In this work, nonlinear and multi-objective mathematical models were developed to determine the process parameters corresponding to optimum weld pool geometry. The objectives of the developed mathematical models are to maximize tensile load (TL), penetration (P), area of penetration (AP) and/or minimize heat affected zone (HAZ), upper width (UW) and upper height (UH) depending upon the requirements.Razvijeni su nelinearni i multi-objektni matematički modeli da bi odredili parametre s optimalnom geometrijom zavarivačke kupke. Cilj razvijanja matematičkih modela je postići maksimalna vlačna čvrstoća, penetracija, područje pretaljivanja i/ili minimalna zona utjecaja topline, širina i nadvišenje zavara ovisno o postavljenim zahtjevima

A. Kokangul

M. Bayramoglu

N. Geren

U. Esme

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U. ESME, A. KOKANGUL, M. BAYRAMOGLU, N. GERENMATHEMATICAL MODELING FOR PREDICTIONAND OPTIMIZATION OF TIG WELDING POOL GEOMETRYReceived – Prispjelo: 2008-06-20Accepted – Prihva}eno: 2008-09-27Preliminary Note – Prethodno priop}enjeINTRODUCTIONTIG welding is an arc-welding process that producescoalescence of metals by heating them with an arc be-tween a non-consumable tungsten electrode and the basemetal 1. Many delicate components in aircraft and nu-clear reactors are TIG welded due to its reliability. Ba-sically, TIG weld quality is strongly characterized by theweld pool geometry as shown in Figure 1. This is becausethe weld pool geometry plays an important role in deter-mining the mechanical properties of the weld 2.TIG welding is a highly non-linear, strongly cou-pled, multivariable process 3, 4, 5. The weld pool ge-ometry and, hence, the quality of TIG welded joints aregreatly dependent on the selection of input control vari-ables such as welding speed (V), welding current (I),shielding gas flow rate (F) and gap distance (G). There-fore, in the TIG welding, engineers often face with theproblem of selecting appropriate and optimum combi-nations of input control variables for the required weldpool quality.In this work, nonlinear and multi-objective mathe-matical models are developed for the selection of the op-timum processes parameters. First, the upper and lowerlimits of the input control variables are obtained and theeffect of the input control variables on the weld poolquality parameters is determined. Then, the mathemati-cal relationships between the input control variables andweld pool quality parameters are obtained. These rela-tionships are considered as objective functions in themathematical models.To the best of our knowledge the optimization prob-lem of the TIG welding using nonlinear andmulti-objective mathematical models has not been in-vestigated previously and applied on real life case studylike in this work.A SYSTEMATIC APPROACHDevelopment of a systematic approach is required toobtain optimum combinations of input control variablesfor the required weld pool quality system. This approachincludes the following steps.I) Identify the process control variables and theirupper and lower limits,METALURGIJA 48 (2009) 2, 109-112 109In this work, nonlinear and multi-objective mathematical models were developed to determine the process pa-rameters corresponding to optimum weld pool geometry. The objectives of the developed mathematical mo-dels are to maximize tensile load (TL), penetration (P), area of penetration (AP) and/or minimize heat affectedzone (HAZ), upper width (UW) and upper height (UH) depending upon the requirements.Key words: TIG welding, nonlinear mathematical models, central composite design.Matemati~ko modeliranje u svrhu predvi|anja i optimiranja zavariva~ke kupke kod TIG zavari-vanja. Razvijeni su nelinearni i multi-objektni matemati~ki modeli da bi odredili parametre s optimalnom geome-trijom zavariva~ke kupke. Cilj razvijanja matemati~kih modela je posti}i maksimalna vla~na ~vrsto}a, penetracija,podru~je pretaljivanja i/ili minimalna zona utjecaja topline, {irina i nadvi{enje zavara ovisno o postavljenim zahtje-vima.Klju~ne rije~i: TIG zavarivanje, nelinearno matemati~ko programiranje, centralno kompozitni pokusISSN 0543-5846METABK 48(2) 109-112 (2009)UDC – UDK 621.791:519.853=111U. Esme, Technical Education Faculty, Department of Mechanical Edu-cation, Mersin University, Tarsus; A. Kokangul, Department of Mechani-cal and Industrial