrence yao et al (2004). In their study on time scale effects material removal by a continous laser beam interaction between heat transfer, oxygen diffusion and gas dynamics is anlysed. They have found that the laser material interactions exhibit significant differences when the interaction time changes from continous to small time scale.The model proposed by them predicts the effects of hydrodynamic instability coupled with oxydation on the striation depth and frequency. They have also reported that the complete thermal interaction plasma formed during the laser material interaction and the melted material as still unclear. The experiments have shown that for a given laser power the cutting speed

17

determines the kerf width in oxygen assisted cutting. The side burning is more ponounced in the oxygen assisted cutting at slow scanning speeds. As the cutting speed is increased the cutting shifts towards higher power densities. A tightly focussed beam is essential to produce a better kerf quality. Chen Jimin Yang Jianhua et al (2007) have in their study on non vertical cutting with lasers in order to avoid interference during 3D laser cutting of thin metal a laser head could not be kept vertical to the surface of a work piece. In such situations, the cutting quality depends not only on “typical” cutting parameters but also on the slant angle of the laser head. In this paper, an experimental design is employed to reduce the number of tests and an artificial neural network (ANN) is set up to describe quantitatively the relationship between cutting quality and cutting parameters in the non-vertical laser cutting situation. A quality point system is used to evaluate the cutting result of the thin sheet quantitatively. Testing of this novel method shows that the calculated “quality point” using ANN is quite closely in accord with the actual cutting result. The ANN is very successful for optimizing parameters, predicting cutting results and deducing new cutting information. Yilbas B.S etal(2008 ) found that laser cutting of wedge surfaces cannot be avoided in sheet metal processing and the quality of the end product defines the applicability of the laser-cutting process in such situations. In their study, CO2 laser cutting of the wedge surfaces as well as normal surfaces is considered and the end product quality is assessed using the international standards for thermal cutting. The cut surfaces are examined by the optical microscopy and geometric features of the cut edges such as out of flatness and dross height are measured from the micrographs. A neural network is introduced to classify the striation patterns of the cut surfaces. It is found that the dross height and out of flatness are influenced significantly by the laser output power, particularly for wedgecutting situation. They have shown that the cut quality improves at certain value of the laser power intensity. The influence of the processing parameters

18

on the dynamic characteristic of supersonic impinging jet in laser cutting is studied numerically. The numerical modeling of a supersonic jet impinging on a plate with a hole is presented to analyze the gas jet–workpiece interaction. The model is able to make quantitative predictions of the effect of the standoff distance and exit Mach number on the mass flow rate and the axial thrust. The numerical results show that the suitable cutting range is slightly different for different exit Mach number, but the optimal cutting parameter for certain exit total pressure is nearly changeless. According to their study better cut quality and capacity can be obtained mainly by setting the suitable standoff distance for a certain nozzle pressure. Laser melting of thick mild steel sheets is considered and effect of cutting parameters on the percentage of kerf width variation is examined by B.S Yilbas(2008). The cutting parameters considered include laser output power, cutting speed, and oxygen assisting gas pressure. A factorial analysis has been carried out to identify the main effects and interactions of the parameters. It is found that laser output power and oxygen gas pressure have significant effect on the percentage of kerf width variation. Regular striation are often observed on the cut surfaces in laser cutting of metals. Although there have been various investigations on the understanding of the striation formation mechanism, the models so far are inadequate for the explanation of the observed phenomena, although it has been established that melt film thickness has a strong influence over the formation of striation. In the two-dimensional analytical model developed by Lee Mein Wee etal(2005) power absorption over the cut front, effect of oxidation and melt film thickness are predicted in the model. The surface profile and roughness of a machined workpiece are two of the most important product quality characteristics and in most cases a technical requirement for mechanical products. Achieving the desired surface quality is of great importance for the functional behavior of a part. The processdependent nature of the surface quality mechanism along with the numerous uncontrollable factors that influence pertinent phenomena, make it important to

19

find a straightforward solution and an absolutely accurate prediction model. Chen Lu(2008) has developed a model for the prediction of surface profile using RBF neural network and future trend are also introduced.

The theoretical models based on heat, mass and energy transfer involve parameters such as viscosity of molten metal, surface tension of the melt. These parameters are temperature dependent and the exact estimation of the temperature produced during the oxygen assisted laser cutting of mild steel itself is a huge task. Further the theoretical non linear optimization using equations with several variables whose constraints are not well defined is a major problem. To overcome these hurdles the present work aims to arrive at optimal cutting conditions through ANN and ANFIS model of the laser cutting process. The experimental knowledge is the driving factor. This approach is extremely useful in online optimization in laser machining applications.

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CHAPTER 3 LASERS

A Laser, in which Light Amplification carried out by Stimulated emission of radiation, is very much different from an ordinary light source in the sense that it produces highly directional, highly monochromatic and intense beam of light. They are mostly used as a powerful beam of electromagnetic wave rather a light beam. Development of high power compact laser such as CO2 laser has resulted in manifold increase in the industrial application of lasers.

3.1.1 Basic Principles In a system at thermal equilibrium the population at each energy level decreases with the increase in energy. If N1 and N2 are the number of atoms in two energy levels E1 and E2 respectively, then from Maxwell-Boltzmann statistics N1= e-E1/kt and N2= e-E2/kt

N2 N1

or N 2

( E2 E1 )

e

N1e

kT

E

kT

21

Thus the population in energy levels depends upon the energy difference

E

and the temperature T. At thermal equilibrium more atoms are in lower energy states than in higher energy states. Further at thermal equilibrium even at higher temperatures the number of atoms in higher energy states can not exceed that of lower energy states. In atoms the electrons in lower energy state upon receiving energy move to the higher state. Conversely the when an electron undergoes transition a from higher energy state to lower energy state it emits energy.

According to Albert Einstein the transition between two energy levels can take place in two different processes. • Spontaneous emission and • Stimulated emission in the presence of a stimulating photon. The emitted photon has the same frequency and polarization state as the stimulating photon. Spontaneous emission is the normal process which is frequently encountered. This requires just an atom rather than an atom plus a photon and consequently the transition probability is usually much higher. The opposite process, absorption, always requires a photon plus an atom and it is the reverse of stimulated emission. In a medium at thermal equilibrium and consisting of identical atoms, if there are N1 atoms in energy state E1 and N2 atoms in energy state E2 the following three processes can occur.

3.1.2 Stimulated absorption Atoms in the lower energy state E1 may absorb energy and jump to the excited state E2. The number of absorptive transition taking place is given by

Rabs

dN1 dt

22

But rate of decrease in population-(dN1/dt) of lower energy state E1 must be same as the rate of increase in population (dN2/dt) of state E2.

dN 2 dt

Rabs

B12 N1ρ( )

B12 is Einstein’s coefficient for induced absorption and ( ) is the energy density of incident radiation. Stimulated emission: Incident photon of appropriate energy may trigger the atoms in energy state E 2 to undergo a transition to E1. The rate of this type of emission known as stimulated emission is given by Rsti

B21 N 2 ρ( )

B21 is the Einstein’s coefficient for stimulated emission. 3.1.3 Spontaneous emission Excited atoms in energy state E2 may undergo a transition spontaneously to a lower energy state E1 either by radiative or non-radiative emission. During the transition the excess energy is given up in the form of photons. The rate of spontaneous transition is given by

Rsp

dN 2 dt

A21 N 2

A21 is the Einstein’s coefficient for spontaneous emission. Under thermal equilibrium B12 ( ) N1

A21 N 2

A21 ( )

N1

N2

B21 ( ) N 2

B12 B 21

B12

23

( E2 E1 )

As N1 N 2

e

N1

( )

A21

kT

h

e

N2

and

(E2

E1 ) h

kT

1 B12

h

e

B 21

kT

B12

To maintain thermal equilibrium, the system must release energy in the form of electromagnetic radiation. For consistency with Planck’s law B12=B21 and A21 B 12

The condition

8 h

3

3

c3

B12=B21 implies that the coefficients of absorption and

stimulated emission are numerically equal. The probability of downward and upward transitions is equal. However for lasing action there must be large stimulated emission. The ratio of spontaneous transitions to stimulated transitions is given by

R1

B21 ( ) N 2

1 A21 N 2

h

e

kT

For h kT 12, R1 6 X 10

1

6

This means that in the optical region the spontaneous emission dominate over the stimulated emission. The ratio of stimulated transition to absorptive transition is given by

24

R2

B21 ( ) N 2 , as B 21 B12 ( ) N1

At thermodynamical equilibrium N 2 N 1

B12 , R 2

N2

N1

1 , hence the absorptive transitions

dominate over the spontaneous transitions. If more atoms are in the excited state then N2>N1, the photons are more likely to cause large stimulated emission. Thus to achieve the lasing action the population in the excited state must be greater than the population in the lower energy state ie the population inversion must be achieved. The radiation density in the medium must be very high.

3.2Population inversion When a well collimated monochromatic beam falls on an absorbing E2

medium with a transition E2E1 & dI ( x) dx

I ( x)

or I

I oe

x

,

E1 h

is considered, then

is the absorption co-efficient.

If spontaneous emission and scattering losses are neglected, then dN dt

N1 ( ) B12

interval and As B12 But Hence

N 2 ( ) B21 , N

is the photon density per unit frequency

is the energy density. B21 ,-

dN dt ( N1

dN dt

( N1

N 2 ) ( ) B21

( )c , n-refractive index and c- velocity of light nh B h N 2 ) 21 , if c

will increase exponentially. I

is negative, then N2>N1 and beam intensity I o e kx . K=

- is the small signal gain. To get positive gain population inversion must be achieved. For this energy must be supplied from pumping process. Pumping results in non thermal equilibrium. When population inversion is achieved, a

25

few randomly emitted photons trigger stimulated emission of more photons which in turn results in a cascading effect to produce large number of photons.

3.3 Pumping In lasers two types of pumping are employed namely direct pumping and indirect pumping. In direct pumping the excitation flux is transferred directly to upper laser level from a suitable target state. The direct pumping has several disadvantages such as the lack of direct path between the lower level and the higher level and non availability of good source of pumping flux. Though there are three pumping schemes, two level pumping scheme does not produce laser radiation. Three and four level pumping schemes are widely used in gas lasers which are widely used in many industrial applications.

3.3.1Three level pumping scheme

Fig.3.1 Three level pumping scheme

Fig.3.1 shows a three level system. In a three level pumping scheme the lower level is either a ground state or whose separation from the ground state is rather small. Normally the population distribution in the three levels is as per Boltzmann law. When the atoms are exposed to intense pumping

26

radiation, the atoms are pumped to the higher level E3.Though some of the excited atoms undergo spontaneous transition to the. ground state, most of the atoms undergo non radiative transition to the metastable state E2.As no spontaneous transition takes place from E2 to E1 the atoms begin to accumulate at the metastable state and population at E2 exceeds that of E1.The situation is now ripe for the incident photon to cause stimulated emission. Pumping efficiency would be better if the level E2 is band instead of a discreet level. The terminal of the laser transition is the ground state. In order to inverse the population more than half of the ground state atoms must be lifted to higher level. A three level system thus requires a large pumping power to achieve population inversion. Three level system can only produce pulsed output.

3.3.2Four level pumping scheme

Fig 3.2 – Four level pumping scheme

Fig 3.2 shows the four level pumping scheme. The terminal laser level is well above the ground state. Also E2-E1>>kT. This ensures that at thermal equilibrium population in E2 is negligible. The pumping energy transfers the

27

atoms to the level E4 which has a short life time. The atoms then undergo spontaneous transition to metastable level E3. The laser transition now takes place from metastable state E3 to nearly vacant terminal laser level E2. Unlike a three level system the lower transition level in four level systems is not the ground state. As E2 is almost vacant, population inversion can be achieved by pumping a few atoms to the upper level. Hence less pumping energy is required for four level systems. As population inversion can be sustained, four level system can be operated in continuous mode. 3.4Laser pumping sources Laser pumping sources are the means by which energy is transferred into the laser gain medium to produce the required population inversion. These pumping sources generally consist of either electrons flowing within the medium or light being absorbed by the medium. 3.4.1Electron pumping Electron pumping is used primarily in gaseous or semiconductor gain media. In gases, many electrons are produced when a few initial electrons within the gain medium are accelerated by an electric field within the medium and these many electrons then collide with neutral atoms, exciting those atoms to higher-lying energy levels and even ionizing some of the atoms (removing an electron). The freed electrons are also accelerated, producing an avalanche of electrons and therefore an electrical current within the medium. The electrons lose their energy by transferring it to the atoms during the collision process. Some of the lasers operate on a pulsed basis, applying a large amount of current for a short period of time. Others operate on a continuous (cw) basis, using a much smaller but continuous current. In semiconductors, the electrons flow through the semiconducting material by applying a voltage across the pn junction with the positive voltage on the side

28

of the p-type material. This leads to recombination radiation when the electrons combine with the holes in the junction. The heat loading of the semiconductor limits the current. 3.4.2Optical pumping Optical pumping of lasers generally applies to the pumping of liquid (dye) lasers and to dielectric solid-state lasers and is provided by either flashlamps or other lasers.The most common types of flash lamps used for pumping lasers are narrow, cylindrical quartz tubes with metal electrodes mounted on the ends, filled with a gaseous species such as xenon that serves as the radiating material within the lamp. A voltage is applied across the electrodes of the flash lamp and current flows through the gas, populating excited levels of the atoms within the gas that radiate and produce intense light emission. The process is similar to that of electron excitation of lasers described above except that a population inversion is not produced and the radiating material of the lamp radiates via spontaneous emission, rather than by stimulated emission as in the case of a laser gain medium. The pumping wavelength of the flashlamp is determined by the gaseous medium inserted within the flashlamp tube. Xenon is the most common species because of both its radiating efficiency and its emission of a broad spectrum of wavelengths from which to choose in matching the lamp emission to the pumping absorption bands of the laser.

3.5Amplification and Gain When the population inversion is achieved, a photon of appropriate energy can stimulate the emission of a cascade of photons. This process is called amplification. The gain of the light amplification system which simply represents the amount of stimulated emission a photon can generate when it travels a certain distance in the active medium is given by

29

G

1 dI I dx

To achieve higher gain the light has to travel a very long distance. Keeping in mind the design and size of the laser, the light is made to travel long distance by bouncing it back and forth between a set of mirrors. The mirrors which act as optical resonators are essential to maintain a large radiation density. The mirrors used for bouncing may be either flat or curved.

3.6Threshold condition As the light bounces back and forth between the optical resonators it suffers both loss and amplification. The losses occur due to the scattering , diffraction and partial transmission at the output mirror. But for good lasing action the amplification must balance the losses in the active medium. The light traveling between two mirrors M1 and M2 of reflectivity r1 and r2 separated by a distance L. Intensity increase is given by I ( L)

I 0e(

)L

Amplification obtained between the mirrors is given by G

r1r2 e (

)2 L

In order to balance the losses in the medium by the gain, the amplification factor 1 1 ln 2 L r1r2

The amplification factor depends on the extent of pumping. The threshold condition for the laser to start oscillate is given by

30

th

1 1 ln 2 L r1r2

represents the distributed losses.

3.7 Growth of beam and saturation If significant gain is provided along the length of the gain medium, the spontaneous emission emitted in the elongated direction will grow at a rate dependent upon the amount of gain available as it moves through the length of the medium. The emission that starts at one end and transits to the other end will have grown by a factor of between 0.02 (2%) and 10 (1,000%) in a single pass, depending upon the type of laser. However, even the high factor-of-10 growth available in some lasers is not sufficient to produce a powerful unidirectional laser beam in one pass. Hence, mirrors are placed at both ends of the medium, forming a cavity to redirect the beam back and forth through the amplifier and thereby allow the beam to continue to grow until a point of beam saturation is achieved. At somewhere between 2 passes (dye lasers) and 500 passes (He-Ne lasers), the beam will have become so intense within the laser cavity that there won’t be sufficient atoms in the upper laser level within the gain medium to accommodate all of the impinging photons. This is the condition of beam saturation, and the intensity of the beam is known as the saturation intensity. In this saturation, the length L is the effective length of many passes through the amplifier so we will define that length as LT. At that point the beam will have grown by a factor of approximately When it reaches that intensity it will settle down to a stable value (as long as the pumping continues) in which the conversion from pump power to laser photons reaches an equilibrium balance. In the case of the He-Ne laser, that requires 500 passes through the amplifier. If more pump power is applied, above the value where the saturation intensity is reached, more laser power will be produced and will be available through the output mirror of the laser.

31

The condition for a population inversion and thus amplification within the amplifier was given by N2/N1 > 1. However, even though gain might exist within the amplifier, the laser still might not develop a beam if the gain is not sufficiently high to overcome losses within the laser cavity. The laser mirrors won’t have 100% reflectivity and there might be additional losses such as scattering and reflective losses at windows and other optical elements placed within the cavity.

3.8 Critical population inversion Minimum population inversion density required to give rise to lasing action and to sustain it is called critical population inversion. It is given by

N th

4

2

c L

0 2

From the above equation it follows that to achieve the critical population inversion the atoms must have a narrow line width. This sets the condition on the atoms which can be used. It is also clear that the laser condition becomes more difficult to satisfy as the laser frequency increases. Life

of the photons

must be as large as possible. 3.8.1 Laser rate equations of four level lasers In the ideal four level laser shown in the figure 3.3. E1 is the ground level, E4 is the pumping level, E3 and E2 are are upper and lower pumping levels. Atoms are pumped from the ground level to pumping level from where they make a non radiative transition to E3.Acumulation of atoms in E2 is not possible if E2 has a very small life time

2

in comparison with the life time

of E3.This ensures the population inversion between E3 and E2

3

32

Fig 3.3 Four level laser . If N2 and N3 are the atomic densities of the levels E2 and E3, R2 the rate at which the atoms are pumped to level E4 from where they make a non radiative transition to E3.Then R2 is essentially is the rate at which the atoms are arriving at E3.The rate R2 at which the atoms are pumped into level E2 is detrimental to population inversion. However in gas lasers this transition is unavoidable. The decrease in the number of atoms in E3 is partly due to the stimulated transition to E2 given by W32(N1-N2) that results in laser light., the spontaneous emission to the level E2 given by N2A32. The rate equation for N3 is

dN 3 dt

R2

W32 ( N 3 N 2 )

N 3 A32

The rate equation for N2 is dN 2 dt

R1

W32 ( N 3 N 2 )

N 3 A32 N 3 A21

33

In the steady state

R2

Therefore

R1

AsR 2

dN 2 dt

dN 3 dt

0 and

0

N 2 A21 R1 , R2

R2 A21

N 2 A21 and N 2

1

Also N 3

It follows that N 3

N2

N2

0 if A21

R2

W32

A32 or

A32 A21 A32

21

32

The above condition must be satisfied to achieve the population inversion and to realize the lasing action.. The threshold value is given by

1 Nt h

Since A32

N3

A21 ,1

N th

As N 3

N2

8

Rth

th

A32 A21

A32

1 and Rth

Nth 32

2 0

A32 A21

32 th

v

v2

N 2 is equal to N th under steady state condition , the total power per unit

volume required at threshold is Pth

N th 32

E 4 , hence the threshold power is

34

8 v0 2

Pth

th

v

v2

E4

Optimum output power The gain coefficient of a laser ( ) is proportional to ( N 3 0

1

w32 A32

where

0

R and R A32

R2 (1

N2 )

A32 ) is the gain coefficient in A21

absence of the feedback. The power emitted by the laser is given by Pc

N thVW32 hv

V is the volume of the lasing material. The amount of spontaneous light generated by the lasing material when it is at just threshold is given by Ps

8

N t hVA32 hv

3 0 3

h vV tc

sp

32

Ps is called critical fluorescence. It gives the total power generated within the cavity by the atoms due to stimulated emission. In practice only a fraction of the total power is coupled out of cavity as laser beam through the output mirror. If the transmission coefficient of the output mirror is increased the light output power increases but it also results in the cavity losses. For a given pumping rate there exists an optimum output coupling which results in maximum output power. Po

8

3 o 2

hvo vA 32

To

2 To

oL

i

sp

Po

I s ATo

2 oL 1 To i

1

35

where Is

8

3 o 2

hvo v

is called saturation intensity.

32 sp

For optimum power

Po To

0 , from this the optimum condition for mirror

transmission that yields maximum power output can be shown to be

To

opt

2

o L i

i

The power at optimum coupling is given by Po

opt

IsA 2

oL

i

2

thus the power levels are higher within the cavity than the outside.

3.9 Laser beam properties Laser beam properties such as direction and divergence of the beam, the beam profile, and the wavelength and frequency characteristics of the laser within the wavelength region of the laser gain bandwidth are determined largely by the laser mirrors. The factors determining those properties include mirror curvature, surface quality, and reflectivity, separation and location. The structure holding the mirrors must be a secure, vibration-free. The unique electromagnetic wave properties produced by the mirrors are referred to as modes.

