active metal transfer control by utilizing enhanced droplet … · 2016-01-13 · enhanced active...

Introduction

As has been reviewed in the firstpart of this paper (Ref 1) and previousstudies the detaching peak current inconventional pulsed gas metal arcwelding (GMAW-P) needs to be higherthan the spray transition current to

produce desired one drop per pulse(ODPP) transfer Further the ODPPtransfer in conventional GMAW-Placks controllability and robustness(Refs 2ndash5) The original version of theactive metal transfer control method isthus proposed in which the liquiddroplet is effectively excited into oscil-lation by reducing the welding current

from a peak level referred to as the ex-citing peak to the base By applyinganother peak current referred to as thedetaching peak at the moment whenthe droplet is moving toward the weldpool the electromagnetic force andthe downward momentum of the oscil-lating droplet are effectively matchedin phase and the droplet can bedetached by their combined action Asa result the needed detaching currentis reduced and unexpected droplet de-tachment or mass accumulation can beavoided such that the metal transferrobustness is improved (Refs 6 7)

As an evolution to the original activecontrol process the enhanced activedroplet oscillation is proposed in Ref 8by using the current waveform shownin Fig 1 and denoted as Wave1 Thedroplet growing and exciting pulses arenow divided by a base period while theyare coupled together in the original ac-tive metal transfer control It is thismodification that significantly increasesthe amplitude of the excited droplet os-cillation (Ref 8) Active metal transfercontrol which utilizes such enhancedoscillation referred to as the enhancedactive metal transfer control has beenexperimentally studied in the first partof this investigation (Ref 1) by usingWave2 which inherits from Wave1 byinserting a relatively low detachingpulse with a phase delay to the excitingpulse as shown in Fig 2 The enhanceddroplet oscillation enables the needed

WELDING RESEARCH

SEPTEMBER 2014 WELDING JOURNAL 321-s

SUPPLEMENT TO THE WELDING JOURNAL SEPTEMBER 2014Sponsored by the American Welding Society and the Welding Research Council

Active Metal Transfer Control by Utilizing EnhancedDroplet Oscillation Part II Modeling and Analysis

The model suggests that the exciting phase delay and exciting peak duration can be fixedas long as the droplet size is controlled to be relatively small

BY J XIAO G J ZHANG W J ZHANG AND Y M ZHANG

ABSTRACTEnhanced active metal transfer control is experimentally studied in the first part

of this investigation Differing from the original active control the enhanced activemetal transfer control uses a modified current waveform which increases the amplishytude of the excited droplet oscillation and the needed detaching current is thus muchmore reduced The experimental study reveals that strongest droplet oscillation andmaximum enhancement on the droplet detachment require optimal selections ofthree waveform parameters the exciting peak duration and excitingdetaching phasedelay In this study a numerical model on the dynamic droplet oscillation and detachshyment is developed based on the massshyspring system The spring constant and dampshying coefficient in the model are experimentally calibrated Analysis on the effects ofthe key waveform parameters on the droplet oscillationdetachment gives a compreshyhensive understanding on the mechanism of the droplet excitation and detachmentGiven other waveform parameters the optimal value of the three key parameterscan be calculated from the model The accuracy of the model is verified by comparshying the modeling results with the corresponding experimental ones The modeling reshysults suggest that the exciting phase delay and exciting peak duration can be fixed aslong as the droplet size is controlled to be relatively small as desired by the dropspray transfer In addition an analytical model has been established through approxishymations and its adequate accuracy in predicting the optimal detaching phase delayhas also been verified

KEYWORDSbull Enhanced Active Control bull Metal Transfer bull Droplet Oscillation bull Theoretical Model bull MassshySpring System

J XIAO is with the State Key Laboratory of Advanced Welding and Joining Harbin Institute of Technology China and the Institute for Sustainable Manufacturshying University of Kentucky Lexington Ky G J ZHANG is with the State Key Laboratory of Advanced Welding and Joining Harbin Institute of Technology ChinaW J ZHANG and Y M ZHANG (yumingzhangukyedu) are with the Institute for Sustainable Manufacturing and Department of Electrical and Computer Enshygineering University of Kentucky Lexington Ky