Engineering, Cukurova University, Adana; M.Bayramoglu, N. Geren, Department of Mechanical Engineering,Cukurova University, Adana, TurkeyFigure 1. Weld pool geometryII) Identify the quality parameters,III) Construct mathematical models,IV) Develop a design matrix ,V) Conduct experiments,VI) Obtain mathematical relationships,VII) Apply the constructed models.Input process control variablesThe independently controllable parameters affectingweld pool geometry and the quality of the weld pool V,I, F and G were selected as input control variables.Weld pool quality parametersIt is possible to present the quality of welding geom-etry with the TL, P, AP, HAZ, UW and UH. These pa-rameters are important weld quality parameters and allof them are considered in this study.Constructing mathematical modelsThe engineer would like to determine the level of in-put control variables according to the only one weldpool quality parameter such as maximizing TL. In thefirst nonlinear model, as seen below, let us maximize TLunder the upper and lover limit of the input control vari-ables indicated with “U” and “L” indices, respectively.Maximizing TL (1)Constraints:VLVVU (2)IL IIU (3)FLFFU (4)GLGGU (5)By the same way four models are constructed for therest of the weld pool quality parameters.In same cases the engineer would like to consider afew objectives simultaneously. Let’s in the next modelmaximize TL and minimize HAZ simultaneously. Toconstruct the necessary multi-objective model the fol-lowing procedures are applied:Step 1. Find the maximum level of TL (TLopt) underthe (2-5) constraints.Step 2. Find the minimum level of HAZ (HAZopt) un-der the (2-5) constraints.Step 3. In addition to the (2-5) constraints add the fol-lowing constraints.TLTLopt (6)HAZHAZopt (7)It is not expected that all of these constraints be satis-fied simultaneously. The right hand sides are variableconstraints with flexibility, which are managerial goalsto be approached as closely as possible. To this end, con-straints (6) and (7) are equated as follows:TL+ S S1 1  = TLopt (8)HAZ + S S2 2  = HAZopt (9)where S1 and S2 represent the positive components, S1and S2 represent the negative components and S1 , S1 ,S2 , S2 0  .Step 4. It is aimed to approach TLopt and HAZopt withthe same percent as closely as possible in which case theobjective function is not measured in a common unit. Toensure this the following constraint is added:(HAZopt) S1= (TLopt )S2 (10)The objective function is:Min S1 S2 (11)The optimization problem aims to find the optimalvalue of the multi-objective function (11) under (2-5),(8), (9) and (10) constraints. By the same way, for eachof the other combination of objectives a multi-objectivemathematical model can be constructed. In order to ap-ply the constructed optimization models, a design ma-trix must be developed.Development of the design matrixThe design matrix should be depending on the upperand the lower limits of the predetermined input controlvariables. The selected design matrix is a four-level,four-factor, central composite rotatable response sur-face design consisting of 90 sets of design matrix. Itcomprises response surface design (RSD) plus 18 centerpoints. All welding variables at their intermediate level(0) constitute the center points. The upper limit of a vari-able was coded as +2 and the lower limit as –2. Thecoded values for intermediate values were calculatedfrom the rotatable central composite design of DesignExpert 6.0 as given in Table 1.Table 1. Lower and upper limits of factorsWeldingparametersMin.value (-2)Max. va-lue (+2)Low level(-1)High le-vel (+1)Travel speed,V (mm/s)1,07 3,55 1,69 2,93Current, I(A) 20 150 52 117Gas flowrate, F(l/min)8 12 9 11Gap distance,G(mm)1 4 1.75 3.25For each of determined combination of the inputcontrol variables (V, I, F and G) perform the TIG weld-ing and determine the value of the quality parameters us-ing conducting experiments.Conducting experimentsThe experimental set up was designed and con-structed to control the linear movement of the torchalong the weld pad center line. The experiments wereconducted according to the design matrix at random or-der to avoid systematic errors infiltrating the system.110 METALURGIJA 48 (2009) 2, 109-112U. ESME et al.: MATHEMATICAL MODELING FOR PREDICTION AND OPTIMIZATION OF TIG WELDING POOL...Weld pools were laid on the joint to join thin