36

3.10 Shape of gain medium The goal of constructing a laser is to capture most of the spontaneous emission that is normally emitted in all directions within the gain medium and redirect it into a single direction. This is done with the assistance of the gain or amplification that can be initiated within the medium. It is desirable to have the gain medium be of an elongated shape so that the gain, which is length dependent, will operate primarily in that one elongated direction. Hence, most laser gain media are long, narrow devices with mirrors located at the ends.

Fig 3.4 Basic components of a laser

The useful power from the laser is obtained by locating a partially transmitting “output” mirror at one end of the amplifier so that part of the beam “leaks out” of the mirror cavity as shown in figure 3.4. The initial gain in the amplifier must be greater than the loss of the transmitting mirror (plus other mirror and cavity losses) or the beam will not develop.

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3.11 Longitudinal cavity modes When the beam is developing within the mirror cavity, traveling back and forth, certain wavelengths within the gain bandwidth of the laser tend to be more enhanced than others. These are wavelengths (or frequencies) in which the light beam in the cavity forms a standing wave. Such an effect occurs when an exact number of half-wavelengths of the light fit within the separation distance between the mirrors. Typically there will be several hundred thousand wave peaks for each standing wave that occurs within the cavity. Hence, each standing wave must have a wavelength such that an integral number of oscillating waves fits in the space separating the mirrors. If more than one standing wave is present, each standing wave (longitudinal mode) will be separated in frequency from the next one by a fixed exact amount that depends upon the laser cavity length d. That frequency separation

v between

longitudinal modes can be obtained by dividing the speed of light c by twice the cavity length or

In Figure 3.5, several of these modes are shown occurring within the frequency bandwidth of a typical gas laser. Typically, the separation in frequency is of the order of 500 MHz Each discrete standing wave is referred to as a longitudinal mode associated with the laser cavity and by the cavity stability (free of vibrations).

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Fig 3.5 Longitudinal modes occurring within the gain bandwidth of a typical gas laser.

Fig 3.6 Two longitudinal modes occurring within a laser cavity

39

Figure 3.6 shows two such modes within a cavity. There will always be at least one longitudinal mode and there could be many more, depending on the frequency or wavelength bandwidth of the laser gain medium. If more than one longitudinal mode is being generated, they will be indistinguishable unless a spectrum analyzer is used to analyze the beam. They all travel in the same direction, and their color will be indistinguishable because their wavelengths (frequencies) are so similar, as indicated above. The frequency width of a single longitudinal mode can be very narrow, typically in the range of 106 to 108 Hz, determined by the mirror reflectivity (higherreflecting mirrors produce narrower

3.12 Transverse modes

Fig 3.7 Two transverse modes occurring simultaneously within a laser cavity. The on-axis mode is the TEM00 mode. The angled mode is actually rotationally symmetric and would produce a doughnut spot on the wall. The presence of more than one longitudinal mode involves many light beams traveling exactly the same path through the amplifier but differing in wavelength depending upon the total number of wave cycles that fit between the mirrors. Contrary to this, different transverse modes involve slightly different optical paths through the amplifier and thus have slightly different

40

directions when they emerge from the laser as shown in Figure3.7. Because of the different optical path lengths, they also have slightly different frequencies. Each of these stable modes evolves because the light traveling that particular pathway recurs exactly from one round trip of the beam to the next, therefore developing into a steady beam. The lowest-order transverse mode, known as the TEM00 mode, travels down the central axis of the laser gain medium. Higher-order modes have slightly diverging beams. The TEM11 mode, for example, if it were the only mode present, would appear as a doughnut-shaped beam when projected onto a screen. Complex patterns can be present if several transverse modes are operating. In most cases, closely located transverse modes differ in frequency by a smaller value than do adjacent longitudinal modes that follow the same path through the amplifier. The TEM00 mode has a beam-intensity profile in the direction transverse to the direction of propagation that is described by a Gaussian function as given by the following expression.

I

Ioe

2r 2 w2

where I0 is the intensity on the beam axis at any location, r is the perpendicular distance from the beam axis, and w is defined as the beam waist. The beam waist, varying along the axis of the laser, is defined such that the intensity has fallen to 1/e2 of the intensity on axis. It turns out that 86.5% of the energy is contained within the beam radius in which r = w. The TEM00 mode is often the desired mode because it propagates with the least beam divergence and can be focused to the tightest spot. It can generally be obtained by placing an adjustable aperture within the laser cavity and decreasing the aperture diameter until only the TEM00 mode remains.

41

Some of the properties of the lasers which make them versatile are monochromaticity, spatial and temporal coherence, directionality, focusability, brightness and intensity.

3.13 Collimation Lasers produce the most collimated light of any type of light source. Gaussian beams have a minimum beam waist w0 that occurs somewhere between the laser mirrors as shown in the fig 3.8. The beam then gradually expands from that location. If the laser mirrors have the same radius of curvature, the minimum beam waist occurs exactly halfway between the mirrors.

Fig 3.8 Collimation of Gaussian beam If the minimum beam waist is known, the beam waist w(z) at any distance z from where the minimum occurs can be determined from the following equation.

w( z )

wo 1

z wo 2

1 2 2

-----1

42

where

is the wavelength of the beam. The expanding beam has a curved

wavefront with a radius of curvature R(z) given by

R( z )

The beam angular spread

z1

wo 2 z

2

-------2

in radians at distances well beyond the laser

mirrors can be expressed as 2 ------3 wo

it can be seen that a larger w0 and a shorter wavelength

gives a smaller

angular beam divergence.

Producing the most collimated light or the least divergent light, is determined by the cavity mirror properties including the radii of curvature of the mirrors and the separation between mirrors as indicated in equations 1,2 and 3. For the smallest beam divergence, w0 must be large, as you can see from equation 3. Also, the rays of the laser beam are the most parallel when the beam is at the location of the minimum beam waist w0. This parallelism of the beam can be realized by using a semi-confocal resonator arrangement for the laser cavity with the flat mirror as the output mirror.

3.14 Monochromaticity Monochromaticity refers to how pure in frequency or wavelength the laser beam is. If the laser is operating in a single longitudinal mode, as most solid-state and semiconductor lasers do, the actual laser linewidth can be significantly narrower, the width of a single longitudinal mode beam. For most

43

applications requiring a single narrow wavelength, most lasers would normally provide a sufficiently narrow frequency output bandwidth, of the order of 109– 1011 Hz. This would represent a bandwidth that is less than 0.1% of the frequency or wavelength of the beam itself. Linewidths of the order of 1 MHz (106 Hz) or less can be obtained by operating with a single longitudinal and single transverse mode (TEM00). The narrowing is enhanced by choosing highly reflecting mirrors, constructing a very stable mirror cavity in conjunction with the amplifier by eliminating vibrations of the mirrors and other cavity elements, and providing temperature stability.

3.15 Coherence Coherence refers to the how much in phase various portions of a single laser beam are. The closeness in phase of various portions of the laser frequency bandwidth is referred to as temporal or longitudinal coherence. The closeness in phase of different spatial portions of the beam after the beam has propagated a certain distance is referred to as spatial or transverse coherence. This phased relationship determines how readily the various portions of the beam can interfere with each other, after the beam has propagated a specific distance, to produce such effects as diffraction of light and related applications such as holography. The coherence length is used to describe the beam propagation distance over which the beams stay in phase. For longitudinal or temporal coherence, the coherence length the wavelength

C

is related to

and the total frequency bandwidth of the laser vL by

is the actual bandwidth of the laser beam given in wavelength units.

44

For transverse or spatial coherence, the transverse coherence length

t

is related

to the laser wavelength , the laser source diameter at its origin s, and the distance r the beam has propagated from its origin, by the following relationship.

3.16 Intensity Intensity or irradiance is the power of the laser beam divided by the cross-sectional area of the beam. It is thus typically given in watts per square meter (W/m2). It is a measure of the amount of energy that can be applied to a specific region within a given amount of time. It is one of the two most important parameters in using the laser for materials processing applications such as welding, cutting, heat treating, ablating, and drilling, or for laser surgery. The other important parameter is the laser wavelength, since the amount of absorption of all materials is dependent upon the wavelength of the light. For the application where the deep penetration of laser is needed, a laser wavelength in which the material has a relatively low absorption would be selected. Other applications might require a shallow penetration in order to control the quality of the edge to be left after the process is completed, such as drilling very small holes. Thus, a wavelength region of high absorption would be chosen for the laser. A general rule is that absorption is very high for most materials at ultraviolet wavelengths and decreasing at longer wavelengths.

3.17 Radiance Radiance is a parameter that includes the beam intensity and takes into account the beam divergence angle. Laser beam divergence is usually given in milliradians (mr) because of the very low divergence of most lasers.

45

Radiance becomes useful when a beam must be propagated over a reasonable distance before it is used or where the divergence can affect the focusing ability of the beam. Since most materials applications do not involve the tightest focusing possible for a given beam, intensity is usually the more important parameter.

3.18 Focusability Many applications of lasers involve their ability to be focused to a very small spot size. Perhaps one of the most demanding applications is in focusing the small diode laser in a compact disk player. To store as much information as possible on each disk, that information must be included in the smallest grooves possible on the disk. The width of the grooves is determined by the ability of a laser beam to access a single groove without overlapping into adjacent grooves. Hence, the diameter of the spot size to which the laser beam can be focused becomes a very important parameter. The smallest diameter that can be obtained with a focused laser, assuming that a single TEM00 mode can be obtained from the laser, is approximately the dimension of the wavelength of the laser and is given by the following expression.

d min

4 f'

in which the f’ is the focal length of the lens used for the focusing divided by the useful diameter of the lens. If the laser beam is less than the actual lens diameter, the beam diameter is used instead of the lens diameter in determining the f’. The effective f’focusing lens (ratio of focal length to laser beam diameter intercepted by the lens) must be of the order of unity to obtain such a small focus. Most lasers, however, can be focused relatively easily to spot diameters

46

of the order of 0.1–0.2 mm. Extra care must be taken in terms of beam quality and lens focal length to obtain smaller spot diameters.

3.19 Laser types Based on the lasing medium, lasers are classified into solid, liquid and gas lasers. They may be of either continuous wave mode (CW) or pulsed wave mode. In continuous mode laser output is uninterrupted whereas in pulsed wave mode it is interrupted. Solid state laser such as Nd – YAG laser is used mainly for etching and machining works. CO2 laser which is a molecular gas laser is extensively used in sheet metal work. The more recently developed chemical oxygen iodine laser provides a very high power laser beam and Femto second lasers which provides a very short duration laser pulse are also used extensively in sheet metal work.

3.19.1 The Carbon Dioxide Laser

Carbon dioxide laser is a molecular laser which produces an output of frequency 10.6 m in the far infrared region. The lasing medium CO2 laser is a mixture of CO2, helium and nitrogen. Excitation of CO2 is achieved by increasing the vibrational energy of the molecule and pumping is done using AC or DC electrical discharge. Carbon dioxide lasers offer the advantage of continuous power capability, high efficiency and ease of construction. The axial flow CO2 lasers are widely used for metal cutting applications.

47

Fig 3.9 Modes of vibration of CO2 molecule In CO2 molecule, the individual atoms are bound by a force which acts much like that of central force. Molecules vibrate as they lack fixed orientations within the molecule. They are able to rotate and spin because they are in a gaseous state. Modes of vibration of CO2 molecule is shown in fig 3.9. These states, as in electronic states, are quantized. The CO2 molecule is a triatomic molecule consisting of 2 oxygen atoms covalently bonded to a central carbon atom. The molecule has 2 stretching vibrational modes, labeled v1 and v3 and a bending mode, v2l. , l is the quantum number for vibrational angular momentum. v1 is the number associated with the symmetrical stretching of the molecule. v2l is that associated with bending and v3 with asymmetrical stretching. The energy level diagram of CO2 laser is shown in figure 3.10.

48

Transitions between vibrational energy levels results in photon emission in the infrared, while transitions between rotational states emit photons in the microwave region. Necessary mechanisms for operation of the CO2 laser are: 1. Excitation of N2 vibration by electron impact 2. Transfer of vibrational energy from N2 to the nearly resonant v3 mode of CO2 3. Laser transition from v3 to v1 mode. 4. Sharing of population between v1and 2v2 modes and relaxation within the v2lmanifold 5. The vibrational energy in the v2l manifold converted into translational energy by collisions with He. For CO2, even values of v2l and v3 along with l = 0, are symmetric. If v3 is odd, v2l is even, and l = 0, the vibrations are asymmetric. So, the state 001 (written as v1, v2, v3) is composed of only odd-spin particles. 100 and 020 states, however, have only even-spin members. Members of the same vibrational state are distributed amongst available energy levels as determined by Boltzmann statistics and are in thermal equilibrium. A potential difference of 18,000 V across the plasma causes electron collisions with N2 which excites these molecules to their lowest state. The energy of this state is very close to the 001 and 002 levels in CO2 (the n=1 state of N2 excites the 001state of CO2 and the n=2 state of CO2 excites the 002 state of CO2).

In

turn, this energy is transferred to the CO2 molecules and results in populating their upper levels. This occurs because the restoring force constant of N2 is almost identical as that of the CO2 molecule. The helium's presence in the gas is largely to maintain the plasma discharge, but also helps to depopulate the lower energy levels.

49

Fig 3.10 CO2 laser energy level diagram

001 to 100 and 001 to 020 are the most important energy level transitions allowing emissions of 10.6 m and 9.4 m respectively.

The 100 and 020

vibrational levels depopulate quickly. As was earlier stated, Helium is not only present to maintain the plasma, but also as a depopulation mechanism. When in one of these lower energy levels, a collision between CO2 and He atoms results in a transfer of the energy to the He atom. The infrared transitions are relatively slower than this depopulation, thus a population inversion is the result. Each of the vibrational modes of CO2 has an associated characteristic frequency of vibration (w) along with (as can be seen above) allowed energy levels. These vibrational energy levels can be approximated by the quantum mechanical simple harmonic oscillator hb= h/(2pi):

50

Ev

hbWo (v 1 / 2)

Where Wo

k u

1

2

is the classical vibration frequency

Though the modes are slightly inharmonic, the vibrational energy of CO2 can be closely approximated by E(n1, n2, n3) = hcw1 (n1+1/2) + hcw2 (n2+1/2) + hcw3 (n3+1/2) In considering the rotational energy of CO2, by visualizing the molecule to be rotating about it's center of mass (the carbon atom) as show in figure 3.11.

Fig 3.11 Rotational mode of CO2 molecule The rotational energy spectrum of CO2 has the same character as that for diatomic molecules and the rotational energy levels are thus approximated by: EJ = hcBeJ(J+1), J = 0, 1, 2, 3, ...

E

J ( J 1) 2 , 2I

J

0,1,2,3 - - - - - - - - - -

where the rotational constant Be for the CO2 molecule is Be = 0.39 cm-1. The difference between energy levels is E

J 2 I

The CO2 molecule may be in a particular vibrational mode, but also in a rotational state. The photon resulting from a transition has angular momentum

51

which must be conserved. Because of this, molecules in a particular vibrational and rotational state may undergo a transition between states only if the change in J is +1 or -1. If the transition results in J changing by +1, the transitions are said to belong to the 'P' branch. If J changes by -1, the transitions belong to the 'R' branch. This results in a branched structure in the emission spectrum.If the energy difference between the lowest (001) state and the lowest (100) state is E0, then the energy of the transition between the J=7 (001) state and the J=6 (100) state will be of energy:

E

Eo

( J ) 2 I

E

Eo

(7 ) 2 I

Likewise, transitions where the change in J is +1, the energy released will be

E

Eo

( J ) 2 I

Fig 3.12 Transition between Vibrational rotational states

From this, there exists the possibility of transitions with energy just greater than Eo and with energy just less than Eo. Thus, there is the branching in the emission spectrum. R(6) refers to the transition from the J=7 rotational state of the (001) vibrational mode to the J=6 state of the (100) mode. The laser emission demonstrates two distinct bands, each having P and R branches.

52

These correspond to the series of transitions from the (001) to the (100) and from the (001) to the (020) vibrational modes. Because the difference between adjacent rotational levels increases as J increases, the separation between emission lines in the P and R branch will increase as well. The branches will be centered around frequency v0 (corresponding to E0).

3.19.2 Axial flow CO2 laser It is one of the most widely used laser for metal cutting work

Fig 3.13 Axial flow CO2 laser

The axial flow CO2 laser system is shown in fig 3.13.Three gases CO2, N2 and He are mixed and fed into one end of a discharge tube at a pressure of a few bars. The gas flows down the end of the tube in about one second and is pumped out the far end with a mechanical fore pump. An electrical discharge is maintained between the metallic end flanges of the tube. The ballast resistance is required because of the negative dynamic resistance of the discharge. A fully reflecting mirror on the left and a partially transmitting mirror on the right helps

53

to fold the beam for amplification. Output couplers, also known as front mirrors, are designed to reflect a portion of the beam back into the laser resonator for continuous amplification while transmitting a portion of the beam to the outside for use. Therefore, the substrate material must be transmissive at the required wavelength of 10.6µm. Germanium and Gallium Arsenide substrates are commonly used for low to medium powered systems. The more expensive Zinc Selenide material is required for higher powered lasers because of its lower absorption at 10.6 microns. Rear mirrors are designed to reflect all or nearly all of a laser beam back through the laser gas mixture for amplification. The inside surface is given a highly reflective (99-100%) coating. In the 100% reflective case, inexpensive silicon can be used as the substrate material and the outside surface does not need polishing or coating. Some rear mirrors, however, are designed to transmit a small (0.5 - 1.0%) percentage of the beam to a power detector for real-time beam monitoring. These mirrors must have a transmissive substrate (Ge is the most common) and the outside surface usually has an antireflective coating. The laser radiates in the far infrared at 10.6 microns. CO2 lasers in the cw mode powers can reach as high as 15 kW.

3.20 Attributes of Laser Energy Fields i)Time attribute Acceleration, velocity and feed rate which are a part of time attribute of laser energy field can greatly affect the interaction between energy and material. Heat Affected Zone for pulsed laser is normally smaller than that of CW laser processing. For femtosecond (10-15 second) scale laser processing, the heat affected zone becomes almost zero at different wavelengths for the same material. These were attributed to the ultra-short laser pulse duration. When laser beam acts on the material, laser energy is first absorbed by

54

electrons. The absorbed energy then flows through the free electron subsystem, then transferred to the lattice. In this way laser energy is transferred to the ambient target material. For femtosecond pulse, laser pulse duration is shorter than the electron cooling time. Electrons are heated instantaneously and electrons in negligible time transfer their energy to lattice ions. When this energy intensity is high enough, which is often true for ultra-fast pulsed lasers, those ions get energy high enough to break the bonding of lattice structure, they break off instantly without having time to transfer their energy to their neighboring lattice ions, thus direct solid-vapor transition occurs. Heat conduction into the target can be neglected, heat affected zone is greatly reduced. The temporal distributions of energy fields can be continuous or discrete, both have their advantages and disadvantages relative to their applications. Both CW lasers and pulsed laser are used widely in laser machining. ii) Spatial Attribute Spatial distribution of energy field is directly related to energy acting area, acting location and relative position between tool/energy sources and parts. Due to high coherence the laser beam can be transmitted over long distances with very small divergence. Highly focused beams can act locally with high intensity, which are used for precision material removal, defocused laser beams are used for surface treating, laser forming, etc. In laser machining the relative position between laser source and parts is important. Optical fibers are used to carry high power laser beams in beam delivery system. They provide a flexible medium for laser energy transmission.

55

iii) Frequency Attributes The frequency refers to the characteristic frequency of energy field. The characteristic frequency of energy field is important because materials may respond very differently to energy fields at different frequencies. Visible laser beams are easily absorbed. The IR lasers require a higher energy density as they are not easily absorbed.

For this reason lasers at different wavelengths are

used to process different materials. iv) Amplitude Attributes Frequency and amplitude are closely related. Amplitude or magnitude is a direct measure of energy field intensity. Optical filters, polarizers, attenuators, beam expanding and focusing systems are used to modulate laser intensity and spatial distribution. With their help, one can match the laser power output to very different applications in the same time without disturbing the laser source. A certain value exists for the interaction between material and energy field, below this value, the interaction may be simple and linear, beyond this value, it becomes nonlinear. This indicates that some kind of transition mechanism between frequency and amplitude exists, or we can say, the effect of amplitude adjustment is somewhat equivalent to the effect of frequency adjustment. when pulse lasting time is very small, peak energy intensity goes up far beyond the safety operation range of normal optical amplification systems. The energy of ultrafast lasers are highly concentrated in time domain, but their frequency distributions are much broader than normal laser systems. The different frequencies travel at different speeds through an optical medium. In free space, different components in a broad band laser pulse travel at nearly same speed, time concentration of the beam is kept. When normal optical components are in the optical path, long wavelength light components travel through the medium faster than short wavelength components, thus pulse lasting time is stretched, energy intensity is lowered.

56

Special optical devices like chirped mirrors or special prism pairs are used to compensate the pulse spreading, they allow short light components pass through faster than long wavelength components, so they compress pulse lasting time.