Xiao 9-14_Layout 1 81514 344 PM Page 321

detaching current to be further reducedThe lower limit of the detaching currentis determined to be not only muchlower than the spray transition currentbut also significantly lower than that ofthe original active metal transfercontrol The experimental study also in-dicates that the enhanced active metaltransfer control process is sufficientlyrobust (Ref 1)

Figure 3 demonstrates a typicaldroplet excitation and detachment inthe enhanced active metal transfercontrol where Wave2 is used Ig = 80 ATg = 20 ms Ib = 30 A Tp1 = 2 ms Ie =120 A Te = 3 ms Tp2 = 32 ms Id = 125A and Td = 4 ms Tb = 20 ms Ifd = 175A Tfd = 5 ms It can be seen that thedroplet is first elongated by the excit-ing pulse (Frames 2ndash4) and goes intooscillation in the base period called de-taching phase delay (Frames 5ndash7) and

then the droplet is accelerated and de-tached by the detaching pulse of only125 A4 ms (Frames 8ndash13)

To better depict the excited dropletoscillation the following conceptsdefined in the experimental study arerepeated here 1) The moment at whichthe excited droplet reaches itsmaximum elongation is referred to asthe elongation peak moment 2) Themoment at which the droplet changesits moving direction from upward(toward the wire) into downward (awayfrom the wire) during the preoscillationor the main oscillation is referred to asthe oscillation reversing moment Ascan be seen from Fig 2 the followingcurrent waveform parameters need tobe properly selected to first maximizethe droplet oscillation amplitude andthen maximize the consequentenhancement on the droplet

detachment in the enhanced activemetal transfer control

1 Exciting phase delay Tp1 ie thebase duration between the growingand exciting pulse As has beenverified in Ref 8 the droplet may beexcited to a preoscillation during Tp1as long as the growing current is highenough to preelongate the droplet Ifthe exciting pulse is synchronized withthe downward momentum during thepreoscillation the main oscillationafter the exciting pulse can be furtherenhanced If the growing current issufficiently low for example80 A used in the experimental study

the selection of Tp1 will not affect themain droplet oscillation significantlyIn this sense this parameter was notdiscussed in the experimental studybut it will be analyzed in this theoreti-cal study

WELDING RESEARCH

WELDING JOURNAL SEPTEMBER 2014 VOL 93322-s

Fig 1 mdash Current waveform for enhanced droplet oscillation Fig 2 mdash Current waveform for enhanced active metal transfercontrol

Fig 3 mdash Typical metal transfer of enhanced active control 1 ms per frame ER70Sshy608shymm wire 15Lmin argon gas flow 6shymm arc length 6shymm wire extension beadshyonshyplate welding

Fig 4 mdash Illustration of massshyspring model ofdroplet oscillation


2 Exciting peak duration TeExperimental study on the enhanceddroplet oscillation demonstrates thatthere is an optimal exciting peak dura-tion for achieving strongest droplet os-cillation when the exciting peak currentis the same (Ref 8) Exciting peak dura-tion greater than the optimal value isnot recommended because it not onlyreduces the oscillation amplitude butalso increases the heat input

3 Detaching phase delay Tp2 ie thebase duration between the exciting anddetaching pulse This is the most impor-tant parameter for the enhanced activemetal transfer control because it deter-mines if the droplet oscillation can beeffectively utilized In particular if thedetaching pulse starts exactly at the os-cillation reversing moment the result-ant detachingphase delay is called feature detachingphase delay denoted as Tp2 It hasbeen verified that the feature detach-ing phase delay is the optimalselection for full utilization of thedroplet oscillation

Experimental determination of the

optimal value of these parameters istime-consuming and costly such thatit will not be preferred in manufactur-ing Hence a theoretical model on thedynamic droplet oscillationdetachment is needed Such a modelwould be highly appreciated since itenables to predict the criticalwaveform parameters in a cost-effec-tive way Further the model will give adeeper scientific understanding on themechanism of the droplet oscillationand detachment Together with the ex-perimental work conducted in the firstpart of this investigation the theoreti-cal modeling and analysis complete afull study on the enhanced activemetal transfer control