stainlesssteel plate with the experimental setup. AISI type 304stainless steel plates of 1,2 mm thicknesses were used asa workpiece material. The specimens were joined usinga single pass welding with AWS A 5.12-80 EW Th-2thoriated (red color code) tungsten electrode with 1,6mm diameter and argon as shielding gas. The chemicalcompositions of the used work-piece material obtainedfrom spectra analysis are given in Table 2. The weldedjoints were sectioned to produce specimens for examin-ing the quality parameters (UW, UH, P and AP) of weldpool shape in the welded specimens.Table 2. Composition (%) of used AISI 304 steelC Mn P S Si Cr Ni Cu0,08 2,0 0,04 0,03 1,0 19 10,5 0,02These specimens were prepared by the usual metal-lurgical polishing methods and etched with Marble’setching reagent (CuSO4 + HCl + H2SO4). Macrographswere then taken for each cross section using stereo mi-croscope with 50X lens. In macro examinations of thespecimens, MOTIC stereo microscope with image cap-ture device mounted on top of the lens section of the mi-croscope was used.The weld pool profile was outlined by using Im-age-pro Plus 4.5 and NIH ImageJ software. The spatialcalibrations were made on the macrographs before themeasurement. The line drawings of the pool profileswere then used to take measurements on UW, UH, P, APand HAZ. Tensile load values were recorded from ten-sile testing of the specimens prepared in accordancewith the EN 895 Standard. Tensile test specimens weretaken from the weld bead according to the transversetensile test method. Tensile test specimens were pre-pared in such a way that the weld zones were centered inthe gage length. At the same time, heat affected zonewas placed in the gage length perpendicular to the weld.Mathematical relationshipsThe suitable mathematical relationship such as a sec-ond-degree response surface quadratic model (seen be-low) should be selected for each of the process qualityparameters according to the experimental results.Y = b0 + b1V + b2I + b3F + b4G + b11V2 + b22I2 + b33F2 +b44G2 + b12VI + b13VF +b14VG + b23IF + b24IG + b34FGwhere “b” values are the coefficients of the models.These values can be calculated using Design Expert 6.0software.Application of the systematic approachThe working range is decided upon by inspecting theweld pool for a smooth appearance without any visibledefects such as surface porosity and undercut. The com-binations of the input control experimental runs for eachof the 90 combination of the input control variables, andfor each of the combination value of the weld quality pa-rameters are obtained and used. Thus, these will allow tothe estimate the effects of the input control variables onthe weld pool quality parameters mathematically. Thebest mathematical relationship obtained for weld poolquality parameters (HAZ, TL, UW, P, UH and AP) arerepresented in Table 3.The obtained mathematical relationships were testedindividually using ANOVA analysis. The test results forHAZ are as follows: The value of the multiple coefficientof R2 is obtained as 0,92, which means that the explana-tory variables explain 92 % of the variability in responsevariable. Adjusted R-square is generally the best indica-tor of the fit quality and it is obtained as 0,90. The test re-sults of the obtained mathematical relationships showthat the model fits well to the observations.The constructed nonlinear mathematical model(maximizing TL (1) under 2-5 constraints) can be con-structed as follows using the determined upper andlower limits of the input control variables (as seen inFigure 1).Maximizing TL,1,07V3,55 (12)20I150 (13)8F12 (14)1G4 (15)METALURGIJA 48 (2009) 2, 109-112 111U. ESME et al.: MATHEMATICAL MODELING FOR PREDICTION AND OPTIMIZATION OF TIG WELDING POOL...Table 3. The mathematical relationships for process quality parametersHAZ=4,2573-2,2532V+0,0781I+0,027766F + 0,1975G+0,3520V2-0,000124I2 – 0,001384F2-0,021331G2-0,006771VI–0,001035VF+0,000108VG-0,000127IF+0,001067IG-0,004653FGTL=9,80665+238,03487V+8,1026I+25,05345F+4,03510G-75,10057V2-0,039932I2-1,78092F2-7,09131G2-0,059916VI+3,34043VF+0,73102VG-0,041339IF+0,04571IG+1,17361FGUW=3,30265+0,44806V+0,089617I+0,20048F+0,074331G-0,21720V2–0,000128I2-0,015592F2–0,004656G2-0,011222VI+0,008071VF-0,027623VG+0,000131IF+ 0,000763IG+0,010625FGUH=0,083160-0,14708V+0,004271I+0,026577F-0,039542G+0,029239V2–0,000001237I2-0,00090246F2+0,0029819G2-0,00076736VI-0,0011008VF+0,00603239VG-0,00008549IF– 