3.21CO2 laser as a cutting tool CO2 Lasers provide thermal energy in the form IR radiation to the work, they remove material by melting to be blown away by an assist gas jet. Some times the process involves direct vaporization or ablation, unlike traditional machining which depends on mechanical stresses produced by tools to break the bonds of materials. It offers the advantage of being localized, noncontact machining and is almost reacting-force free. It is to a great extent dependent upon the optical properties of material than its mechanical properties. The forces in laser machining are of micro scale. The photon pressure on target material is negligible for bulk material. Laser as a cutting tool remove material in very small amount. Hence, the kerf in laser cutting is usually very narrow and results in very little material loss. Laser cutting however results in small removal rate compared with traditional machining. Lasers can cut the work pieces along lines or curves. Thin work pieces may be difficult to cut by other means, while laser cutting is suitable because of its non contact nature. The efficiency of laser cutting system apart from depending on the laser used and the power delivered by the laser depends on the optical and chemical properties of the material being processed. Most of the materials absorb laser radiations in the lower wavelengths readily than the infrared radiations. Lasers have been used to cut a wide range of materials. CO2 laser and Nd:YAG laser are the most popular laser in cutting. Laser cutting can be of two types. First is the direct Evaporative Laser Cutting, in which laser provides the latent heat until the material reaches vaporization point and ablate in vapor state. This process is used in cutting organic materials such as paper, cloth or

57

polymers. As the materials have poor thermal conductivity, a non reactive gas jet maybe used to reduce charring. The second is fusion Cutting. Laser energy melts the target material and the gas jet blows the molten material away. In this way the requirement on laser energy is lower compared with vaporization cutting. The gas jet can be reactive or non reactive. If the gas jet is reactive, the laser heat the material, laser heating combined with exothermic chemical reaction with the assisting gas provides the necessary melting of the target material, this is called Reactive Laser Cutting. This helps to further reduce the necessary laser energy. The cutting process depends on the absoptivity of materials which itself is strongly dependent on laser wavelengths. In general, metal cutting requires higher average laser power than laser cutting of nonmetals due to their higher reflectivity and thermal conductivity, and oxygen assisted cutting is more commonly used. Plastics, polymers, rubber, papers, wood, composites, stones and crystals have been successfully machined by lasers at infrared wavelengths. Some materials such as composites and gemstones can be readily cut with high quality using lasers, while they may be difficult to handle using other techniques. UV lasers can even cut polymers with negligible heat affected zone, because the phone energy is comparable to the bonding energy of the material, photochemical instead of photo thermal ablation dominates. In general, non-metallic cutting requires less energy than metal cutting, direct vaporization cutting is used. Metals are highly reflective of infrared energy. Hence the initial absorptivity can be as low as 0.5% to 10%. But the focused laser beam quickly melts the metal surface and the molten metal can have an absorption of laser energy as high as 80%. Fusion cutting assisted with gas jet is used. Non-metallic materials are good absorbers of infrared energy. They also have lower thermal conductivity and relatively low boiling temperatures. Thus the laser energy can almost totally transmitted into the material at the spot and instantly vaporize the target material. Vaporization cutting is commonly used, non reactive gas jet is used to prevent charring. In

58

laser cutting negligible thermal and mechanical distortions are produced. Scanning speed of laser source, laser power, focal position, gas jet alignment and gas composition, and laser material interaction influence the cutting quality. Another factor that affects the surface quality is the surface tension of the molten metal. The chemical action at the surface will change the surface tension and the geometry of the resolidified melt puddle. If too large a power is used it is found that the molten slag gets deposited on the kerf edge and this results in a poor surface and a wider cut. In oxygen assisted laser cutting , the process can be speeded up as the energy released during the reaction of the oxygen with the iron. But the same can result in a poor surface as the repeated oxidation cycles tend to produce deeper striations on the kerf. For cutting process different lasers can produce different effects. The CO2 laser is not suitable for cutting copper and aluminum. However they are effective in processing steel, ceramics and plastics. Nd-YAG lasers are useful for processing all types of materials. But their speed of processing is very small compared to CO2 lasers. Femtosecond lasers can process a material with least heat affected zone. The usefulness of a laser for cutting is dependent on both the material being cut and the frequency at which the laser delivers the power.

3.22 Laser material interaction

When laser beam strikes on the target material, a part of the energy is reflected and a part of it is absorbed. The absorbed energy heats up the target materials. When laser beam acts on the material, laser energy is first absorbed by free electrons. The absorbed energy then propagates through the electron subsystem and then transferred to the lattice. In this way laser energy is transferred to the target material. The laser cutting front is shown in the figure 3.14.

59

Fig 3.14 Laser cutting front The Depth of Heat Penetration, which is the distance that heat can be transferred to during the laser pulse is given by D

4 t

where D is the depth of heat penetration,

is the diffusivity of materials, t is

the pulse duration. Thermal conductivity of the material being processed, density, heat capacity and thermal diffusivity are the important parameters that influence heat conduction into the target material. Since laser machining involves strong phase changes, melting temperature, vaporization temperature, latent heat of melting and vaporization latent heat must be taken into account.

60

When a laser beam is incident in a material, surface absorption of laser energy is given by A=1-R-T where A is the surface absorptivity, R is reflection, T is transmission. For opaque material, T=0, then A=1-R. This relation should be revised for 2D or 3D conditions when the direction of incident beam relative to the target material is important. Then all energy balances should be considered in the normal direction of the surface. Under the action of laser irradiation, the surface quickly rises up to melting temperature. This melting range expands through heat conduction. If the laser intensity is too high, surface starts vaporizing before a significant melting depth of molten material is formed. In laser machining, however, vaporization is preferred. When higher intensity laser beam is focused on the surface of target material, the surface quickly reaches vaporization temperature, both melting and vaporization exist, materials are removed through direct vaporization ablation or through hydrodynamic ablation. Hydrodynamic ablation here means that material is removed as liquids. For laser pulse durations longer than microseconds, hydrodynamic ablation is the dominate component over pure vaporization ablation. When vaporization occurs, it generates a pressure than acted on the molten material, this pressure is called recoil pressure. Also there exists strong temperature gradients in the molten material, usually the center is hotter than the outer because of the profile of the laser intensity. The recoil pressure and temperature gradient drives the molten material out as liquids. The combination of vaporization and hydrodynamic ablation cut target material in depth. When the laser intensity is even higher, the vaporized material will be ionized and plasma is formed. This makes the situation much more complex. If the plasma density is low, laser energy can still be transmitted to the material without obvious absorption, if the plasma density is high enough, absorption by

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the plasma should be considered. This absorption is related to the electron density in the plasma. When the plasma density reaches certain value, high absorption and reflection happens. Then laser energy cannot be effectively transmitted to the target material, thus laser and material interaction is decoupled. When the plasma expands, dissipates and rarefies, then laser energy can reach the surface again, another cycle of plasma generation, decoupling and dissipation happens. In this process, shock wave is generated. So the highest laser intensity is optimum for laser machining. Transmission of Laser energy in the target material is governed by the Lambert law: I ( z)

I oe(

az )

Where z is the distance from the surface, a is the absorption coefficient, I0 is the intensity at surface. The laser energy transmitted to the material at depth z: I ( x, y , z , t )

AI o (t )e(

az )

SP( xy )

Where A is the coefficient considering surface reflection and plasma absorption, I0(t) is the temporal distribution of laser intensity, a is the absorption coefficient, SP is the spatial distribution of the laser intensity. For a Gaussian beam with beam radius r and for a material with A=1-R, R is the reflectivity, we have:

I ( x, y , z , t )

(1 R) I o (t )e(

az )

e(

x2

y2 r2

z2

)

Analytical solutions can be got only for simple conditions. The laser machining process involves many physical processes, such as melting, vaporization, radiation and convection heat transfer. In order to get analytical solutions, usually only pure conduction is considered. The basic 3D partial differential equation for heat conduction in a stationary medium is:

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2

2

T x2

T is temperature, t is time,

2

T y2

T z2

1 T t

is the thermal diffusivity. In above equation, we

assumed no heat generation and constant thermal properties, phase changes, convection and radiation are neglected. Heat conduction of a moving heat source is of interest because in laser cutting laser beam is in relative movement to the part. In case of constant scanning velocity, the erosion front and the resulting temperature distribution is constant relative to the coordinate system fixed at the laser beam. A steady temperature field is assumed. T t

T x x t

v

T x

Where v is the scanning velocity. From this relation, we can rewrite the 3D heat conduction as following: 2

2

T x2

2

T y2

T z2

v T x

Assuming a temperature distribution in the following form, the 3D equation can be simplified. vx

T

To e

Where T0 is the initial temperature and

2

( x, y, z )

is an unknown function of x, y and z.

Take it into the governing equation we have: 2

2

2

x2

y2

z2

v 2

2

0

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For constant scanning velocity, the solution of the above equation is only spatial dependent. For 2D heat conduction problems, assuming that the heat T z

flows only in the x and y-directions, so that

2

2

x2

y2

0 , the equation becomes

2

v 2

In cylindrical coordinates, the governing equation becomes: d2 dr 2

1d r r

v 2

2

0

Similarly, the boundary conditions are dT dx

0 for x

dT dy

0 for y

If a circle of area with radius r=(x2+y2)1/2 is considered, the heat source takes the form: for

, 2 rk

dT dr

q' for r

0 here q' is a linear heat source.

The cylindrical governing equation with the above boundary equation is known as the modified Bessel function of the second kind and zero order, we denote it as K0(vr/2 ). Then we can express the solution as following:

T T0

q' e 2 rk

vx 2

ko

vr 2

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A more rigorous analysis gives(George Chryssolouris,1991) the following expression for the cutting depth in terms of surface temperature Ts, cutting speed v and spot diameter d

s

2aP vd (c p (Ts To ) L)

The cutting depth is thus proportional to p/vd, which is the energy per unit area of the workpiece.

3.23 The role of assist gas in laser cutting The assist gas plays very important role in many laser cutting process. In laser cutting, it provides the necessary mechanical force to eject the melt from the cut zone and cools the cut zone by forced convection. Inefficient removal of the molten layer can lead to deterioration in cut surface. When a reactive gas such as oxygen is used, it also delivers additional exothermic energy through chemical reaction between the assist gas and the molten material. This chemical reaction produces additional energy that enhances the cutting process. This extra energy can be beneficial to cut thick sections of materials. The use of oxygen though is useful to enhance the speed of cutting results in a poor surface. Oxygen assisted cutting is characterized by the striations on the kerf. The surface quality can be improved by using the inert gas such as argon . Use of nitrogen also results in a better surface. Apart from the type of gas that is used the pressure at which it is blown in the narrow kerf also plays an important role in laser cutting.

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CHAPTER 4 Artificial Neural Network and ANFIS

4.1Artificial neural network Artificial neural network is a massively parallel distributed processor which can store the experimental knowledge. They offer sophisticated modeling techniques capable of modeling extremely complex functions and are highly effective for non linear optimization problems. The main blocks of ANN are network architecture which refers to arrangement of neurons into layers and pattern of interconnection between the layers, setting of weights and activation function.

Fig 4.1 Single neuron with bias

In its simplest form, a neural network would consist of a single input to a single neuron, which outputs a single output. A more useful form of this single neuron network is diagrammed in Figure 4.1. In this case, a single neuron with several inputs generates a single output. The figure illustrates the network consisting of a set of inputs p, weighted connections W, a single neuron with transfer

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function f, and output O. It also adds the concept of an applied bias B. Where B is used to offset or bias the output of the neuron. B is also sometimes viewed as another weighted connection to a fixed input value of 1. A mathematical description of the single-neuron network is: O = f(Wp + B) The transfer function of the neuron is determined when the network is initially designed and created. Determination of the transfer function is one of the basic issues the network designer has to address.

Fig 4.2 Neural network interconnections

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Fig 4.3Transfer functions

The neurons are connected to one another by a communication link, which are associated with weights. Fig 4.2 shows the ANN interconnections. It receives a number of inputs. Each input comes via a connection that has a weight. Each neuron also has a single threshold value. The weighted sum of the inputs is formed, and the threshold subtracted, to compose the activation of the neuron. The activation signal is passed through a transfer function to produce the output of the neuron. Fig 4.3 shows the transfer functions. A simple network has a feed forward structure, the signals flow from inputs, forwards through any hidden units, eventually reaching the output units. Such a structure has stable behavior. A typical feedforward network has neurons arranged in a distinct layered topology. The input layer is not really neural at all, these units simply serve to introduce the values of the input variables. The hidden and output layer neurons are each connected to all of the units in the preceding layer. Again, it is possible to define networks that are partially-connected to only some units in the preceding layer; however, for most applications fully-connected networks are better. The process of modifying the values of the weights to realize the desired output is termed as training the network. In supervised training, which used in this thesis, the network is provided with a series of inputs and the output is compared with the expected response. The weights are adjusted according to the learning algorithm until the expected output is realized. Multilayer Perceptrons are the most popular network architecture. Each unit performs a

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biased weighted sum of their inputs and pass this activation level through a transfer function to produce their output, and the units are arranged in a layered feedforward topology. The network thus has a simple interpretation as a form of input-output model, with the weights and thresholds the free parameters of the model. Such networks can model functions of almost arbitrary complexity, with the number of layers, and the number of units in each layer, determining the function complexity. Important issues in Multilayer Perceptrons design include specification of the number of hidden layers and the number of units in these layers. The experimental knowledge is the basis of the modeling process. Hence the cutting process is done several times for collecting sufficient data. This process involves i. Identification of the key process variables ii. Collection of experimental data iii. Neural network development and iv. Model validation.

4.2 Training algorithm The flow chart of fig.4.4 shows the training procedure. Once the number of layers, and number of units in each layer, has been selected, the network's weights and thresholds must be set so as to minimize the prediction error made by the network. This is the role of the training algorithms. The experimental values are used to automatically adjust the weights and thresholds in order to minimize this error. This process is equivalent to fitting the model represented by the network to the training data available.

The error of a

particular configuration of the network can be determined by running all the training cases through the network, comparing the actual output generated with the desired or target outputs. The differences are combined together by an error

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Fig. 4.4 Flow chart for training algorithm

function to give the network error. The algorithm progresses iteratively, through a number of epochs. On each epoch, the training cases are each submitted in turn to the network, and target and actual outputs compared and the error calculated. This error is used to adjust the weights, and then the process repeats. The initial network configuration is random, and training stops when a given number of epochs elapses, or when the error reaches an acceptable level, or when the error stops improving.

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The weights between layers are initialized randomly. Input pattern is presented to the input layer. For the jth node in the hidden layer Input tj =

W ji oi , where Wji is the weight connecting node i with node j

oi is the output of ith input node Output of the jth node in the hidden layer is o j = f inputj

where f is a sigmoid transfer function given by f x

1 1 exp

x

and x = input j Error for each output node is computed using the equation k

dk ok f ' inputtk

The error for each node in the hidden layer is computed using the equation j

f ' inputt k

kw kj

Weights between the layers updated by the rule wkj t 1

wkj t

kok

where wkj(t+1) and wkj(t) are the weights connecting nodes k and j at iterations (t+1) and t respectively.

is the learning rate.

The back propagation algorithm is a gradient descent optimization procedure, which minimizes the overall error E for all the input patterns P. E=

1 2P

dk ok p

2

k

Where dk is the expected output. Though many algorithms are available for training the ANN, modern secondorder algorithms such as conjugate gradient descent, quasi Newton and Levenberg-Marquardt are substantially faster. In back propagation, the gradient vector of the error surface is calculated. This vector points along the line of steepest descent from the current point, so we know that if we move along it a

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short distance, we will decrease the error. A sequence of such moves will eventually find a minimum. Gradient descent algorithm performs gradient descent on the error surface. It calculates the direction of steepest descent on the surface, and jumps down the surface a distance proportional to the learning rate and the slope, picking up momentum as it maintains a consistent direction. Actually, the descent is calculated independently on the error surface for each training case, and in random order, but this is actually a good approximation to descent on the composite error surface. Other MLP training algorithms work differently, but all use a strategy designed to travel towards a minimum as quickly as possible. More

sophisticated

techniques

for

non-linear

function

optimization have been in use for some time. These methods include conjugate gradient descent, quasi-Newton, and Levenberg-Marquardt which are very successful forms of two types of algorithm: line search and model-trust region approaches. They are collectively known as second order training algorithms. A line search algorithm works as follows: pick a sensible direction to move in the multi-dimensional landscape. Then project a line in that direction, locate the minimum along that line, and repeat. An obvious choice is the direction of steepest descent. Actually, this intuitively obvious choice proves to be rather poor. Having minimized along one direction, the next line of steepest descent may spoil the minimization along the initial direction.A better approach is to select conjugate or non-interfering directions - hence conjugate gradient descent. The idea here is that, once the algorithm has minimized along a particular direction, the second derivative along that direction should be kept at zero. Conjugate directions are selected to maintain this zero second derivative on the assumption that the surface is parabolic. If this condition holds, N epochs are sufficient to reach a minimum. In reality, on a complex error surface the conjugacy deteriorates. Quasi-Newton training is based on the observation that the direction pointing directly towards the minimum on a quadratic surface is

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the so-called Newton direction. This is very expensive to calculate analytically, but quasi-Newton iteratively builds up a good approximation to it. QuasiNewton is usually a little faster than conjugate gradient descent, but has substantially larger memory requirements and is occasionally numerically unstable. Instead of following a search direction, assume that the surface is a simple shape such that the minimum can be located directly - if the assumption is true. Try the model out and see how good the suggested point is. The model typically assumes that the surface is a nice well-behaved shape , which will be true if sufficiently close to a minima. Elsewhere, the assumption may be grossly violated, and the model could choose wildly inappropriate points to move to. The model can only be trusted within a region of the current point, and the size of this region isn't known. Levenberg-Marquardt uses a model that assumes that the underlying function is locally Levenberg-Marquardt is typically the fastest of the training algorithms, although unfortunately it has some important limitations, specifically: it can only be used on single output networks, can only be used with the sum squared error function, and has memory requirements proportional to W2 where W is the number of weights in the network. Levenberg-Marquardt, quasi-Newton, and conjugate gradient descent all perform calculations using the entire data set, so increasing the number of cases can significantly slow each epoch, but does not necessarily improve performance on that. There are two primary methods of implementing a neural network system. One is in dedicated hardware, and the other is to simulate the network on a digital computer. Because of the obvious cost and flexibility concerns, the latter is the most common method. Also, because of the programming complexities of implementing the various

types of networks, it is often

advantageous to use a commercially available packaged set of routines to design and develop a neural network system. One such package is the Neural Network Toolbox from The Mathworks, Inc. This package runs under The

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Mathworks’ MATLAB program and extends its capability to include many of the functions necessary to design and implement a neural network system. MATLAB and the Neural Network Toolbox provide the capability to design many different types of neural network systems for a variety of applications. The Neural Network Toolbox User’s Guide includes chapters on applications in control systems, linear and adaptive filters, and others (Demuth and Beale). There is a Graphical User Interface (GUI) for interacting with the routines in the toolbox, or command-line access for use in MATLAB’s programming and scripting capability. The Toolbox also includes an extensive set of built-in transfer functions to use in defining the neurons of the network, and many different training routines are included for use. All of the neural network development and implementation discussed in this thesis was done using MATLAB and the Neural Network Toolbox. Utilizing the functionality of the Neural Network Toolbox, defining a new neural network becomes a fairly simple series of one line commands with a few parameters to create, train, and simulate the specific characteristics and behavior of the network. The toolbox has commands to create perceptrons, radial basis networks, competitive, feedforward networks, and many others.

4.3 Adaptive neuro –fuzzy inference system(ANFIS) Nonlinear system identification is becoming an important tool which can be used to improve control performance and fault-tolerant behavior. Among the different nonlinear identification techniques, methods based on neuro-fuzzy models are extensively used in industrial applications. Neuro-fuzzy modeling is the hybrid of neural networks and qualitative fuzzy models. The tools for building neuro-fuzzy models are based on combinations of algorithms from the fields of neural networks, pattern recognition and regression analysis. The present work has made use of the experimental knowledge to arrive at a

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adaptive neuro fuzzy inference system(ANFIS) to predict the surface finish Ra for different values of key input parameters. Soft

computing

is

a

practical

alternative

for

solving

computationally complex and mathematically intractable problems. The main components of soft computing namely fuzzy logic and neural network have shown great ability in solving complex nonlinear system identification problems Neural networks possess a variety of alternative features such as massive parallelism, distributed representation and computation, generalization ability, adaptability and inherent contextual information processing. On the other hand, fuzzy sets constitute the oldest and most reported soft computing paradigm. They are well suited to modeling different forms of uncertainties and ambiguities, often encountered in real life. The objective of the synergy (using neural networks and fuzzy logic) through ANFIS has been to overcome the weaknesses in one technology during its application, with the strengths of the other by appropriately integrating them. An ANFIS is proposed as a core neurofuzzy model that can incorporate human expertise as well as adapt itself through repeated learning. An adaptive network is a multi-layer feedforward network in which each node (neuron) performs a particular function on incoming signals. The form of the node functions may vary from node to node. In an adaptive network, there are two types of nodes: adaptive and fixed. The function and the grouping of the neurons are dependent on the overall function of the network. Based on the ability of an ANFIS to learn from training data, it is possible to create an ANFIS structure from extremely limited or no mathematical representation of the system. In sequel, the ANFIS architecture can identify near-optimal membership functions of fuzzy logic for achieving desired input-output mappings. The network applies a combination of the least square method and the back propagation gradient descent method for training fuzzy inference system (FIS) membership function parameters to emulate a given training data set. The system converges when the training and checking

75

errors are within the acceptable bound. This architecture has demonstrated high performance in many applications.. In this paper, the ANFIS structure is considered for dynamic modeling of laser cutting of mild steel using CO2 laser. An ANFIS consisting of a set of TSK-type fuzzy IF-THEN rules is used to map the system inputs to outputs. This hybrid combination enables to deal with both the verbal and the numeric power of intelligent systems. As is known from the theory of fuzzy systems, different fuzzification and defuzzification mechanisms with different rule base structures can lead to various solutions to a given task. The fuzzy regions are parameterized and each region is associated with a linear subsystem. Owing to the fuzzily defined antecedents, the nonlinear system forms a collection of loosely coupled multiple linear models. The degree of firing of each rule is proportional to the level of contribution of the corresponding linear model to the overall output of the model. Gaussian membership functions with product inference rule were used at the fuzzification level.. The fuzzifier outputs the firing strengths for each rule. The vector of firing strengths is normalized. The resulting vector is defuzzified by utilizing the first-order Sugeno model.