Objective and MethodThe task now is to establish a theo-

retical model on the dynamic dropletoscillation and detachment under thecurrent waveform shown in Fig 2 Themodel will be used to predict the criti-cal waveform parameters in the

enhanced active metal transfer controlwhen other waveform parameters aregiven

1 Optimal exciting phase delay Tp1which is the time interval between thereversing moment of thepreoscillation and the end moment ofthe growing pulse

2 Optimal exciting peak durationTe under which the end moment ofthe exciting pulse is exactly theelongation peak moment

3 Optimal detaching phase delayTp2 which is the time intervalbetween the reversing moment of theexcited droplet oscillation and the endmoment of the exciting pulse

Mass-spring system has been widelyused to model the pendant droplet os-cillation in GMAW under different cur-rent conditions (Refs 9ndash12) It is alsoused in this study to model theenhanced droplet oscillation anddetachment under the waveform as Fig2 shows The modeling is facilitated bythe following assumptions 1) thedroplet shape is symmetric 2) thedroplet motion in the wire redial direc-

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Fig 5 mdash Variation of l as the function ofhalf angle

Fig 6 mdash Illustration of damping coefficientmeasurement

Fig 7 mdash Damping coefficient measuredfrom experiments 1ndash4

Fig 9 mdash Structure of the Simulink programFig 8 mdash Correlation between spring constant and droplet mass


tion is negligible and 3) the physicalproperty of the liquid metal is constant

Figure 4 shows the mass-springmodel for pendant droplet oscillationin GMAW The surface tension acts asthe spring force Fs Since the dropletvolume increases continuously untilits detachment the oscillation systemvaries with time and follows the gov-erning equations

(1)

F(t) = Fem + Fd + Fg (2)

where x represents the droplet masscenter displacement to the wire tip inthe wire axial direction m b and k arethe mass damping coefficient andspring constant respectively F is theaxial force exerted on the droplet in-cluding the electromagnetic force Femthe plasma drag force Fd and thedroplet gravitational force Fg

The droplet mass is proportional to

the wire melting speed (Ref 13)

m(t) = middotvm (t)dt (3)

where represents the mass densityof the wire vm the wire melting speedwhich is the function of the weldingcurrent and wire extension (Ref 13)

vm = C1I(t)+C2 pr le(t)I2(t) (4)

where I represents the welding currentle the wire extension r the wire resis-mx bx kx F t ( )+ + =

WELDING RESEARCH


Fig 10 mdash Modeling results of the dynamic droplet oscillation and detachment A mdash Droplet displacement B mdash resultant detaching forceC mdash droplet gravitational force D mdash electromagnetic force E mdash inertia force F mdash droplet radius

A B

C D

E F


tivity plus C1 and C2 the meltingconstants The first term in the rightside of Equation 4 represents the arcanode heat while the second term repre-sents the wire resistive heat Hencegiven the current waveform the dropletmass can be obtained

The droplet gravitational force Fg isgiven by

Fg(t) = m(t)g = 4frasl3 rd (t)3rg (5)

where rd is the droplet radius and rprimerepresents the density of liquid droplet

The electromagnetic force Fem isgiven by (Ref 14)

(6)

(7)

where1113110m0 is the magnetic permittivityI is the welding current rw the wire ra-dius and θ is the half angle subtendedby the arc root at the center of thedroplet Figure 5rsquos graphicalillustration of Equation 7 shows that ldoes not change significantly when thehalf angle ranges from 90 to 150 degso that the selection of the half anglewould not significantly influence themodeling results and the half angle is fixed at 120 deg in this study Refer-ences 15 and 16 also use constant half

angle to calculate the electromagneticforce for the same reason

The plasma drag force is given by(Ref 15)

(8)