0,0001254IG+0,001736FGP=0,64397+0,066087V+0,006967I+0,018256F+0,087894G-0,019590V2–0,000039I2-0,000551F2-0,014174G2+0,000502VI-0,003323VF-0,020984VG+0,000052IF+ 0,000333IG-0,001458FGAP=3,94256-0,19505V+0,084155I+0,21323F+0,46969G-0,019710V2–0,000126I2-0,012660F2-0,10079G2-0,013011VI-0,020638VF-0,048323VG+0,0005887IF+0,000453IG +0,012111FGThis kind of mathematical models can be solved us-ing optimization software such as LINGO 8.0,MATLAB 7.0. The obtained global optimal solution isgiven in Table 4.In addition, by the same way for the other quality pa-rameters five nonlinear models are constructed andsolved under the same constraints (12-15). Their resultsare shown in Table 4. In case of the engineer aims tomaximize TL and minimize HAZ simultaneously underthe same constraints, the constructed multi-objectivemathematical model (minimizing Eq. 11 under the con-straints of 2-5, 8, 9 and 10 ) will be as follows: As seenin table 4, the TLopt is obtained as 12963,21 N, andHAZopt as 1,92 mm. Therefore, the objective function(11) and the constraint 8, 9 and 10 will be as follows:Min S S1 2  (16)TL+ S S1 1  =12963,21 N (17)HAZ+ S S2 2  =1,92 mm (18)1,92 S1 =12963,21 S2 (19)For this multi-objective mathematical model the globaloptimal solution is given in Table 4. By the same way asseen in Table 4, five multi objective models are con-structed and solved. When the engineer considers morethan one objective, then the optimum values will be de-creased compared to the optimal values obtained usingone objective model. The engineer can then easily findthe optimal level of input control variables using the re-sults given in Table 4 according to the objectives.CONCLUSIONSA systematic approach has been developed and em-ployed in this study for the optimization problem of theTIG welding process parameters. The mathematical re-lationships between input control variables and weldpool quality parameters are obtained using the results ofexperiments. Six nonlinear and five multi-objectivemathematical models are constructed and solved underthe predetermined limits of the input control variablesusing the obtained mathematical relationships as objec-tive functions.This developed systematic approach can also beadopted for other type of arc welding processes.REFERENCES1 S.C. Juang, Y.S. Tarng, H.R. Lii, A comparison between theback propagation and counter-propagation networks in themodeling of the TIG welding process, J. Mater. Process.Technol. 75 (1998) 54–62.2 S.C. Juang, Y.S. Tarng, Process parameter selection for op-timizing the weld pool geometry in the tungsten inert gaswelding of stainless steel, J. Mater. Process. Technol., 122(2002) 33–37.3 Y.M. Zhang, R. Kovacevic, L. Li, Characterization and re-al-time measurement of geometrical appearance of the weldpool, Int. J.Mach. Tools Manuf., 36 (7) (1996) 799–816.4 Y.S. Tarng, H.L. Tsai, S.S. Yeh, Modeling, optimizationand classification of weld quality in TIG welding, Int. J.Mach. Tools Manuf. 39 (9) (1999) 1427–1438.5 H.B. Cary, Modern Welding Technology, Prentice-Hall,Englewood Cliffs, NJ, 1989.Note: The responsible translator for English Language is U. Esme, Tech-nical Education Faculty, Mersin University, Tarsus, Turkey.112 METALURGIJA 48 (2009) 2, 109-112U. ESME et al.: MATHEMATICAL MODELING FOR PREDICTION AND OPTIMIZATION OF TIG WELDING POOL...Table 4. Results of mathematical modelsObjective Input process control variables Optimal ValuesVmm/sIAFI/minGmmTLNHAZmmUWmmUHmmPmmAPmm2Max TL 1,73 96,59 8 1 12963,2 6,93 10,02 0,27 1,26 9,67Min HAZ 3,41 20 12 1 8666,5 1,92 3,92 0,04 0,81 3,22Min UW 3,55 20 12 1 8331,0 1,93 3,75 0,05 0,79 3,11Min UH 2,81 20 8 2,04 9721,1 2,21 4,95 0,02 0,86 4,47Max P 1,07 117,3 12 3,07 11688,6 9,42 12,08 0,30 1,34 11,43Max AP 1,07 150 12 1 11036,2 10,27 13,08 0,39 1,21 12,86Multi objective nonlinear modelsMax TLMin HAZ2,41 20 9,41 1 10375,6 2,30 5,15 0,05 0,91 4,81Max TLMin HAZ,UW2,72 20 12 1 9901,0 2,09 4,64 0,03 0,88 4,03Max TLMin HAZ, UW, UH2,99 20 12 1,17 9488,6 2 4,40 0,03 0,85 3,70Max TL, PMin HAZ, UW, UH,2,69 20 8 1,62 9969,5 2,22 5,03 0,03 0,88 4,61Max TL, P, APMin HAZ,UW, UH,2,49 21,02 8 2,19 10162,5 2,45 5,31 0,03 0,90 4,94

Matematičko modeliranje u svrhu predviđanja i optimiranja zavarivačke kupke kod TIG zavarivanja

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