4.3.1 Takagi-Sugeno model Takagi–Sugeno(1977) model is widely used in data driven modeling. In this model the consequent( then part) is a linear function of input variables. Ri: If x is Ai then yi=aiTx + bi,

i= 1,2,3---------------K, where ai is the

consequent parameter vector bi is scalar offset. This model is a combination of linguistic description with standard functional regression. The antecedents describe the fuzzy regions in the input space in which the consequent functions are valid. The output y is computed by taking the weighted average of the individual rules’ contributions.

76

K

K

i ( x) yi y

i 1 K

i ( x) ai T x bi

i 1 K

i ( x) i 1

i ( x) i 1

Where i(x) is called the degree of fulfillment of the ith rule. The antecedent fuzzy sets are defined in such a way that they describe partly overlapping, distinct regions of the input space. Sugeno model is a piece wise linear approximation of a non linear function. At the computational level, a fuzzy system can be seen as a layered structure similar to artificial neural networks. In order to optimize parameters in a fuzzy system, gradient descent training algorithms can be employed. Hence, this approach is usually referred to as neuro-fuzzy modeling. Both prior knowledge and process data can be used to construct neuro-fuzzy systems. Prior knowledge can be of a rather approximate nature. Two main approaches to the integration of knowledge and data can be distinguished: 1. Expert knowledge is formulated as a collection of if–then rules. In this way, an initial model is created. The parameters of this model namely the membership functions, consequent parameters are then fine-tuned by using process data. 2. Fuzzy rules

are constructed by using numerical data. In this case, the

advantage of using a neuro-fuzzy model is the possibility to interpret the obtained result.

In neuro-fuzzy models, the structure selection process involves the Selection of input variables. This involves not only the physical inputs but also insight in the process behavior and the purpose of the modeling. Automatic data-driven

77

selection can then be used to compare different structures in terms of some specified performance criteria. Number and type of membership functions, number of rules: These two structural parameters are mutually related (for more membership functions more rules must be defined) and determine the level of detail, called the granularity, of the model. The purpose of modeling and the amount of available information (knowledge and data) will determine this choice. Automated, methods can be used to add or remove membership functions and rules. The ANFIS system generated in the MATLAB allows for generation of a standard Sugeno style fuzzy inference system or a fuzzy inference system based on sub-clustering of the data (MathWorks, 1995)

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CHAPTER 5 EXPERIMENTATION

5.1 Laser cutting system The present work involves cutting of mild steel sheets using carbondi-oxide laser. The assist gas used is oxygen. The mild steel is cut at different laser powers, assist gas pressures and cutting speeds. The cutting work is done using the state of the art CO2 laser cutting facility at Government Tool room Training Centre, Mysore ( A govt. of India recognized research facility). High power Rofin Sinar CO2 laser is used for metal cutting. Figure 5.1 shows the block diagram of the laser cutting machine assembly. Apart from the CO2 laser tube, it consists of a dc power unit, assist gas unit, a mother board for the cutting machine, a computer, a control panel display and the x- y motion assembly. The dc power supply provides the power for all the sub units. The assist gas unit provides pure assist gas which is used to remove the debris generated during the cutting process. The focusing of the beam on the work is done with the help of helium-neon laser. The entire unit is housed in a chamber with complete protection to prevent beam leakage. Extra care is taken for CO2 laser as it delivers extremely high power output in the infra red region.

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Fig 5.1 Block diagram of laser cutting system

As the beam is invisible sufficient precaution is necessary during the cutting operation. All polished objects which can reflect IR radiation must be kept away from the machine unit.

Fig .5.2 Rofin sinar CO2 laser cutting machine used in the present work

80

Figure 5.2 shows the laser cutting head. It consists of optical wave guide, focusing lens and a coaxial nozzle to supply assist gas during the cutting operation. The important units of laser cutting assembly are beam delivery system, lenses, mirrors used in the waveguide and the polarizer.

5.1.1Beam Delivery

Fig. 5.3 Laser cutting head with beam delivery system The single most important factor in CO2 laser processing is the beam quality. A laser-driven manufacturing process must deliver sufficient laser power to the work area at the correct time. Industrial lasers can function in multimode operation, and good beam quality in all modes is very important. The beam delivery from the actual laser to the work is thus an important process. A good laser with poor beam delivery will not produce the desired result. The laser beam delivery system is shown in figure 5.3. A laser beam delivery system comprising a mirror telescope, beam-bending mirrors, and a phase shifter delivers the beam properly to the laser cutting head. Once the beam is delivered to the laser head, a focusing lens centers the point correctly to the work area.

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The laser beam is enclosed in a protective pathway that must be filled with a gas to slightly above atmospheric pressure. Any particulate matter or particles that may absorb the laser light must be removed from this pathway. Compressed air is a common choice for this application, but air can contain moisture, particles, and oils that can decrease the laser beam's power and shorten the beam-bending mirror's life. Nitrogen is preferable because it is contaminant-free and does not affect the mirrors adversely.

5.1.2 Lenses Meniscus lenses Laser cutting work requires a sharply focused laser beam on the work. This is ensured by the lens system. Small and focused spot size is important as cutting requires high-power density and power density is a strong function of spot size. Metal cutting jobs demand narrow kerf widths and heat affected zones, which can only be achieved with a tightly focused beam. The lenses made from materials with high transmittivity in far infra red region such as zinc selinide, gallium arsenide and germanium are preferred in CO2 laser. Focal point, spot size and power density depends upon the material to be cut. For highly reflecting surfaces the beam is kept off focused. For mild steel cutting the beam is focused to the top of the work whereas for stainless steel the beam is focused to the base or middle region of the work. The design variables that will affect the performance of a CO2 laser lens are: focal length, diameter, shape, material and coating. Focal length affects both spot size and depth of focus. In general, a shorter focal length will produce a smaller focused spot and a shorter depth of focus. The specified focal length is a compromise between desired spot size, penetration depth and workpiece clearance. Higher power laser used in the present cutting work require larger diameter lenses to prevent thermal overload. At any given focal length, a larger diameter lens will yield a smaller focused spot if the incoming beam is expanded to fill the larger lens..

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Meniscus lenses have one concave surface and one convex surfaceas shown in figure 5.4 . A meniscus lens creates a smaller beam diameter thus reducing the spherical aberration and beam waste while precision cutting or marking. The lens also provides a smaller spot size that creates the same power output in a smaller area. In order to achieve the smallest spot size the concave side of the lens is mounted face down, pointing towards the material being cut

Fig 5.4 Meniscus lens

Although meniscus lenses are more expensive than Plano-Convex lenses, they offer a higher accuracy of the cut therefore increasing cutting speed by up to 5%.

5.1.3 Plano convex lens

Plano-convex is the simplest lens shape. It is used in such applications where achieving the smallest spot size is not critical, or at relatively long focal lengths when more complex shapes would not be beneficial. Plano-Convex lenses have one flat surface and one outward curving surface as shown in the figure 5.5. The lens should be oriented with the flat side toward the workpiece and the convex side toward the laser. These lenses are more suitable for high turnover cutting applications.

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Fig 5.5 Plano convex lens

When cutting steel and other thick materials, a Plano-Convex lens provides a greater width of the cut enabling the laser's Oxygen assist to enter and ease the cutting process. In addition, Plano-Convex lenses give a greater depth of field needed to maintain a taper less edge when cutting thicker materials. In the present work the laser beam is focused on the work using a lens system with a focal length of 190mm. The laser system used for the current work uses metallic mirrors in beam delivery system as they have high reflectivity. They are also not susceptible to thermal damages and can withstand large power density. There are two main categories of laser mirrors: internal and external. Internal mirrors are used to generate, maintain and amplify the laser beam by forming a reflective “resonator” around the excited CO2 gas mixture. External mirrors are used to deliver, manipulate, split and focus the laser beam. Most mirrors have flat reflective surfaces, but some have curved surfaces designed to reduce beam divergence. The design of the substrate material and coating of a CO2 laser mirror is primarily determined by its function. Water cooled polished copper mirrors are used in CO2 laser systems. Loss of mirror efficiency negatively affects the process. It also increases costs by delivering the beam at less than the required power level. Mirrors have finite lives that vary depending on the process and the required power level. The typical life is about 300 hours. A 10 percent decrease in mirror efficiency can increase electrical power consumption

84

for the same operation by as much as 15 to 20 percent. Internal Mirrors, also known as front mirrors, are designed to reflect a portion of the beam back into the laser resonator for continuous amplification while transmitting a portion of the beam to the outside for use. Therefore, the substrate material must be transmissive at the required wavelength of 10.6µm. Germanium and Gallium Arsenide substrates are commonly used for low to medium powered systems. The more expensive Zinc Selenide material is required for higher powered lasers because of its lower absorption at 10.6 microns. Rear mirrors are designed to reflect all or nearly all of a laser beam back through the laser gas mixture for amplification. The inside surface is given a highly reflective (99100%) coating. In the 100% reflective case, silicon can be used as the substrate material and the outside surface does not need polishing or coating. Some rear mirrors, however, are designed to transmit a small (0.5 - 1.0%) percentage of the beam to a power detector for real-time beam monitoring. These mirrors must have a transmissive substrate (Ge is the most common) and the outside surface usually has an antireflective coating.

5.1.4 Circular polarizers A photon traveling in space has oscillatory motion. This causes the oscillations in electric field and results in the generation of electromagnetic waves. This wave represents the direction and magnitude of the electric field vector of the photon as a function of time. Due to the coherence of the laser beam, the photons have identical phase, amplitude and direction. Thus the electric vectors for all the photons are aligned in the same direction. This results in the linear polarization of the laser beam. In metal cutting , the cutting speed, the width of the cut and the curvature of the kerf depends on the amplitude and beam scanning direction. The depth of cut will be more if the electric vectors are oriented in the direction of scanning of the beam. A shallow cut results if the electric vector is 900 out of phase with the

85

scanning direction. The directional effect produced in the cut kerf can be eliminated by using a circularly polarized light.

Fig 5.6 Circular polarizer assembly used in the cutting machine Fig 5.6 shows circular polarizer assembly. Mirrors coated with birefringent material are used for this purpose. Figure shows an arrangement of four mirrors aligned at 450 to the incident beam. Such an arrangement is effective in producing a circularly polarized beam.

5.1.5 Gas Impurities Laser gas contamination can cause beam degradation. Contaminates can be found in the gases themselves, introduced by the systems that deliver the gases to the resonator, or created by the laser reaction. Most commonly used lasing gases for industrial lasers are combinations of helium, nitrogen, and CO2. All gases can contain impurities caused by the actual gas production. As soon as a laser is activated, a series of reactions begins to occur within the resonator,

86

separating the CO2 molecules into carbon monoxide and oxygen. This creates many thousands of parts per million of carbon monoxide and oxygen within the resonator—many more than are present even in the source product. Introducing these impurities at this level can cause the laser resonator to overheat if other impurities are present. The main impurities present in the gases that can cause problems in the laser resonator are moisture and total hydrocarbons. Moisture destabilizes the beam, absorbs into the coating of some optics, and decreases power output. Hydrocarbons reduce the laser's power, which limits its ability to amplify power. CO2 lasers, used for cutting ferrous and nonferrous materials require beam generation gases - generally mixtures of helium, nitrogen and carbon dioxide that are stimulated electrically and emit radiation in the form the laser beam. Beam Delivery System Purge gas is the gas that protects the laser beam within the transmission tube from particles and moisture that could damage mirrors or spread the beam. The assist gas is the gas that acts to remove material melted by the laser. The assist gases and supply equipment must be able to supply the gas to the laser head at the desired pressure and flow. Most lasers can process various materials using different assist gases. Air, oxygen, and nitrogen are the most common. Argon and helium are required for titanium and some other materials. Each assist gas has advantages and disadvantages for each material. Oxygen is the common choice for carbon steel; air and nitrogen can be used to cut carbon steel, stainless steel, and aluminum. Gas must be supplied continuously, without interruption. If the assist gas is contaminated, oxidation can occur at the cut and cause problems if further processing. Therefore, contamination-free delivery of the process gas is also essential. But particles and vapors in the beam delivery tube can disrupt the beam and reduce power and defuse the beam. The transmission tube must be purged with dry nitrogen and be particle-free. The higher the beam quality, the higher the cut quality, and the more cost effectively the part is produced.

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5.1.6 Nozzles Figure 5.7 shows the nozzles used in laser cutting. From a gas dynamics point of view, these nozzles are very different. The supersonic minimum length nozzle provides the best compromise of shock-free flow, nozzle length, and ease of manufacture. This nozzle has a conical contracting section, followed by an expansion part that is matched to the expansion of the compressed gas at a particular pressure, resulting in smooth flow without significant shocks. True Laval nozzles, which have a shaped contracting section and a smooth throat section, can produce a completely shock-free flow, but they are impractical because of their high cost as no mass manufacturing technique exists.

Fig 5.7 Different types of laser cutting nozzles.

Sonic nozzles are the commercial standard for industrial laser cutting. Laser cutting with low assist gas pressures is rarely performed on thin section material, as gains in processing speed and quality are obtained by operating at higher inlet pressures. Increasing the inlet gas pressure causes the

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free jet to become under-expanded at the nozzle exit resulting in shock wave formation

5.2 Melt removal mechanisms The capability to eject the molten metal from the cut kerf depends upon the drag from the gas assist jet along the cutting front. There are two types of drag, viscous drag (due to tangential stresses) and pressure drag(due to normal stresses). Viscous drag prominent where the surface area parallel to the flow direction is large compared to the projected area normal to the flow. Pressure drag arises when the cut front is at an angle to the gas jet where a shear force will act upon it and a pressure gradient will be formed. Viscous drag and pressure drag are of the same order of magnitude in laser cutting, and are the main forces of melt and vapor removal from the kerf.

5.3 Kerf flow conditions According to the theory of gas dynamics, the best condition for material removal from the cut is when the exit pressure at the bottom of the kerf is equal to the ambient pressure. If it is much higher than the ambient pressure, not only is the viscous force reduced, due to the decrease in gas velocity approaching the kerf bottom, but also the flow leaving the kerf will rapidly accelerate due to supersonic expansion. As a result, the molten material is ejected in divergent directions not along the tangential direction of the cut front. Some molten metal is forced towards the sidewalls and forms dross attached to the bottom of the cut edges. Therefore, an exit flow condition with maximum gas velocity and a pressure equal to the ambient pressure is desired to achieve a cut edge without dross. Industrial use of nozzles for laser cutting is generally limited to sonic (conical) nozzles operating at high inlet gas pressures. The flow conditions in the kerf for the supersonic nozzle are similar to that of the sonic

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nozzle, at low pressure, even though the nozzle flow conditions are totally different. At higher pressures the shock over the work is stronger than for a conical nozzle, because the impact flow velocity is higher (fully expanded supersonic flow). Shock waves destroy kinetic energy (turning it into heat), resulting in a slower flow inside the kerf than for conical nozzles, which is quite contrary to intuition. The supersonic nozzles also generate less expansion of the flow within the kerf, which reduces the force upon the kerf front, which acts upon the molten layer. It can therefore be expected that supersonic nozzles will result in a lower maximum cutting speed, at lower nozzle pressure. Plasma formation above the work absorbs some of the laser radiation reducing the energy available to generate the melt. The flow from the supersonic nozzles has been expanded before it exits the nozzle resulting in lower pressure and density above the work than the sonic nozzle. This reduction in density limits the plasma formation allowing more of the laser power to reach the work. For thin steel sheet cutting, supersonic nozzles do not improve the melt removal conditions, as the advantages of the supersonic flow are completely lost on impact with the work surface. Also, cutting speeds are not limited by the melt removal rate. Increasing gas pressure does not necessarily increase cutting speeds, as at high pressures the formation of plasma above the kerf reduces the incident laser energy on the work limiting melt production and cutting speeds. Dross-free cuts are achieved when the flow out of the kerf is fully expanded. For thin sheet, both conditions are most easily achieved with a conventional conical nozzle. The mild steel plates without any oxidation is prepared for the study. The research work is focused on the oxygen assisted cutting process of mild steel with thickness 3mm and 5mm. In the experiment laser power up to 3 kw, assist gas pressure up to 7.5 bar is used. Cutting speed is varied from 2500mm per minute to 5000mm per minute. Beam is focused on the top surface of the mild steel.

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5.4 Surface finish measurement

Fig 5.8 Mitutoya surf test meter used for Ra value measurement

To assess the quality of the cut surface, the centre line average Ra is considered as the important parameter. The surface quality analysis is done using digital Mitutoya surftest shown in the figure 5.8. The surface assessment is done soon after the cutting to avoid any damage to the cut surface that might arise from atmospheric moisture. The advanced cut surface analysis is carried out by using scanning electron microscope. Key hole structure

is

photographed. To assess the surface damage the regions with transverse cracks are studied with greater magnification of SEM. Micro metallurgical changes in the cut surface is studied using electron diffused X-ray analysis. Cut surface etched and photographed to study the heat affected zone which might result due to extreme heat generated by laser metal interaction.

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5.5 Scanning electron microscope (SEM) and EDAX studies:

Fig 5.9 Scanning electron microscope used in the present study (Leica 440i)

In scanning electron microscope(SEM), the electrons emitted by a thermionic cathode are focused by an electrostatic lens arrangement into a beam of 1 to 1000nm diameter. The electron beam is then made to scan the surface using the deflection coils. At the point where the electrons are incident, the secondary electrons with an energy of about 100eV are emitted. These electrons are accelerated by a potential of 10 kV towards a detector. The secondary electron signal is sensitive to the topographical details of the surface

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scanned and gives excellent lateral resolution. This signal can be coupled to a computer to get a display on the CRT. The incident electron beam may interact inelastically with the atoms to knock off an inner shell electron. The de excitation of the atom produces X-ray photons with energy characteristic of de ionizing transition. The X-rays emitted is therefore a characteristic of the element ionized. In energy dispersive X-ray(EDAX) analysis an energy dispersive spectrometer(EDS) attachment is used in the scanning electron microscope.

EDS arrangement consists of liquid nitrogen cooled lithium

detector which is placed very close to the specimen. The X-ray photons produce a current which proportional to its energy in the detector. The current is used to produce the voltage pulses which are counted in a multichannel analyzer. The EDAX spectrum consists of sharp peaks corresponding to the different elements present in the surface scanned. The height of the peaks indicates the abundance of the element. In laser cutting of mild steel with oxygen assist gas the micro metallurgical changes produced by the extremely high temperature produced can be studied using the EDAX analysis. In our study the mild steel sample which is electrically conducting is put straight into the microscope.. The size of the specimen for SEM is 12.5 mm diameter. Cleanliness of the top surface is important. in SEM as the topmost surface layers of the sample are scanned. Care is taken to prevent the surface oxidation of the kerf due to

atmospheric moisture.

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CHAPTER 6 Results and discussion

6.1 Effect of laser power on surface finish Though laser machining is a non contact process the extreme heat produced during the process affects the cut surface quality. If oxygen is used as an assist gas the additional heat generated during oxidation process will produce further deterioration of the cut surface. Surface quality also depends on the assist gas pressure. One of the functions of the assist gas is to remove the molten metal before the resolidification takes place. At higher gas pressures this task is achieved quickly and consequently the surface quality will be better. The surface quality assessment is done using centre line average Ra . The Ra value for different specimens cut under different conditions is measured using the Mititoya digital Ra meter. The surface quality assessment is also done using the SEM photographs and micrograph. The results are depicted in the following graph. It shows the change in Ra value with laser power while the assist gas pressure are kept constant. The cutting speed is kept at 3000 mm per minute.

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Fig.6.1 Variation of surface finish with laser power at different assist gas Pressures

It can be seen in fig. 6.1 that Ra increases with laser power. Laser machining being a thermal process, increase in laser power in Oxygen assisted cutting leads to a surface that has poor quality. The cutting front maintains a very thin liquid layer and a relatively thicker gas boundary layer. The gas layer is partly impurity gases left behind by the oxygen and product of Chemical reaction. The incomplete melt removal followed by solidification results in poor surface.

6.2 Effect of cutting speed on surface finish One of the key attributes of laser machining is the cutting speed. The metal cutting is done by keeping the work stationary and moving the laser head over it in order to initiate the cut. If the cutting speed is less it results in a greater kerf but the surface damage will be more. If the cutting speed is very

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high , it results in insufficient exposure of the work to the laser beam and may not initiate the cutting process at all. In laser metal cutting a compromise has to be arrived between the material removal rate and the quality of the cut surface.

800 watts 1400 watts

Ra value in microns

3.6

3.2

2.8

2.4

2.0 2800

3200

3600

4000

4400

Cutting speed in mm/min

Fig.6.2 Variation of Ra value with cutting speed

Fig. 6.2 shows the variation of Ra with cutting speed. The Ra values are measured at fixed assist gas pressure. The experiment on mild steel has shown that the Ra value decreases marginally with the increase with the cutting speed.

6.3 Effect of assist gas pressure on Ra value Assist gas is mainly used to remove the molten metal from the kerf. If it is not realized properly there is a chance that the metal may resolidify to close the kerf. Apart from this the oxygen assist gas undergoes exothermal reaction with iron to generate additional temperature.

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800 Watts 1200 Watts 1400 Watts

3.6

Ra value in microns

3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.8 5.5

6.0

6.5

7.0

7.5

8.0

Assist gas pressure in bars

Fig.6.3Variation of Ra with assist gas pressure

Fig.6.3 shows the variation of Ra value with assist gas pressure. At a given laser power, as the pressure is increased the molten metal gets removed from the kerf by the jet of assist gas before the resolidification takes place. The design of the nozzle through which the assist gas is blown coaxial to the laser beam also affects the surface quality of the cut kerf.

6.4 Effect of laser power on kerf width Laser machining is characterized by narrow kerf. This is because of fine focusing of the laser beam. The kerf width is a measure of the material removal rate at a given cutting speed and laser power. Following table shows the kerf width values obtained at different laser powers. Cutting speed is held constant at 3000mm/minute.

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5.5 bars POWER

6 bars

6.5 bars

7 bars

7.5 bars

Kerf width in mm

1000

0.326

0.332

0.342

0.39

0.412

1100

0.366

0.37

0.375

0.403

0.424

1200

0.39

0.396

0.4

0.41

0.43

1300

0.41

0.421

0.43

0.451

0.466

1400

0.431

0.453

0.461

0.477

0.485

1500

0.44

0.47

0.49

0.509

0.52

1600

0.464

0.49

0.52

0.53

0.541

Table 6.1 - the kerf width at different laser powers.

0.6

Kerf width in mm

0.55

0.5

0.45 5.5 bars 6 bars 0.4

6.5 bars 7 bars 7.5 bars

0.35

0.3 1000

1100

1200

1300

1400

1500

1600

Laser power in watts

Fig 6.4 Kerf width variation at different laser powers

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Fig 6.4 shows the variation of kerf width laser power. Being an extreme thermal process, in laser metal cutting with oxygen assist gas, the kerf width is found to increase with the laser power. SEM studies revealed the side burning in some regions. Kerf opened up appreciably at higher pressures. At lower pressures, the dross height at the exit end of the laser beam is found to be more. 6.5 Effect of assist gas pressure on the kerf width The assist gas used in laser machining primarily to remove the molten metal from the kerf . It impinges on the metal surface co axial to the laser beam. The nozzle shape plays an important role in the effect of assist gas pressure on the kerf width.

Assist gas pressure In bar

800 w

1000w

1200w

1400w

1600w

Kerf width in mm

5.5

0.218

0.326

0.39

0.431

0.464

6

0.231

0.332

0.396

0.453

0.49

6.5

0.261

0.342

0.4

0.461

0.52

7

0.3

0.39

0.41

0.477

0.53

7.5

0.344

0.412

0.43

0.485

0.541

Table 6.2 Variation of kerf width assist gas pressure.

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0.6

Kerf width in mm

0.5

0.4

0.3 800 w 1000w 0.2

1200w 1400w 1600w

0.1

0 1

2

3

4

5

Assist gas pressure in bars

Fig.6.5 Variation of kerf width with assist gas pressure

Fig.6.5 shows the variation of the kerf width with the assist gas pressure. It is seen that though the kerf width increases with the increase in assist gas pressure, at higher laser powers the kerf is wider but variation is not much. This is because the molten metal expelled from the kerf by the gas jet quickly.

6.6 Effect of cutting speed on kerf width Fig.6.6 shows the variation of kerf width with cutting speed. The laser power and assist gas pressure are kept constant. Kerf width is found to decrease with the increase in cutting speed. As the cutting speed is increased, the interaction time between the laser beam and the metal decreases. This results in the decrease in the kerf width. However if the cutting speed is made very high the interaction between the laser beam and the metal may not be sufficient to initiate the cut.

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Kerf width in mm

0.6 800 Watts 1400 Watts

0.5 0.4 0.3 0.2 0.1 3.0k

3.5k

4.0k

4.5k

5.0k

Cutting speed in mm / min

Fig.6.6Variation of kerf width with cutting speed

In laser cutting the cutting speed affects the material removal rate as well as the quality of the cut surface. If the speed is enhanced the laser material interaction becomes less. This may result in the lesser surface damage. But the kerf width becomes less due limited exposure of the work to the laser beam. A balance has to be achieved between increasing the material removal rate and producing a surface of better quality. 6.7 Key hole technique Key hole technique is employed in laser cutting. The cut is first initiated and then the laser beam is moved in the desired direction. SEM photograph of the keyhole is shown in the fig 6.7.1 and fig 6.7.2. When the cut is initiated the beam is focused for a longer duration on an area equal to spot size. As a result the opening of the mouth on the front side of the key hole is visible.

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Fig 6.7.1 SEM photograph of the front side of the keyhole

Fig.6.7.2 SEM photograph of the back of keyhole

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As the heating reduces exponentially with the thickness the kerf becomes narrow at the base. The dross sticking to the exit of the beam is also visible.

Fig.6.8 SEM photographs showing periodic striations

Fig 6.9 Enlarged view of the striation showing lateral cracks

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In the fig 6.8 characteristic striations produced in the laser cutting with oxygen assist gas is visible. Mild steel is known to be combustible under a high purity oxygen jet and striations have been found to be more pronounced in laser reactive gas cutting than in inert gas cutting. The sequence of ignition, burning assisted by high pressure gas and extinction causes the striation pattern. The striations affect the quality of the surface. Striations are fairly periodic and it depends on the oxidation cycle.

Fig 6.10.1 SEM photograph of the kerf showing transverse cracks

Fig. 6.10.1 is the SEM photograph of cut surface. The lateral cracks are seen in this photograph. These are due to thermal stresses produced during the cutting.

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Fig 6.10.2 SEM photograph of the kerf showing surface damage

Fig 6.10.3 SEM photograph of 5mm thick mild steel cut using laser

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Fig 6.10.4 SEM photograph of the work etched across the thickness

In Fig. 6.10.2 overlaid flow of molten metal which is due to the flow of assist gas is observed. Fig 6.10.3 shows a finely distributed and solidified layer with honeycomb surface characteristics and finer cavities. Such a deposition is mainly controlled by cutting conditions. These cavities control the surface characteristics and surface finish. Fig. 6.10.4 is the etched section across the thickness of mild steel cut with laser. The depth of HAZ is least with little damage deep at the cross section with a laminated type of separation of the work material. This is mainly due to heating and subsequent cooling of the surface by the assist gas.

6.8 White layer A hard, featureless and brittle white layer is observed at the cut surface in laser cutting. A dark overtempered layer is seen immediately below the white layer. A typical white layer from laser cutting can be seen in the figure 6.11.1 and figure 6.11.2. These two photographs are taken for samples

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cut with different laser powers. White layer formation results from complex mechanical, thermal and metallurgical processes.

Fig 6.11.1 SEM photographs showing white layer when the laser power is 1200watts

Fig 6.11.2 SEM photographs showing white layer when the laser power is 800watts

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Three main factors responsible for white layer formation are (1) thermally-induced phase transformation due to extreme cutting temperature, (2) mechanical grain refinement arising from severe plastic deformation and (3)surface reaction with the environment. White layer will affect the cut surface quality. It is detrimental to the performance of the machined part and can reduce its corrosion resistance and fatigue life.

6.9 Heat affected zone

6.12 Micrograph near the surface of the work exposed to the laser beam

One of the important features of laser cutting is the flow of heat through the bulk metal predominantly in the vertical direction. The lateral conduction is very limited. Further the laser beam is moved over the metal surface at a high speed. Though the temperature of the metal where the laser beam lands is raised to a very high value, the lateral damage is found to negligible. This is established in the study of heat affected zone.

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6.13Micrograph away from the surface of the work exposed to the laser beam

Micro structure is determined by polishing and etching the surface near the cut edge and away from it using 2% Nital and the cut surface. The SEM photographs taken near the surface exposed to laser beam and away from it are shown in the figures 6.12 and 6.13. Structure is found to be predominantly ferrite with some cementite. The grain size is found to be ASTM no 7 both near the cut edge and away from it.

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Fig.6.14 Digitized image of the cut kerf showing lateral cracks and surface damage

Fig. 6.14. shows the digitized image of the cut kerf. It can be obtained by scanning the cut surface and converting it into a matrix consisting of 1’s and 0’s using the image processing tool box of Matlab. Such surface can be used to assess the surface damage caused by the laser beam. The transverse cracks and surface damage caused by the laser beam can be seen. The red colored lines indicate deeper striations. Striation frequency which indicates burning and extinction cycles caused by the exothermal reaction which takes place when oxygen assist gas is used can be determined using the digitized image.

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CHAPTER 7 Process modeling

7.1Development of ANN model ANN is used to model the laser cutting process. Two separate models were developed for

the kerf width and Ra value prediction. A

comprehensive model which can simultaneously give the kerf width and Ra value is designed to model the process. For the present study, the laser power, cutting speed and assist gas pressure are identified as key input parameters. Kerf width and surface finish (Ra) are identified as key output parameters. The study is restricted to 3mm mild steel specimen. Figure 7.1 shows the block diagram of the ANN model scheme. The neural network model development requires a large number of experimental data. The experimental data can be divided into two sets, a training dataset and a test dataset. The ANN model is developed using only the training data. The test data are then used to check the behavior of the model when presented with previously unseen data.

Choosing a proper model and adjustment of the

parameters of a model so as to minimize a certain fit criterion determines accuracy of modeling. Supervised learning technique is used for the development of ANN model. The input pattern is presented to the first layer (input layer) and is processed to the hidden layer by a with a sigmoid transfer function to produce an output which in turn becomes an input to the neurons of

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Fig 7.1 The neural network model scheme

output layer to obtain the actual output. The difference between the actual output and the desired output yields an error signal. This error signal depends on the connection weights and biases used in the network layers. The main purpose of the learning process is to minimize this error by updating the values of those weights. The algorithm recalculates the weights at the last layer and continues computing the error and updating weights moving backward, toward the input layer, until the input layer is reached. The training for all input-output patterns will be repeated until the error between the actual output from the neural network and the desired output diminishes to a specified bound or other stopping criteria are met. Different artificial neural network architectures used

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to model oxygen assisted CO2 laser cutting of mild steel. Two different networks were developed to predict the surface finish Ra and kerf width separately. Both had 3-9-7-1 architecture with three input nodes, two hidden layers, one with nine and the other with 7 neurons. The output layer had one neuron. The sigmoidal transfer function was employed for the first and second hidden layers. A linear transfer function is employed between the second hidden layer and the output. Gradient descent learning rule was employed. The trained networks could model the process fairly accurately. However ANN model that could model the process and give both outputs simultaneously is preferred.

Fig.7.2 Multilayer perceptron with 3-9-7-1 architecture

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3-9-7-1 architecture shown in fig.7.2 is used in the model to study the relation between the surface finish and laser power, assist gas pressure and cutting speed.

Fig.7.3 Multilayer perceptron with 3-9-7-1 architecture

Similar model shown in fig.7.3 is used to study the relation between the surface finish and laser power, assist gas pressure and cutting speed. These two models produced good results for predicting the kerf width and Ra value. However the model with two hidden layers and two outputs could not predict the values of the output parameter accurately. Several networks with only one hidden layer were tried to model the process.

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Fig. 7.4 MLP with 3-12-2 architecture

A model with only one hidden layer shown in fig.7.4 is found to be suitable to study the relation between kerf width, surface finish, laser power, assist gas pressure and cutting speed. In this model laser power, assist gas pressure and cutting speed are taken as input parameters and the kerf width and surface finish(Ra) are taken as output parameters. Unlike the other two models in this model only one hidden layer is employed. Sigmoidal transfer function is employed for the input to hidden layer. Linear transfer function is employed between the hidden layer and output layer. The network training is performed using three different learning rules namely conjugate gradient, LevenbergMarquardt and quasi Newton method. Details of these algorithms are

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elaborated in the appendix 1. The implementation was done using the Matlab 6.1. The instruction for training are listed below

Simple commands to create 3-12-2 ANN with Levenberg-Marquardt training algorithm: The input data is a 112 X 3 matrix consisting of power, pressure and cutting speed values obtained from the experiment. Input p = [power,pressure,cutting speed]. The target output used comprises of the outputs kerf width and Ra value. It is a 112 X 2 matrix Target t=[kerf width, Ra value]. net=newff(minmax(p),[12, 2],{'tansig' 'purelin'},'trainlm'); net.trainParam.show = 5; net.trainParam.epochs = 100; net = train(net,p,t); This command would train the network for the toolbox default of 100 epochs (one epoch is a single pass through all the inputs and targets) utilizing the other toolbox default parameters for training function, performance function and other parameters. If desired, or necessary, all of the default parameters can be changed to meet the specific needs of the application under study. During the learning process the network is trained to mimic the process to the desired accuracy. Whether a network is well trained or not can be ascertained from the graph of number of epochs and the mean square error. In a well trained network the mean square should decrease towards the goal. After training , the network is tested using a set of values. The over learning of the network is avoided as it leads to pattern recognition rather than the intelligent modeling of the actual process.

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Fig 7.5Mean square error versus number of epochs for Levenberg – Marquardt Algorithm

Simulating network behavior when presented with a set of inputs is accomplished using the command: sim(net,p); which presents the network the inputs p and calculates the output of the network based on its current definition.

Simple commands to create 3-12-2 ANN with Conjugate gradient training algorithm: net=newff(minmax(p),[12, 2],{'tansig' 'purelin'},'traincgf'); net.trainParam.show = 5; net.trainParam.epochs = 100; net = train(net,p,t);

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Simulating network behavior when presented with a set of inputs is accomplished using the command: sim(net,p);

Fig 7.6 Mean square error versus number of epochs for Conjugate gradient Algorithm

Simple commands to create 3-12-2 ANN with Quasi Newton training algorithm: net=newff(minmax(p),[12, 2],{'tansig' 'purelin'},'trainbfg'); net.trainParam.show = 5; net.trainParam.epochs = 100; net = train(net,p,t); Simulating network behavior when presented with a set of inputs is accomplished using the command:

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sim(net,p);

Fig 7.7 Mean square error versus number of epochs for Quasi Newton Algorithm

The trained network is used to simulate the laser cutting of 3 mm mild steel. The effects of laser power, assist gas pressure and cutting speed on kerf width and surface finish can studied using the ANN model. The values of the kerf width and surface finish Ra as predicted by the ANN on the basis of the three different algorithms are given in appendix 2.

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7.2 Comparison of the results obtained from ANN with experimental values Actualkerfwidth LMprediction CGprediction BFGprediction

Kerf width in mm

0.4

0.3

0.2

0.1

0.0 800

900

1000

1100

1200

1300

1400

Power in watts

Fig 7.8 Comparison of results for kerf obtained from the three algorithms

Expt LM CG BFG

3.0

Ra value in microns

2.5

2.0

1.5

1.0

0.5

0.0 5.8

6.0

6.2

6.4

6.6

6.8

7.0

7.2

7.4

7.6

Assist gas pressure in bar

Fig 7.9 Comparison of results for Ra obtained from the three algorithms

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The trained ANN’s were presented with inputs to simulate the kerf values. The values obtained from three different learning algorithms are presented in the bar graph. It is quite evident that the values generated by Levenberg-Marquardt algorithm are closer to the experimental values. This is also evident from the mean square error versus number of epochs curve. Convergence is faster and more accurate in case of Levenberg-Marquardt algorithm. The kerf width predicted by other two algorithms is found to deviate from the actual values. Bar plot for the comparison of kerf width and Ra values obtained from three different algorithms are shown in the figure 7.8 and fig 7.9. The deviation from the experimental values is again found to be least for Levenberg-Marquardt algorithm. The gradient descent algorithm predictions were also in good agreement.

7.3 Comparison of ANN predicted and experimental values of kerf widths at different laser power and pressures. Cutting speed = 3000mm/minute.

Fig 7.10 ANN predicted values of kerf width with laser power

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Figure 7.10 shows the kerf width values predicted by three different algorithms at different laser powers and pressures. Cutting speed is kept constant at 3000mm/minute. The values predicted by Lavenberg-Marquardt algorithm at four different pressure values were found to be in good agreement with experimental values. However the divergence of the predicted values are found to be much less pronounced at higher pressure. It can be seen that an increase in the assist gas pressure results in a wider kerf when the cutting speed remains same.

7.4 Comparison of ANN predicted Ra values with experimental values at different laser powers.

Fig 7.11 Variation of Ra value with laser power at a constant Cutting speed of 3000mm/minute

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Figure 7.11 shows the variation of the surface finish Ra with laser power at different assist gas pressures. The cutting speed is kept constant at 3000mm/minute.It is seen that the Ra value is minimum for a particular laser power of 1000watts. Power for minimum Ra is found to be same for different pressures. At this power the melt film thickness is such that the assist gas flushes out the molten metal before resolidification. The predicted values from Levenberg-Marquardt algorithm is very close to the experimental values.

7.5 Comparison of ANN predicted Ra values with experimental values at different pressures.

Fig.7.12 Variation of Ra value with assist gas pressure (cutting speed = 3000m/minute)

Fig 7.12 shows the variation of the Ra value with assist gas pressure at constant cutting speed and laser power. Ra value variation with pressure is

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not much pronounced. At all powers its value is found to decrease marginally with the assist gas pressure. In this case also the ANN predicted value with Levenberg-Marquardt algorithm is in close agreement with experimental values.

7.6 Comparison of ANN predicted Ra values with experimental values at different cutting speeds.

Fig.7.13 Variation of kerf width cutting speed ( pressure =6 bar)

Figure 7.13 shows the variation of kerf width with cutting speed at constant pressure and power. As the cutting speed is increased the kerf width decreased. Increase in speed will reduce the concentration of the heat at a particular spot. But if the speed is increased beyond a particular value, the laser

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may not initiate the cut due to insufficient exposure of the work to the laser beam. It is also seen that the values predicted by an ANN with LevenbergMarquardt algorithm is very close to the actual values.

Fig.7.14 Variation Ra value with cutting speed and laser power

The relationship between the three input parameters and the output parameter can be displayed using a 3-D surface plot. Figure 7.14 shows the 3-D surface plot of Ra value versus cutting speed and laser power used to study the effects of cutting speed and laser power on surface finish. At higher cutting speeds the surface plot shows a downward slope indicating lower material displacement with better surface quality. At the same time an increase in the laser power in oxygen assisted cutting leads to a poor surface finish. At higher laser powers and cutting speeds the Ra value is high. Too high a cutting speed will lead to insufficient laser material interaction and higher laser power leads

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to more turbulent flow of the material..It is observed that a laser power of 1030 watts and a cutting speed of 3000mm/minute produce a relatively good surface finish.

Fig.7.15 Contour plot of kerf width with laser power and cutting speed.

Kerf width, laser power and cutting speed contour plot shown in fig 7.15 reveals that higher laser powers and smaller cutting speed produces a wider kerf. But smaller speed and larger laser power will result in poor surface finish as discussed above. A compromise has to be achieved keeping in view the surface quality and kerf width. A laser power of 1030 watts and cutting speed of 3000mm/minute resulted in a narrow kerf.

126

Fig 7.16 Variation of Ra value with laser power and assist gas pressure

The variation of Ra value with laser power and assist gas pressure is depicted in surface plot shown in fig.7.16.A small decrease in Ra value was noticed as the power is increased from 800 watt to 1000 watts. This is due to accelerated flow solidification at lower power. Minimum Ra value occurred at about 1030 watts. However a nominal decrease in the Ra value is noticed in the Ra value as the assist gas pressure is increased.

7.7 Comparison of the kerf width obtained from the ANN and regression

method A conventional non linear regression model for the laser cutting problem is developed to predict the values of kerf width and Ra value foe different laser power assist gas pressure and cutting speeds. Details of the model are included in the appendix 3. Following table shows the comparison

127

table between the ANN model with Levenberg-Marquardt algorithm and regression model used for predicting the kerf width..

Laser power in watts 800 800 800 800 800 900 900 900 900 1000 1000 1000 1100 1100 1200 1200 1200 1300 1300 1400 1400

Assist gas pressure in bar 6 6.5 7 7.5 6 6 6.5 7 6.5 6 6.5 6 6 6.5 6 7 7.5 6 7 7 7.5

Cutting speed in mm/min 3000 3000 3000 3500 3500 3000 3000 3000 3500 3000 3000 4000 3000 3000 3500 3500 4000 4000 3500 4000 4500

Expt 0.231 0.261 0.3 0.32 0.22 0.292 0.299 0.36 0.277 0.332 0.342 0.301 0.37 0.375 0.385 0.395 0.4 0.398 0.434 0.44 0.44

Kerf width in mm Regression L-M model prediction 0.268 0.227 0.282 0.263 0.297 0.295 0.295 0.319 0.253 0.214 0.293 0.28 0.309 0.304 0.325 0.338 0.291 0.29 0.321 0.332 0.338 0.35 0.286 0.296 0.351 0.363 0.369 0.379 0.362 0.38 0.401 0.407 0.398 0.4 0.373 0.401 0.438 0.435 0.452 0.438 0.45 0.439

Table 7.1 Comparison of ANN predicted values of kerf width using L-M algorithm, regression model and experimental values

128

Comparison of kerf width from regression model and ANN model 0.5 0.45

Kerf width in mm

0.4 0.35 0.3 0.25 0.2

Expt Regression

0.15

L-M prediction

0.1 0.05 0 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 Row number

Fig 7.17 Comparison of ANN predicted values of kerf width using L-M algorithm, regression model and experimental values

Figure 7.17 shows the comparison graph for the kerf width predicted by ANN with Lavenberg-Marquardt algorithm and regression model. The ANN predicted values are very close to the experimental values of kerf width.

Statistics

------------------------------------------------------------Dataset

N

Mean

SD

SE

-------------------------------------------------------------Expt Regresion L-M

21 21 21

0.34152 0.34057 0.34048

0.0661 0.06039 0.06724

0.01442 0.01318 0.01467

----------------------------------------------------------------------

129

ANOVA

-----------------------------------------------------------Source DoF

Sum of Squares

Mean Square

F Value

P Value

-----------------------------------------------------------Model 2 Error 60

1.40952381E-5 7.04761905E-6 0.250745619 0.00417909365

0.00169

0.99832

-----------------------------------------------------------Table 7.2 Data statistics - kerf width

Above table shows the comparison statistics for the experimental, regression model predicted and ANN with Levenberg-Marquardt algorithm predicted values.

7.8 Comparison of the Ra value obtained from the ANN and regression methods Laser power in watts 800 900 1000 800 900 1000 800 900 1000 800 900 1000 800 900 1000 800 900 1000 800 900

Pressure in bars 6 6 6 6.5 6.5 6.5 7 7 7 7.5 7.5 7.5 6 6 6 6.5 6.5 6.5 7 7

Cutting speed 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3500 3500 3500 3500 3500 3500 3500 3500

Expt 2.97 2.75 2.66 2.89 2.7 2.62 2.86 2.69 2.62 2.85 2.7 2.64 3.01 2.8 2.73 2.92 2.75 2.67 2.89 2.73

Ra in micron Regression 2.37 2.52 2.67 2.45 2.6 2.76 2.52 2.69 2.84 2.62 2.77 2.93 2.39 2.54 2.69 2.47 2.63 2.78 2.56 2.71

Contd.

L-M 2.94 2.759 2.67 2.873 2.697 2.617 2.88 2.695 2.61 2.855 2.688 2.61 3.01 2.81 2.72 2.9 2.74 2.67 2.895 2.73

130

Continuation Table 7.3

1000 800 900 1000 900 1000 800 900 1000 800 900 1000 800 900 1000 800 900 1000 800 900 1000

7 7.5 7.5 7.5 6 6 6.5 6.5 6.5 7 7 7 7.5 7.5 7.5 6 6 6 6.5 6.5 6.5

3500 3500 3500 3500 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4500 4500 4500 4500 4500 4500

2.63 2.87 2.71 2.6 2.67 2.58 2.73 2.61 2.54 2.7 2.58 2.48 2.61 2.53 2.41 2.77 2.62 2.51 2.67 2.55 2.5

2.86 2.64 2.79 2.95 2.57 2.72 2.5 2.65 2.8 2.58 2.73 2.88 2.66 2.81 2.97 2.43 2.59 2.74 2.52 2.67 2.82

2.646 2.88 2.71 2.63 2.685 2.59 2.75 2.619 2.54 2.69 2.56 2.49 2.657 2.5 2.427 2.78 2.59 2.5 2.672 2.557 2.478

Table 7.3 Comparison of ANN predicted Ra values using L-M algorithm, regression model and experimental values

The Ra values obtained from ANN using L-M algorithm, regression method along with the experimental values are given in the table 7.3. Though the deviation is not much in the values obtained from ANN, regression method showed some deviation.

131

3.5

3

Ra vlue in micron

2.5

2

1.5

Ra expt Ra regression Ra L-M

1

0.5

0 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 Row

Fig 7.18 The comparison of Ra values obtained from experiment, regression analysis and ANN with L-M algorithm

Figure 7.18 shows the comparison of Ra values obtained from experiment, regression analysis and ANN with L-M algorithm. It is evident that the ANN predicted values are in good agreement with experimental values. Following table shows the comparison statistics.

Statistics

-----------------------------------------------------------Dataset

N

Mean

SD

SE

-------------------------------------------------------------------Expt Regression L-M

42 42 42

2.69548 2.66531 2.69533

0.13944 0.15632 0.1389

0.02152 0.02412 0.02143

-------------------------------------------------------------

132

ANOVA

------------------------------------------------------------Source DoF

Sum of Squares

Mean Square

F Value

P Value

------------------------------------------------------------Model 1 2 Error 123

0.0253541176 2.59011737

0.0126770588 0.0210578648

0.60201

0.54931

------------------------------------------------------------Table 7.4 Data statistics – Ra value

Fig 7.19 Relative error between the experimental and predicted values

The relative errors between the experimental and ANN predicted values are given in fig.7.19.The maximum and minimum relative errors are 3.08 % and 0.27%.

133

Fig.7.20 Post regression analysis result

The results of post regression analysis are shown in fig. 7.20.The value of R=0.993 indicates that the prediction error is small. The relationship between the kerf width and the process variables can be established using artificial neural network.

The model predictions are in good agreement with

experimental results with very small relative error.

134

7.9Adaptive neuro-fuzzy inference system (ANFIS) Using the experimental knowledge an adaptive neuro-fuzzy inference system is developed. An ANFIS consisting of a set of Tagaki- Sugeno type fuzzy IF-THEN rules is used to map the system inputs to outputs. This hybrid combination enables to deal with both the verbal and the numeric power of intelligent systems. According to the theory of fuzzy systems, different fuzzification and defuzzification with different rule base structures will result in various solutions to a given task. The fuzzy regions are parameterized and each region is associated with a linear subsystem. Owing to the fuzzily defined antecedents, the nonlinear system forms a collection of loosely coupled multiple linear models. The degree of firing of each rule is proportional to the level of contribution of the corresponding linear model to the overall output of the model. For simplicity, the fuzzy inference system under consideration is assumed to have three inputs and one output. The membership function for both the inputs is set to be 15. The ANFIS block diagram is shown in figure7.21 and structure with first-order Sugeno model containing 15 rules is shown in Fig. Laser power, assist gas pressure and cutting speed are used as inputs and Ra value is taken as the output. Gaussian membership functions are used for the inputs. Details of the Sugeno type ANFIS structure is given in the appendix 4.

Fig 7.21Block diagram of the ANFIS system

135

7.9.1 ANFIS structure

Fig 7.22 ANFIS structure

Figure 7.22 shows the ANFIS structure with first-order Sugeno model containing 15 rules. Laser power, cutting speed and assist gas pressure are taken as input parameters. Ra value is taken as output parameter.

7.9.2 Input membership functions A membership function is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. The only condition a membership function must satisfy is that it must vary between 0 and 1. The function itself can be an arbitrary curve whose shape can be define as a function that suits us from the point of view of simplicity, convenience, speed, and efficiency. The domains of the antecedent variables are partitioned by a number of membership functions. The rule base is

136

then established to cover all the combinations of the antecedent terms. In order to obtain an efficient representation with as few rules as possible, the membership functions are placed such that they capture the non-uniform behavior of the system. Gaussian membership functions with product inference rule were used at the fuzzification level. Fifteen fuzzy rules are used in the ANFIS model for laser cutting. The fuzzifier outputs the firing strengths for each rule. The vector of firing strengths is normalized. The resulting vector is defuzzified by utilizing the first-order Sugeno model. In the present study fifteen Gaussian membership functions are used for each of the inputs namely laser power, assist gas pressure and cutting speed are shown in the Fig 7.23.

Fig7.23.1 Membership functions of input variable laser power

137

Fig 7.23.2 Membership functions of input variable pressure

Fig 7.23.3 Membership functions of input variable cutting speed.

138

7.9.3 Fuzzy rules

Fuzzy sets and fuzzy operators are the subjects and verbs of fuzzy logic. These if-then rule statements are used to formulate the conditional statements that comprise fuzzy logic. Based on the experimental knowledge following fifteen rules are used in the design of the ANFIS. 1. If (laserpower is in1mf1) and (pressure is in2mf1) and (cutting_speed is in3mf1) then (Ra is out1mf1) (1) 2. If (laserpower is in1mf2) and (pressure is in2mf2) and (cutting_speed is in3mf2) then (Ra is out1mf2) (1) 3. If (laserpower is in1mf3) and (pressure is in2mf3) and (cutting_speed is in3mf3) then (Ra is out1mf3) (1) 4. If (laserpower is in1mf4) and (pressure is in2mf4) and (cutting_speed is in3mf4) then (Ra is out1mf4) (1) 5. If (laserpower is in1mf5) and (pressure is in2mf5) and (cutting_speed is in3mf5) then (Ra is out1mf5) (1) 6. If (laserpower is in1mf6) and (pressure is in2mf6) and (cutting_speed is in3mf6) then (Ra is out1mf6) (1) 7. If (laserpower is in1mf7) and (pressure is in2mf7) and (cutting_speed is in3mf7) then (Ra is out1mf7) (1) 8. If (laserpower is in1mf8) and (pressure is in2mf8) and (cutting_speed is in3mf8) then (Ra is out1mf8) (1) 9. If (laserpower is in1mf9) and (pressure is in2mf9) and (cutting_speed is in3mf9) then (Ra is out1mf9) (1) 10. If (laserpower is in1mf10) and (pressure is in2mf10) and (cutting_speed is in3mf10) then (Ra is out1mf10) (1) 11. If (laserpower is in1mf11) and (pressure is in2mf11) and (cutting_speed is in3mf11) then (Ra is out1mf11) (1)

139

12. If (laserpower is in1mf12) and (pressure is in2mf12) and (cutting_speed is in3mf12) then (Ra is out1mf12) (1) 13. If (laserpower is in1mf13) and (pressure is in2mf13) and (cutting_speed is in3mf13) then (Ra is out1mf13) (1) 14. If (laserpower is in1mf14) and (pressure is in2mf14) and (cutting_speed is in3mf14) then (Ra is out1mf14) (1) 15. If (laserpower is in1mf15) and (pressure is in2mf15) and (cutting_speed is in3mf15) then (Ra is out1mf15) (1) Fuzzy inference is the process of formulating the mapping from given inputs to an output using fuzzy logic. The mapping then provides a basis from which decisions can be made. The 3D surface plot gives a surface that represents the mapping of input to output. Surface plots can be of great use for evaluating the effect of process variables on the outputs.

Fig 7.24. 3D Surface plot of laser power, assist gas pressure versus Ra value from ANFIS analysis

140

As there are three input variables , two separate surface plots are obtained to depict the relation between the input and output variables

Fig 7.25. 3D Surface plot of laser power, cutting speed versus Ra value from ANFIS analysis

The surface plot shown in the fig 7.24 gives the Ra values at different laser powers and assists gas pressures. It is evident that the Ra value decreases slightly with the increase in assist gas pressure. From the 3D surface plot of Ra value versus laser power and cutting speed shown in fig 7.25 it is evident that the Ra value tends to decrease in the beginning with an increase in the laser power but after attaining a minimum value it again begins to increase with the increase in laser power.

141

7.9.4 Comparison of ANFIS predicted values and experimental Ra values at different laser powers

Figure 7.26 shows the comparison of ANFIS predicted and experimental values of Ra at different laser powers. The cutting speed is kept at 3000mm/minute and the assist gas pressure is kept at 6 bars. It can be seen that there is very little deviation in the predicted and experimental values.

3.7

Ra value in microns

3.5

3.3

3.1 Expt 2.9

ANFIS

2.7

2.5 800

900

1000

1100

1200

1300

1400

Laser power in watts

Fig 7.26 Comparison of ANFIS predicted Ra values at different laser powers with experimental values (Cutting speed = 3000mm/minute, Assist gas pressure =6 bar)

The ANOVA test on the experimental and ANFIS predicted values are shown in the table.

142

------------------------------------------------------------------------------------------Sum of Mean Source DoF Squares Square Value P Value -------------------------------------------------------------------------------------------Model 1 6.97245714E-4 6.97245714E-4 0.09713 0.76065 Error 12 0.0861415750 0.00717846459 ---------------------------------------------------------------------------------------------One-Way ANOVA Summary Statistics ------------------------------------------------------------------------------- -----------Dataset N Mean SD SE --------------------------------------------------------------------------------------------Experimental 7 3.08714 0.35523 0.13427 ANFIS 7 3.06286 0.33125 0.1252 --------------------------------------------------------------------------------------------Table 7.5 Data statistics – Ra value from ANFIS 7.9.5

Comparison of ANFIS predicted Ra values with experimental values at different assist gas pressures.

3 2.95

Ra value in microns

2.9 2.85 2.8

Expt ANFIS

2.75 2.7 2.65 2.6 2.55 6

6.5

7

7.5

Assist gas pressure in bars

Fig 7.27Comparison of ANFIS predicted Ra values at different pressures with experimental values(laser power=900watts,Speed=3000mm/min)

143

Figure 7.27 shows the Anfis predicted and experimental values of Ra at different pressures of assist gas. Laser power and cutting speeds are kept constant. It can be seen that predicted values agree with experimental values with a slight deviation at 7.5 bar.

7.10 EDX analysis The scanning electron microscope combined with the energy dispersive X–ray spectrometer (SEM–EDX) analysis can be applied directly to a material to determine the composition before and after laser cutting. It is a useful non destructive method of analysis of material at surface. In scanning electron microscope, an incident electron may interact inelastically with atoms causing X radiations to be emitted. An electron of sufficient energy may ionize an atom by ejecting an inner shell electron. De excitation produces either an Auger electron or an X-ray photon with energy characteristics for the deionizing transition. The photon is therefore a characteristic for the element ionized. Table shows the composition of mild steel before and after cutting with laser. Composition

Composition after

Before cutting

cutting

Fe %

99.47

99.21

Mn %

0.07

0.54

Si %

0.08

0.12

Ni %

0.1

0.09

P%

0.06

-

S%

0.03

-

Table 7.6 EDAX analysis – composition before and after cutting.

144

Fig.7.28 EDX spectrum of cut kerf of mild steel

Figure 7.28 shows the EDX spectrum of the cut surface. It is a plot of counts per second versus energy in electron volts. EDAX reveals the micro metallurgical changes that occur very near to the surface. From the EDAX spectrum pickup of manganese and silicon can be seen .

145

CHAPTER 8 Conclusions

8.1 Summary and conclusions Laser cutting process is highly localized, non-contact and is devoid of reactional forces. Negligible physical force is exerted by the laser on the work. This makes it useful for any type of material. The quality of cut surface is very good and a very small amount of material is lost in the process. The cut surface being smooth, finish cut quality can be achieved in single process. Sharp angles, small radius rounds and complex curves can be cut with high speed. ANN model for CO2 laser cutting of mild steel has been developed to predict the kerf width and surface finish Ra. The developed model can help to predict the process variables and enable metal cutting process using CO2 laser to run in optimal conditions based on specific objectives and practical constraints. The developed ANN model offers a very good approach for metal cutting process in planning and optimization by integrating experimental and

numerical

knowledge into one system with user friendly interface. Unlike theoretical models no assumptions are made. The output of the ANN can be used for optimization of the cutting process. The model could easily be extended to a more complete system once more knowledge and data are accumulated to predict accurately the metal cutting using CO2 laser. The result of the present work establishes the fact that well designed and trained ANN is a powerful tool for non linear process modeling. ANN models match the experimental results well, and much higher performance relative to the conventional linear regression method has been achieved. The values predicted

146

by ANN can be implemented on a real machine in an industrial production environment. The neural network system was designed and can be integrated into the real-time, deterministic, level of the machine control system. Apart from conventional regression model, three different network architectures were closely studied, and it was demonstrated that the model with LevenbergMarquardt algorithm is best suited for process modeling and to achieve the desired results. With some analysis of the problem prior to network design, an architecture could be created that is well tailored to the application. This architecture is able to meet system requirements with much less training time.

ANN models for predicting kerf width and surface finish Ra were trained based on the limited experimental data. During the experimental verification for the proposed model, the measured surface roughness and kerf width values deviated from the model prediction for certain pressure and laser power values. The beam quality and assist gas purity are the very important factors that determine the output variables. Beam quality deteriorates with the ageing of the laser. In oxygen assisted cutting the exothermal reaction generates additional heat, the exact value of which is difficult to account. Multivariable complexity of the laser cutting process is itself a great challenge to the modeling. The ANN is a powerful optimization tool. Instead of development of mathematical optimization process and subsequent experimental verification of the results of the same, the present work is intended to arrive at the optimal values of the key process parameters through experimental knowledge. The ANN can be used as an on line optimization tool. Some of the input parameters have diverging influence on the output parameters. Narrow kerf is achieved by using a tightly focused beam with lesser power. The same can be achieved by using a higher cutting speed or higher assist gas pressure. The present study

147

revealed that the Ra value in fact initially decreases with the increase in laser power. After reaching a optimum value at particular power it begins to increase again. It is also observed that the increase in assist gas pressure has little effect on the power for which the surface finish is better. EDAX studies show the micro metallurgical changes that occur at the surface. During the laser cutting, the pickup of some elements like manganese and silicon and loss of some elements at the cut surface in laser cutting. As part of the work an adaptive neuro fuzzy inference system (ANFIS) is evolved. This system uses the basic concepts of fuzzy logic and neural network to predict the output parameters. The ANFIS can be integrated into a real time machine to acheive the desired results. The ANFIS model is developed using Sugeno type function and it is applicable to a system with only one output. This has made to narrow down on the surface finish as the key output parameter. The nuero fuzzy model developed is found to be quite effective in predicting the values of surface finish Ra. The laser power which produced a good surface finish, ie least Ra value is found to be 1031watts for cutting 3mm thick mild steel. While ANN model is used to investigate one aspect of the effect of process parameters laser metal cutting, the scanning electron microscopy revealed the other aspect of the process. The laser metal cutting is an efficient, non contact and extremely fast process. But in our study it is revealed that the process also results in transverse cracks. The cracks which are caused by extreme heat generated in the process have even led to flaking in the cut kerf at certain laser powers. However the heat flow in the horizontal flow is found to extremely small. This is revealed by the microstructure analysis which showed no change in the grain structure. The main feature of the laser metal cutting is the extremely narrow kerf width. In spite of using oxygen as an assist gas, it is observed that kerf width is only of the order of 0.2mm to 0.45 mm. The kerf is found to widen

148

when the assist gas pressure is increased, keeping the laser power and cutting speed constant. The kerf width is also dependent upon the cutting speed. At higher speeds the kerf is found to narrow down. The loss of heat along the thickness of the work results in a tapered kerf. A small but significant white layer is observed in the cut surface. A dark overtempered layer was observed immediately below the white layer. This layer, which is due to the extreme temperature released in the oxygen assisted cutting, is found to affect the quality of the cut surface. In this work, all the accumulated knowledge was used to develop a model which when integrated back into the system provides a very useful tool to control the process of metal cutting with CO2 lasers and process planning decisions. The ANN model developed has predicting capability based on what are recognized as key process parameters. An extensive database when added to the system, will be very useful for on-line optimization, as the ANN model will be accurate with more data/knowledge, and the prediction module is updated. Process planning decisions can made based on the updated prediction models. Such a system could also serve as an adaptive control system to achieve the best process performance of a metal cutting system designed using CO2 laser. Conventional regression model is also developed in this work to compare and assess the advantages of the ANN model. This

thesis

provides

a

scientific,

systematic

and

reliable

methodology to study the process variables and process modeling for carbondi-oxide assisted laser cutting of mild steel. Optimization of process parameters is realized using the ANN technique. The methodology developed improves the process planning and provides an insight into the effects of key process parameters without complex theoretical mathematical analysis. The ANN model enables the user of the laser cutting machine for efficient utilization and effective control of the equipment.

149

8.2 Scope for future work

The ANN model can be iterated by extending the database available, and the prediction module can be constantly updated for finer machine control. A more comprehensive model which includes the type of laser used for cutting such as Nd.YAG, chemical oxygen iodine laser (COIL) can be developed. More input parameters like the thickness of the metal and the type of metal being cut and operations can be included for ANN model. A study on white layer thickness and its control can be included in the model. Development of a universal architecture would also facilitate development of effective universal real-time calculation routine that would not have to be independently implemented for each type of laser. ANFIS could also serve as an adaptive control system to achieve the best process performance for metal cutting with carbon-di-oxide laser. The fuzziness in the estimation of the actual power delivered to the work and other parameters are taken care in this model. Investigation on the structural changes which takes place in the metal due to laser material interaction has to be investigated. More investigations of surface cracks and heat affected zone can be done. Effects of process parameters on the white layer thickness can be investigated. The change that might have taken place in the magnetic properties of the metal can be studied.

150

Appendix 1

Learning algorithms of artificial neural networks used in this thesis

Learning is a process by which the free Parameters of a neural network are adapted through a continuing process of stimulation by the environment in which the network is embedded. The type of learning is determined by the manner in which the parameters changes take place. A prescribed set of well- defined rules for the solution of a learning problem is called learning algorithm

Error backpropagation In a multilayer neural network with one output, if the input data pairs are given by

X k , y *k

k

1,2,..............n

The matrix X having input vectors xk in its rows is represented by X the matrix of desired outputs y*k is represented by the column vector y *

X

Nxp

and

N

.

[ X 1 , X 2 .......... ....... X N ]T , y* [ y1 * .......... ..... y * N ]T

Error is the difference between the desired output and the output of the network. This error is used to adjust the weights by minimizing following function

J

1 N 2 ek 2k 1

where ek

y *k

yk

Here summation runs over all neurons in the output layer of the network. This method has the task of continually search for the bottom of cost function in

151

iterative manner Minimization of the cost function J with respect to free parameters of the network leads to so-called Method of Gradient Descent. Wk ( n

J ( n) ,η is called learning rate and it defines the proportion

wk ( n)

1)

of error for weight updating (correction). The learning parameter has a profound impact on the performance of convergence of learning As the network outputs are nonlinear in weights ,the training of a ANN is a non linear optimization problem. The following different methods can be applied for this purpose i)

Gradient descent method

ii)

Levenberg – Marquardt method and

iii)

Quasi Newton method

LINE SEARCH The learning algorithms involve a sequence of steps through weight space. Each of these steps can be considered in two parts. First the direction in which to move and secondly, how far to move in that direction. This direction of search provides minimum of the error function in that direction in weight space This procedure is referred to as line search and it forms the basis for several algorithms which are considerably more powerful than gradient descent. Suppose at step n in some algorithm the current weight vector is n w and we wish to obtain a particular search direction n p through weight space. The minimum along the search direction then gives the next values for the weight vector as Wn E( )

1

Wn E (Wn

n p n -----

(1) where the parameter

is chosen to minimize

p n ) . The line search represents a one dimensional minimization

problem. This minimization can be performed in a number of ways. A simple approach would be to proceed along the search direction in small steps, evaluating the error function at each step (position), and stop when the error starts to increase. It is possible, however, to find much more efficient

152

approaches. This includes the issue that whether local gradient information is preferable in line search. The search direction is towards the minimum is obtained through proper weight adjustments, this involves search through negative direction of gradient information. To apply line search to the problems of error function minimization, we need to choose a suitable search direction at each stage of the algorithm. Note that at minimum of the line search E (Wn

pn )

0

Gives g (Wn 1 )T . pn

0

where g n

E Wn 1

Thus the gradient of the new minimum is orthogonal to the search direction. Choosing successive search directions to the local (negative) gradient directions can lead to the problem of slow learning. The algorithm can then take many steps to converge, even for a quadratic error function. The solution to this problem lies in choosing the successive search directions d n to minimize cost function such that, at each step of the algorithm, the component of the gradient parallel to the previous search direction which has just been made zero, is unaltered. Suppose, we have already performed a line minimization along the direction d n , starting from the point w n , to give the new point w

n+1.

Then at

the point w n+1 , we have g n 1 (W ). pn

0

Now the new search dn is such that g (Wn

gives

1

pn 1 )T . pn

pn 1. p

0

0

that is along the new direction, the component of the gradient parallel to the previous search direction remains zero.

153

Conjugate gradient technique The conjugate Descent algorithms have been derived on the assumption of a quadratic error function with a positive definite Hessian matrix. For a non-quadratic error function, the Hessian matrix will depend on the current weight vector, and so will need to be re-evaluated at each step of the algorithm. Since the evaluation of Hessian is computationally costly for nonlinear neural networks, and since its evaluation would have to be done repeatedly, the use the Hessian is avoided. In fact, it turns out that the search direction and learning rate can be found with out explicit knowledge of H. This leads to the CGA. The conjugate method avoids this problem by incorporating an intricate relationship between direction and gradient vector. The conjugate gradient method is guaranteed to locate the minimum of any quadratic function of N variables in at most N steps. But for non-quadratic function, like as the cost function in M.L.P, the process is iterative rather than N-step, and a criterion for convergence is required. The weight vector of the network is updated in accordance with the rule in (1).In conjugate gradient approach, successive steps in Δwn are chosen such that weight correction at previous step is preserved. A direction for yth weight correction is computed and line minimization in this direction is performed, generating w n+1. Successive weight corrections are constrained to be conjugate to those used previously. Interestingly, this is achieved without inversion of Hessian matrix, when compared with Newton’s method. There exists several techniques, most often gradient is involved. Suppose initial weight correction is p (0)= − g (0). Line minimization in the direction p(0) takes weight vector w(0). Beginning with K = 0, we require subsequent gradient to be orthogonal to the previous weight correction direction, that is from pnT . g n

pn [ g n Wn+2.

i

0, i = 1,2,3---------n,

2

g n 1 ] 0 , where [ g n

2

g n 1] is the change of gradient of E from Wn+1 to

154

Using Taylor’s series, we have

E ( w)

E W

E (Wo)

w wo

1 W 2!

2

E and Hessian H = W

Where Jacobian J

E 2 W

Wo

(W

Wo)

w wo

2

2

E 2 W

W Wi W j

Differentiating with respect to W, we get g ( n)

F

H (W Wo )

Therefore pnT Hpn

1

0

which is called the conjugacy condition. Weight correction direction p n+2 and p

n+1

are said to be conjugate to one another in the context of H, or H-

conjugate. There exist several methods of finding conjugate vector p(n +1). Each search direction vector is then computed as a linear combination of the current gradient vector and the previous direction vector. pn

1

gn

1

n 1. pn ,

a path or sequence of search directions pn; n = 1,2,3,--,n in the weight space, time varying parameter. There are different rules to determine

n

defines n

is a

in terms of gradient

vectors gn and gn+1. Each weight correction pn is computed sequentially with po and is formed as the sum of current gradient direction and scaled version of previous correction.

n 1

pnT Hgn

1

T

pn Hpn

In most of the conjugate gradient algorithms, the step size is adjusted at each iteration, and a search is made along the conjugate gradient direction to determine the step size which minimizes the performance function along that line. There are a number of different search functions. Most widely used search functions are

155

i)

ii)

iii)

Fletcher – Reeves

n 1

Powell-Beale restarts Polak – Ribiere

T 1 .g n 1 g nT .g n

gn

n 1

n 1

gn

T 1 .( g n 1 pnT ( g n 1

gn

T 1 .( g n 1 g nT .g n

gn )

and

gn )

gn )

The conjugate gradient algorithms are usually much faster than variable leaning rate backpropagtaion algorithm, although the results will vary from one problem to another. The conjugate Gradient algorithms require only a little more storage than the simple algorithm, so they are often a good choice for networks with a large number of weights. It is important to note that above three expressions are equivalent provided the error function is exactly quadratic. For nonquadratic error functions, the Polak-Ribiere form is generally found to give slightly better results that the other expressions.

QUASI NEWTON METHODS Using the local quadratic approximation, one can obtain directly an expression for the location of the minimum or more the stationary point of the error function. E (W )

Gives W * W

E (W *)

1 (W W *)T H (W W *) 2

H 1g

The vector –H-1g is called Newton step. Unlike the local gradient vector the Newton location for a quadratic error surface, evaluated at any w , points directly at the minimum of the error function provided H is positive definite. Since the quadratic approximation in above algorithm is not exact, it would be necessary to apply W * W H 1g iteratively with the Hessian being reevaluated at each new search. The Hessian for non-quadratic networks is computationally demanding since it required O(NW2 )steps, where W is the

156

number of weights in the network and N is the number of patterns in the data and O(W3 ) operation for inversion in each iteration. The Newton’s step may move towards a maximum or a saddle point rather than a minimum. This occur if the Hessian is not positive definite. Thus the error is not guaranteed to be reduced at each iteration. One of the drawbacks of Newton’s method is that it requires the analytical derivative of Hessian matrix at each iteration. This is a problem if the derivative is difficult to compute. In such cases it may be convenient to iterate according to Wn

1

Wn

G 1g n , where G is an easily

computed approximation to H. For example, in one dimension, the secant method approximates the derivative with the difference quotient a ( n)

g (Wn 1 ) g (Wn ) Wn 1 Wn

Such an iteration is called a quasi-Newton method. If G is positive definite, as it usually is, an alternative name is variable metric method. One positive advantage to using an approximation in place of H is that G can be chosen to be positive definite, ensuring that the step will not be attracted to a maximum of f when one wants a minimum.

LEVENBERG-MARQUARDT ALGORITHM Levenberg-Marquardt is a trust region based method with hyperspherical trust region. This method works extremely well in practice, and is considered to be most efficient algorithm for training medium sized artificial neural networks. Like Quasi- Newton methods, the Levenberg-Marquardt algorithm was designed to approach second order training speed with out having to compute Hessian matrix. When the performance function has the form of a sum of squares, then we have Wn

H

1

Wn H

1

F (W ) , here the Hessian matrix can be approximated as

JT J

and the gradient can be computed as g

J T e , where the Jacobian matrix J

contains the first derivatives of the network errors with respect to weights and

157

biases and e vector of network errors. The Gauss- Newton update formula is given by Wn

1

Wn

( J nT J n ) 1 J nT en , Where (JTJ) is positive definite. The recursion

equation is Wn

1

Wn

( J nT J n

I ) 1 J nT en . Where

is called learning

parameter. The main disadvantage of this algorithm is that it requires large memory. MATLAB also has a reduced memory version of Levenberg – Marquardt algorithm.

158

Appendix 2 Table 2A.1 showing the ANN predicted values using Levenberg and Marquardt algorithm and experimental values.

Power in watts

800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300

cutting Pressure speed in bars in mm/min 6 6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5 7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5

3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000

Actual values Kerf in Ra in mm micron 0.231 2.97 0.292 2.75 0.332 2.66 0.37 2.92 0.396 3.26 0.421 3.47 0.453 3.58 0.261 2.89 0.299 2.7 0.342 2.62 0.375 2.85 0.4 3.18 0.43 3.36 0.461 3.46 0.3 2.86 0.36 2.69 0.39 2.62 0.403 2.84 0.41 3.14 0.451 3.31 0.477 3.41 0.344 2.85 0.4 2.7 0.412 2.64 0.424 2.86 0.43 3.14 0.466 3.3

Predicted value L-M kerf in Ra in mm micron 0.227 2.94 0.28 2.759 0.332 2.67 0.363 2.91 0.392 3.28 0.424 3.457 0.452 3.59 0.263 2.873 0.304 2.697 0.35 2.617 0.379 2.848 0.406 3.216 0.437 3.374 0.465 3.48 0.295 2.88 0.338 2.695 0.373 2.61 0.393 2.83 0.419 3.16 0.448 3.3 0.473 3.4 0.33 2.855 0.374 2.688 0.405 2.61 0.41 2.84 0.429 3.17 0.455 3.29

159

Continuation table 2A.1

1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800

7.5 6 6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5 7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5 6 6 6 6 6 6 6 6.5

3000 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 4000 4000 4000 4000 4000 4000 4000 4000

0.485 0.222 0.267 0.314 0.362 0.385 0.411 0.433 0.247 0.277 0.323 0.37 0.391 0.421 0.452 0.28 0.31 0.36 0.382 0.395 0.434 0.46 0.32 0.35 0.383 0.401 0.411 0.447 0.478 0.19 0.261 0.301 0.356 0.373 0.398 0.412 0.233

3.38 3.01 2.8 2.73 2.96 3.31 3.5 3.61 2.92 2.75 2.67 2.88 3.34 3.41 3.51 2.89 2.73 2.63 2.85 3.2 3.37 3.47 2.87 2.71 3.6 2.86 3.16 3.34 3.44 2.89 2.67 2.58 2.85 3.17 3.34 3.48 2.73

0.478 0.214 0.268 0.32 0.359 0.38 0.414 0.44 0.25 0.29 0.338 0.374 0.395 0.427 0.455 0.285 0.326 0.36 0.388 0.407 0.435 0.46 0.319 0.362 0.393 0.41 0.42 0.442 0.467 0.191 0.244 0.296 0.335 0.367 0.401 0.423 0.227

3.379 3.01 2.81 2,72 2.92 3.32 3.49 3.61 2.9 2.74 2.67 2.89 3.26 3.4 3.51 2.895 2.73 2.646 2.86 3.2 3.35 3.47 2.88 2.71 2.63 2.86 3.18 3.329 3.419 2.884 2.685 2.59 2.84 3.166 3.326 3.496 2.75

160

Continuation table 2A.1

900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000

6.5 6.5 6.5 6.5 6.5 6.5 7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5 6 6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5 7 7 7

4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500

0.269 0.317 0.362 0.381 0.41 0.435 0.26 0.29 0.34 0.37 0.387 0.42 0.44 0.29 0.31 0.373 0.389 0.4 0.413 0.459 0.151 0.23 0.24 0.3 0.33 0.381 0.398 0.2 0.241 0.267 0.32 0.35 0.39 0.421 0.24 0.27 0.31

2.61 2.54 2.8 3.09 3.23 3,39 2.7 2.58 2.48 2.73 3 3.17 3.32 2.61 2.53 2.41 2.67 2.96 3.05 3.19 2.77 2.62 2.51 2.79 3.06 3.34 3.37 2.67 2.55 2.5 2.73 2.96 3.08 3.24 2.61 2.5 2.43

0.267 0.312 0.348 0.38 0.413 0.432 0.26 0.299 0.33 0.36 0.387 0.419 0.438 0.29 0.335 0.366 0.383 0.4 0.43 0.445 0.16 0.21 0.263 0.299 0.33 0.366 0.395 0.203 0.241 0.285 0.319 0.352 0.386 0.425 0.238 0.27 0.308

2.619 2.54 2,792 3.07 3.22 3.4 2.69 3.56 2.49 2.74 3 3.145 3.31 2.657 2.5 2.427 2.67 2.95 3.06 3.2 2.78 2.59 2.5 2.8 3.08 3.23 3.36 2.672 2.557 2.478 2.737 2.97 3.102 3.22 2.61 2.5 2.42

161

Continuation table 2A.1

1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400

7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5

4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500

0.343 0.37 0.402 0.43 0.281 0.298 0.352 0.361 0.38 0.39 0.44

2.69 2.91 3 3.13 2.6 2.9 2.37 2.63 2.87 2.95 3.03

0.332 0.365 0.4 0.42 0.267 0.316 0.348 0.363 0.38 0.412 0.439

2.687 2.906 3.02 3.13 2.59 2.439 2.35 2.625 2.835 2.94 3.04

162

Table 2A.2 ANN predicted and actual values with conjugate gradient learning rule Power in watts

Pressure in bars

cutting speed in mm/min

800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800

6 6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5 7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5 6

3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3500

Predicted value CG Actual values Kerf width in mm

0.231 0.292 0.332 0.37 0.396 0.421 0.453 0.261 0.299 0.342 0.375 0.4 0.43 0.461 0.3 0.36 0.39 0.403 0.41 0.451 0.477 0.344 0.4 0.412 0.424 0.43 0.466 0.485 0.222

Ra in micron

2.97 2.75 2.66 2.92 3.26 3.47 3.58 2.89 2.7 2.62 2.85 3.18 3.36 3.46 2.86 2.69 2.62 2.84 3.14 3.31 3.41 2.85 2.7 2.64 2.86 3.14 3.3 3.38 3.01

Kerf width in mm

0.31 0.35 0.356 0.38 0.39 0.392 0.396 0.35 0.36 0.361 0.38 0.389 0.39 0.36 0.355 0.353 0.365 0.39 0.401 0.413 0.425 0.355 0.4 0.392 0.422 0.45 0.461 0.389 0.35

Ra in micron

2.99 2.99 3.2 3.21 3.216 3.22 3.37 2.99 3 3.03 3.17 3.21 3.21 2.98 2.99 2.992 3.06 3.2 3.21 3.213 3.27 2.81 2.93 2.97 3 3.19 3.208 3.21 2.99

163

Continuation table 2 A.2

900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400

6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5 7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5 6 6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5

3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000

0.267 0.314 0.362 0.385 0.411 0.433 0.247 0.277 0.323 0.37 0.391 0.421 0.452 0.28 0.31 0.36 0.382 0.395 0.434 0.46 0.32 0.35 0.383 0.401 0.411 0.447 0.478 0.19 0.261 0.301 0.356 0.373 0.398 0.412 0.233 0.269 0.317 0.362 0.381 0.41 0.435

2.8 2.73 2.96 3.31 3.5 3.61 2.92 2.75 2.67 2.88 3.34 3.41 3.51 2.89 2.73 2.63 2.85 3.2 3.37 3.47 2.87 2.71 3.6 2.86 3.16 3.34 3.44 2.89 2.67 2.58 2.85 3.17 3.34 3.48 2.73 2.61 2.54 2.8 3.09 3.23 3,39

0.353 0.359 0.36 0.388 0.389 0.391 0.353 0.351 0.362 0.369 0.389 0.391 0.394 0.362 0.355 0.353 0.361 0.366 0.39 0.392 0.34 0.41 0.369 0.366 0.356 0.38 0.39 0.35 0.357 0.35 0.361 0.358 0.39 0.393 0.343 0.352 0.36 0.363 0.372 0.385 0.39

3 3.07 3.13 3.205 3.21 3.33 2.99 3 3.07 3.17 3.208 3.21 3.218 2.98 2.99 3.01 3..08 3.17 3.21 3.28 2.79 2.92 2.97 2.99 3 3.2 3.21 2.99 3 3 3.07 3.02 3.2 3.21 2.99 2.992 3.01 3.13 3.18 3.19 3.21

164

Continuation table 2 A.2

800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400

7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5 6 6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5 7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5

4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500

0.26 0.29 0.34 0.37 0.387 0.42 0.44 0.29 0.31 0.373 0.389 0.4 0.413 0.459 0.151 0.23 0.24 0.3 0.33 0.381 0.398 0.2 0.241 0.267 0.32 0.35 0.39 0.421 0.24 0.27 0.31 0.343 0.37 0.402 0.43 0.281 0.298 0.352 0.361 0.38 0.39 0.44

2.7 2.58 2.48 2.73 3 3.17 3.32 2.61 2.53 2.41 2.67 2.96 3.05 3.19 2.77 2.62 2.51 2.79 3.06 3.34 3.37 2.67 2.55 2.5 2.73 2.96 3.08 3.24 2.61 2.5 2.43 2.69 2.91 3 3.13 2.6 2.9 2.37 2.63 2.87 2.95 3.03

0.36 0.35 0.361 0.369 0.373 0.39 0.398 0.37 0.42 0.37 0.377 0.381 0.393 0.412 0.281 0.31 0.323 0.36 0.361 0.367 0.388 0.35 0.353 0.36 0.363 0.369 0.37 0.381 0.365 0.35 0.361 0.374 0.36 0.36 0.372 0.384 0.41 0.37 0.362 0.366 0.3 0.35

2.98 2.99 3.03 3.1 3.18 3.21 3.28 2.76 2.91 2.97 2.98 2.99 3.08 3.203 2.99 3 2.98 2.99 3.03 3.13 3.21 2.99 2.99 3 3.02 3.07 3.11 3.19 2.97 2.98 3 3.05 3.07 3.11 3.12 2.743 2.9 2.96 2.98 3 3.13 3.21

165

2A.3ANN predicted value and actual value with Quasi Newton learning algorithm

Power in watts

800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400

Pressure in bars 6 6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5 7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5

cutting speed in mm/min 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000

Predicted value Quasi Newton Actual values Kerf width in mm 0.231 0.292 0.332 0.37 0.396 0.421 0.453 0.261 0.299 0.342 0.375 0.4 0.43 0.461 0.3 0.36 0.39 0.403 0.41 0.451 0.477 0.344 0.4 0.412 0.424 0.43 0.466 0.485

Ra in micron 2.97 2.75 2.66 2.92 3.26 3.47 3.58 2.89 2.7 2.62 2.85 3.18 3.36 3.46 2.86 2.69 2.62 2.84 3.14 3.31 3.41 2.85 2.7 2.64 2.86 3.14 3.3 3.38

Kerf width in mm 0.27 0.28 0.36 0.368 0.37 0.375 0.42 0.32 0.33 0.341 0.4 0.41 0.423 0.432 0.33 0.34 0.36 0.41 0.42 0.43 0.437 0.37 0.379 0.381 0.405 0.43 0.44 0.47

Ra in micron 2.65 2.67 3.12 3.13 3.14 3.16 3.64 2.8 2.81 3.25 3.26 3.28 3.38 3.58 2.81 2.83 2.95 3.26 3.23 3.243 3.3 2.73 2.76 2.77 2.781 3.12 3.27 3.3

166

Continuation table2A.3 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400

6 6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5 7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5 6 6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5

3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000

0.222 0.267 0.314 0.362 0.385 0.411 0.433 0.247 0.277 0.323 0.37 0.391 0.421 0.452 0.28 0.31 0.36 0.382 0.395 0.434 0.46 0.32 0.35 0.383 0.401 0.411 0.447 0.478 0.19 0.261 0.301 0.356 0.373 0.398 0.412 0.233 0.269 0.317 0.362 0.381 0.41 0.435

3.01 2.8 2.73 2.96 3.31 3.5 3.61 2.92 2.75 2.67 2.88 3.34 3.41 3.51 2.89 2.73 2.63 2.85 3.2 3.37 3.47 2.87 2.71 3.6 2.86 3.16 3.34 3.44 2.89 2.67 2.58 2.85 3.17 3.34 3.48 2.73 2.61 2.54 2.8 3.09 3.23 3,39

0.33 0.34 0.346 0.352 0.415 0.421 0.44 0.332 0.334 0.34 0.352 0.41 0.42 0.429 0.327 0.33 0.334 0.34 0.38 0.416 0.42 0.347 0.36 0.353 0.35 0.355 0.44 0.49 0.264 0.318 0.33 0.346 0.415 0.416 0.436 0.26 0.29 0.33 0.34 0.37 0.415 0.42

2.8 2,82 2.818 2.91 3.26 3.27 3.6 2.8 2.81 2.88 2.91 3.26 3.27 3.34 2.79 2.81 2.82 2.828 3.08 3.26 3.29 2.68 2.76 2.783 2.796 2.81 3.23 3.33 2.61 2.76 2.81 2.88 3.263 3.266 3.491 2.598 2.69 2.8 2.82 3.03 3.26 3.32

167

Continuation table2A.3

800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400 800 900 1000 1100 1200 1300 1400

7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5 6 6 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 6.5 6.5 7 7 7 7 7 7 7 7.5 7.5 7.5 7.5 7.5 7.5 7.5

4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4000 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500

0.26 0.29 0.34 0.37 0.387 0.42 0.44 0.29 0.31 0.373 0.389 0.4 0.413 0.459 0.151 0.23 0.24 0.3 0.33 0.381 0.398 0.2 0.241 0.267 0.32 0.35 0.39 0.421 0.24 0.27 0.31 0.343 0.37 0.402 0.43 0.281 0.298 0.352 0.361 0.38 0.39 0.44

2.7 2.58 2.48 2.73 3 3.17 3.32 2.61 2.53 2.41 2.67 2.96 3.05 3.19 2.77 2.62 2.51 2.79 3.06 3.34 3.37 2.67 2.55 2.5 2.73 2.96 3.08 3.24 2.61 2.5 2.43 2.69 2.91 3 3.13 2.6 2.9 2.37 2.63 2.87 2.95 3.03

0.26 0.27 0.326 0.34 0.349 0.4 0.42 0.29 0.284 0.325 0.35 0.36 0.387 0.53 0.257 2.26 0.264 0.317 0.36 0.42 0.42 0.259 0.26 0.261 0.29 0.33 0.39 0.419 0.26 0.25 0.26 0.27 0.33 0.34 0.42 0.28 0.278 0.281 0.283 0.343 0.412 0.51

2.59 2.626 2.789 2.81 2.87 3.187 3.28 2.54 2.569 2.716 2.8 2.82 2.85 3.347 2.59 2.56 2.61 2.76 2.97 3.26 3.41 2.59 2.6 2.63 2.68 2.8 3.15 3.3 2.59 2.6 2.67 2.7 2.79 2.81 3.25 2.55 2.57 2.58 2.6 2.75 2.87 2.97

168

Appendix 3

Regression results for the prediction of kerf width Model Definition: Y = exp(a*x1+b*x2+c*x3+d)

Number of observations = 23 Number of missing observations = 1 Solver type: Nonlinear Nonlinear iteration limit = 250 Diverging nonlinear iteration limit =10 Number of nonlinear iterations performed = 5 Residual tolerance = 0.0000000001 Sum of Residuals = -4.20586237694981E-03 Average Residual = -1.82863581606513E-04 Residual Sum of Squares (Absolute) = 7.79299646908178E-03 Residual Sum of Squares (Relative) = 7.79299646908178E-03 Standard Error of the Estimate = 0.020252350700574 Coefficient of Multiple Determination (R^2) = 0.9166017964 Proportion of Variance Explained = 91.66017964% Adjusted coefficient of multiple determination (Ra^2) = 0.903433659 Durbin-Watson statistic = 1.76523276159829

169

Regression result( continued) Variable

Value 8.926905565 E-04

a

0.102683514 -1.157148523 E-04 -2.297598634

b c d

Standard Error 7.543453701 E-05 2.390193514 E-02 3.383416564 E-05 0.151329182

t-ratio

Prob(t)

11.83397674

0

4.296033514

0.00039

-3.420059283 -15.18278632

0.00287 0

68% Confidence Intervals Variable

Value

Lower Limit 8.156568073 E-04 7.827485825 E-02

-1.157148523

3.455144995 E-05

-1.502663023 E-04

0.1270922 8.1163402 E-05

-2.297598634

0.154537361

-2.452135995

2.1430613

8.926905565 E-04

a

0.102683514

b

Upper Limit 9.6972431 E-04

68% (+/-) 7.703374919 E-05 2.440865616 E-02

E-04 c d

90% Confidence Intervals Variable Value 8.926905565 E-04 a b c d

0.102683514 -1.157148523 E-04 -2.297598634

90% (+/-) 1.304338579 E-04 4.132883605 E-02 5.850265581 E-05 0.261663289

Lower Limit 7.622566986 E-04 6.135467836 E-02 -1.742175081 E-04 -2.559261923

Upper Limit 1.0231244 E-03 0.1440124 -5.7212196 E-05 -2.0359353

170

Continued 95% Confidence Intervals

Variable a b c d

Value 8.926905565 E-04 0.102683514 -1.157148523 E-04 -2.297598634

99% Confidence Intervals Variable Value 8.926905565 E-04 a b c d

0.102683514 -1.157148523 E-04 -2.297598634

95% (+/-) 1.57884486 E-04 5.002675024 E-02 7.081490869 E-05 0.316731978

Lower Limit 7.348060706 E-04 5.265676417 E-02 -1.86529761 E-04 -2.614330612

Upper Limit 1.050575042 E-03

99% (+/-) 2.158106669 E-04 6.838104624 E-02 9.679616448 E-05 0.432937657

Lower Limit 6.768798896 E-04 3.430246817 E-02 -2.125110168 E-04 -2.730536291

Upper Limit 1.108501223 E-03

0.152710265 -4.489994361 E-05 -1.980866656

0.171064561 -1.891868782 E-05 -1.864660977

171

Appendix 4

Regression results for Ra value prediction: Model Definition: Y = a*x1+b*x2+c*x3

Number of observations = 97 Number of missing observations = 0 Solver type: Nonlinear Nonlinear iteration limit = 250 Diverging nonlinear iteration limit =10 Number of nonlinear iterations performed = 1 Residual tolerance = 0.0000000001 Sum of Residuals = 1.05088903015429 Average Residual = 1.08339075273638E-02 Residual Sum of Squares (Absolute) = 16.8590315317432 Residual Sum of Squares (Relative) = 16.8590315317432 Standard Error of the Estimate = 0.423498995599593 Coefficient of Multiple Determination (R^2) = 0.0861488353 Proportion of Variance Explained = 8.61488353% Adjusted coefficient of multiple determination (Ra^2) = 0.0667051935 Durbin-Watson statistic = 1.64874554275951

172

Variable

Value

a

1.295527281 E-03

2.032358472 E-04

6.3745018

0

b

0.212862211

4.371774849 E-02

4.8690113

0

c

2.391573232 E-05

6.869864065 E-05

0.3481253

0.72852

68% Confidence Intervals Variable Value

68% (+/-)

Lower Limit

Upper Limit

a

1.295527281 E-03

2.031749 E-04

1.092352404 E-03

1.498702157 E-03

b

0.212862211

4.370463 E-02

0.169157578

0.256566844

c

2.391573232 E-05

6.867803 E-05

-4.476229874 E-05

9.259376337 E-05

90% Confidence Intervals Variable Value

Standard Error

90% (+/-)

t-ratio

Lower Limit

Prob(t)

Upper Limit

a

1.295527281 E-03

3.376154 E-04

9.579118915 E-04

1.63314267 E-03

b

0.212862211

7.262392 E-02

0.140238287

0.285486135

c

2.391573232 E-05

1.141222 E-04

-9.020644953 E-05

1.380379142 E-04

95% Confidence Intervals Variable Value

95% (+/-)

Lower Limit

Upper Limit

a

1.295527281 E-03

4.035248 E-04

8.920025063 E-04

1.699052055 E-03

b

0.212862211

8.680159 E-02

0.126060621

0.299663801

c

2.391573232 E-05

1.364012 E-04

-1.124854187 E-04

1.603168833 E-04

173

99% Confidence Intervals 99% (+/-) Lower Limit

Variable

Value

Upper Limit

a

1.295527281 E-03

5.343273658 E-04

7.61199915 E-04

1.829854647 E-03

b

0.212862211

0.114938333

9.792387831 E-02

0.327800543

c

2.391573232 E-05

1.806155961 E-04

-1.566998638 E-04

2.045313284 E-04

Sum of Squares 1.589302489 16.85903153 18.44833402

Mean Square 0.794651244 0.179351399

Variance Analysis Source Regression Error Total

DF 2 94 96

F Ratio 4.430694423

174

Appendix 5 Weights used in ANN designed using Lavenberg-Marquardt algorithm 26.0636 -4.5909 -12.848 -7.4699 -21.509 -0.0023 0.1229 -0.0334 0.4732 -0.7451 0.6823 -0.0079 -0.0511 -0.0033 -0.0536 0.0454 0.5617 0.0069 -0.1997 0.0026 0.0537 -0.0014 -0.1248 -0.0227 -0.1834 0.0342 0.4517 -0.0062 0.0429 -0.1069 3.5435

0.0145 0.0004 0.0079 -0.0057 -0.0012 -0.0001 0.0001 0.0017 -0.009 -0.0005 0.001 -23.86 0.0004 -10.658 0.0128 1.1895 -0.0005 -26.748 0.0144 12.6863 -0.0019 10.3196 0.0081 -6.2993 0.1101 0.1941 -0.1467 3.6851 0.8169 -0.5369 -2.063 1.9051 3.925 0.0291 0.3423 2.5793 0.0011 0.0001 0.0001 0.0011 -0.0022 -0.0018 0.0052

175

Appendix 6 Details of ANFIS designed to predict the surface roughness: 1. Name sug31 2. Type sugeno 3. Inputs/Outputs [3 1] 4. NumInputMFs [15 15 15] 5. NumOutputMFs 15 6. NumRules 15 7. AndMethod prod 8. OrMethod probor 9. ImpMethod min 10. AggMethod max 11. DefuzzMethod wtaver 12. InLabels laserpower 13. pressure 14. cutting_speed 15. OutLabels Ra 16. InRange [800 1400] 17. [6 7.5] 18. [3000 4500] 19. OutRange [2.58 3.37] 20. InMFLabels in1mf1 21. in1mf2 22. in1mf3 23. in1mf4 24. in1mf5 25. in1mf6 26. in1mf7 27. in1mf8 28. in1mf9 29. in1mf10 30. in1mf11 31. in1mf12 32. in1mf13 33. in1mf14 34. in1mf15 35. in2mf1 36. in2mf2 37. in2mf3 38. in2mf4 39. in2mf5 40. in2mf6 41. in2mf7

176

42. in2mf8 43. in2mf9 44. in2mf10 45. in2mf11 46. in2mf12 47. in2mf13 48. in2mf14 49. in2mf15 50. in3mf1 51. in3mf2 52. in3mf3 53. in3mf4 54. in3mf5 55. in3mf6 56. in3mf7 57. in3mf8 58. in3mf9 59. in3mf10 60. in3mf11 61. in3mf12 62. in3mf13 63. in3mf14 64. in3mf15 65. OutMFLabels out1mf1 66. out1mf2 67. out1mf3 68. out1mf4 69. out1mf5 70. out1mf6 71. out1mf7 72. out1mf8 73. out1mf9 74. out1mf10 75. out1mf11 76. out1mf12 77. out1mf13 78. out1mf14 79. out1mf15 80. InMFTypes gaussmf 81. gaussmf 82. gaussmf 83. gaussmf 84. gaussmf 85. gaussmf

177

86. gaussmf 87. gaussmf 88. gaussmf 89. gaussmf 90. gaussmf 91. gaussmf 92. gaussmf 93. gaussmf 94. gaussmf 95. gaussmf 96. gaussmf 97. gaussmf 98. gaussmf 99. gaussmf 100. gaussmf 101. gaussmf 102. gaussmf 103. gaussmf 104. gaussmf 105. gaussmf 106. gaussmf 107. gaussmf 108. gaussmf 109. gaussmf 110. gaussmf 111. gaussmf 112. gaussmf 113. gaussmf 114. gaussmf 115. gaussmf 116. gaussmf 117. gaussmf 118. gaussmf 119. gaussmf 120. gaussmf 121. gaussmf 122. gaussmf 123. gaussmf 124. gaussmf 125. OutMFTypes linear 126. linear 127. linear 128. linear 129. linear

178

130. linear 131. linear 132. linear 133. linear 134. linear 135. linear 136. linear 137. linear 138. linear 139. linear 140. InMFParams [95.46 900 0 0] 141. [95.46 1000 0 0] 142. [95.46 1300 0 0] 143. [95.46 800 0 0] 144. [95.46 800 0 0] 145. [95.46 1300 0 0] 146. [95.46 1400 0 0] 147. [95.46 1000 0 0] 148. [95.46 1200 0 0] 149. [95.46 800 0 0] 150. [95.46 1200 0 0] 151. [95.46 1400 0 0] 152. [95.46 800 0 0] 153. [95.46 1100 0 0] 154. [95.46 900 0 0] 155. [0.2386 6.5 0 0] 156. [0.2386 6 0 0] 157. [0.2386 7 0 0] 158. [0.2386 6 0 0] 159. [0.2386 7 0 0] 160. [0.2386 6 0 0] 161. [0.2386 7.5 0 0] 162. [0.2386 6 0 0] 163. [0.2386 7.5 0 0] 164. [0.2386 7.5 0 0] 165. [0.2386 6 0 0] 166. [0.2386 7 0 0] 167. [0.2386 6 0 0] 168. [0.2386 6 0 0] 169. [0.2386 6.5 0 0] 170. [238.6 3000 0 0] 171. [238.6 3000 0 0] 172. [238.6 3500 0 0] 173. [238.6 3000 0 0]

179

174. [238.6 3000 0 0] 175. [238.6 4000 0 0] 176. [238.6 4500 0 0] 177. [238.6 4000 0 0] 178. [238.6 4000 0 0] 179. [238.6 3500 0 0] 180. [238.6 3500 0 0] 181. [238.6 4000 0 0] 182. [238.6 3500 0 0] 183. [238.6 3000 0 0] 184. [238.6 3500 0 0] 185. OutMFParams [-0.002178 0 0.001562 0] 186. [0 0 0.0007354 0] 187. [0.001858 0 0.0002831 0] 188. [0.001124 0 0.0007349 0] 189. [-0.001687 0 0.001405 0] 190. [0 0 0.0008261 0] 191. [0 0 0.0006706 0] 192. [0 0 0.000636 0] 193. [0 0 0.0007391 0] 194. [0 0 0.0008233 0] 195. [0 0 0.0009952 0] 196. [0 0 0.0008265 0] 197. [0 0 0.0008552 0] 198. [0 2.491 -0.00387 0] 199. [0 0 0.0007555 0] 200. Rule Antecedent [1 1 1] 201. [2 2 2] 202. [3 3 3] 203. [4 4 4] 204. [5 5 5] 205. [6 6 6] 206. [7 7 7] 207. [8 8 8] 208. [9 9 9] 209. [10 10 10] 210. [11 11 11] 211. [12 12 12] 212. [13 13 13] 213. [14 14 14] 214. [15 15 15] 200. Rule Consequent 1 201. 2 202. 3

180

203. 4 204. 5 205. 6 206. 7 207. 8 208. 9 209. 10 210. 11 211. 12 212. 13 213. 14 214. 15 200. Rule Weigth 1 201. 1 202. 1 203. 1 204. 1 205. 1 206. 1 207. 1 208. 1 209. 1 210. 1 211. 1 212. 1 213. 1 214. 1 200. Rule Connection 1 201. 1 202. 1 203. 1 204. 1 205. 1 206. 1 207. 1 208. 1 209. 1 210. 1 211. 1 212. 1 213. 1 214. 1

181

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VITAE

K.J.Muralidhara was born in Mysore, in May, 1958. He received his B.Sc. degree in 1979 and M.Sc. degree in 1981 in Solid state Physics from Mysore university. After Working for a brief period as research assistant in Indian Institute of Science, Bangalore, he took up the job of lecturer in Physics at Sarada Vilas College, Mysore. He continues to work in the same college as Professor and head of the department of Physics. He is engaged in the inter disciplinary research work since 2004.

188

List of Papers presented at International conferences. 1 Ranganth B J , Muralidhara K J , Viswanath G , ‘ A Study on Characteristics of Machining with CO2 Laser’

Proceedings of

International conference on recent advances in material processing technology , National Engg College, Kovilpatti, (India) 23-25 Feb 2005. PP 585-591, Allied Publishers, New Delhi.

List of Papers presented at International conferences. 1 Ranganth B J , Muralidhara K J , Viswanath G , ‘ A Study on Machining with CO2 Laser’ Proceedings of 19th National conference of production engineers, Institute of Engineers (I) ,Mysore,25-26oct 2004, pp 31-34. 2

Ranganth B J , Muralidhara K J , Viswanath G , ‘ A Study on process variables in metal cutting CO2 Laser’

Proceedings of national

conference on emerging trnds in thermal engineering,competitive manufacturing and management, St Joseph College of Engg, Chennai, 24-25 Jan 2005. 3 Ranganth B J , Muralidhara K J , Viswanath G , ‘Laser – an emerging Technology for modern machining’’ Proceedings national conference on emerging technology in advanced manufacturing , KLN College of Engg, Madurai, 4th -5th Feb 2005. (CD) 4

Ranganth B J, Muralidhara K.J , ‘Optimizing Process Parameters in Laser Machining of mild steel using Artificial Neural network model’ Proceedings of national conference on Recent advances in Mechanical Engg, PA College of Engg , Mangalore (India) , 1st-3rd Dec 2005. PP71.

189

5 Dr Ranganath B.J and Muralidhara K.J, “On the applications of Artificial neural network for laser cut surface conference

on

emerging trends in

Engineering sciences, JSS college,

analysis”, National

Physics, Electronics and Ooty Road Mysore,25&26

September 2006,pp233-236, Allied Publishers, New Delhi.

List of papers published on this thesis National 1. B.J.Ranganath, K.J.Muralidhara,&G.Viswanath, ‘A Study of the Characteristics of Machining with CO2 Laser’ Manufacturing Technology Today(NICMAP),vol-4.Issue- 6 pp3-7 (2005). International 1. B.J.Ranganath, K.J.Muralidhara.‘Artificial Neural Network for predicting kerf width in oxygen assisted CO2 laser cutting of mild steel’ JUSPS.Vol18(1),pp5-8(2006). 2. B.J.Ranganath, K.J.Muralidhara, ‘A study on the effect of process parameters on surface quality and material removal in oxygen assisted co2 laser cutting of mildsteel’, JUSPS.Vol20(2)M,pp551-555(2008). 3. Ranganth B J , Muralidhara K J , Viswanath G ,“A Study on Characteristics of Machining with CO2 Laser”, Proceedings of International conference on recent advances in material processing technology , National Engg College, Kovilpatti, (India) 23-25 Feb 2005. PP 585-591, Allied Publishers, New Delhi.

190