(9)

where Cd is the aerodynamic drag coef-ficient Ap is the area of the drop seenfrom above and rp and vp are the den-sity and velocity of the arc plasmaSince the plasma velocity in GMAWis not available the plasma velocitywas assumed to be 100 ms which isthe same as that in GTAW and thevalue of Cd was calculated to be 044(Ref 15) For a less-developed plasmajet 10 ms plasma velocity was usedand the value of Cd was also calculatedto be 044 The calculation in Ref 15shows that the equilibrium dropletsize of a steel electrode with plasmavelocity of 10 ms and 100 ms are al-most the same Thus in this study100 ms plasma velocity is used andthe drag coefficient is thus 044

According to the dynamic force bal-ance theory on the metal transfer (Ref9) the droplet will be detached whenthe following criteria are satisfied

Fem + Fd + Fg + Fin gt Fs (10)

where Fin is the inertia force generatedby the oscillation Fin = ndashma and a rep-resents the droplet acceleration Fs isthe surface tension given by

Fs = 2rw g (11)

where g is the surface tensioncoefficient

Calibration of Model Coefficients

To solve Equation 1 the spring con-stant k and damping coefficient b needto be determined first It is the majordifficulty in our modeling effortbecause the spring constant anddamping coefficient may change withthe droplet mass even if the wire (ma-terial diameter) and shielding gas aregiven The theoretical models on thespring constant and dampingcoefficient are briefly reviewed hereThe spring force and spring constantin the axial direction are expressedusing the potential energy generatedby the surface tension and surface areaof a droplet in Ref 9

dU = gdS = Fsdx k = dFsdx

where U is the potential energy Fs thespring force g the surface tension co-efficient and S the drop surface areaGiven the droplet massvolume thespring constant can be calculated Un-fortunately the accuracy of this modelis not satisfactory (minimum 11error and 38 maximum in thedroplet oscillation frequency) and it isthus not used in this paper Howeverit indicates that the droplet springconstant is related to the dropletmassvolume Reference 9 also gives

( ) ( )

( )

=μ

π

+ λ⎡

⎣⎢

⎤

⎦⎥

F tI t4

lnr t

r

em0

2

d

w

( )

( )

λ = θ minus minusminus θ

+minus θ + θ

1n sin14

11 cos

2

1 cosln

21 cos2

F12

C Ad d p p p2= ρ ν

A r rp d2

w2( )= π minus

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Fig 11 mdash T p1 under different Tg Fig 12 mdash Droplet dynamic response to exciting pulse


the damping coefficient as

b = 3m Vx2

where m is the viscosity coefficient V isthe droplet volume and x the dropletdisplacement Using this model thedamping coefficient is calculated to beonly at the order of 10ndash5 ndash 10ndash4 Nmwhich does not match the real dampingobserved in the experiments Hencethis model for calculating the dampingcoefficient is also not used in this studyThereby the two key variables need tobe experimentally calibrated

Calibration Method

The calibration proceeds with the as-sumption that the droplet mass duringthe first free oscillation period after theexciting pulse is constant since the wiremelting rate at the base current is suffi-ciently low and the concerned periodthe first oscillation cycle after the excit-ing pulse is only a few milliseconds Asa result the droplet mass during thefirst oscillation cycle can be consideredequal to the droplet mass m0 measured

at the end moment of the excitingpulse In this case the droplet mass mthe damping factor b the spring coeffi-cient k and the axial force F during thefirst oscillation cycle all becomeconstant m0 b0 k0 and F0 respectivelyTherefore Equation 1 is simplified intoa constant coefficient ordinarydifferential equation

(12)

This constant coefficient equationhas an analytical solution as follows

(13)

and A along with are coefficients de-termined by the initial dropletdisplacement and velocity dependingon the exciting parameters Based onEquation 13 the damping coefficientand spring constant can be calculated

since the droplet mass oscillation pe-riod and amplitude all can bemeasured from the experiments Byadjusting the initial droplet mass m0the correlation between bk and m canbe determined

To perform the calibrationexperiments 1 ndash4 are conducted by usingWave1 The experimental system andconditions are the same with thatdescribed in the first part of this investi-gation 08-mm ER70S-6 welding wire15 Lmin pure argon shielding gas 6-mm wire extension and bead-on-platewelding of mild steel (Ref 1) The initialdroplet masssize is controlled byadjusting the growing duration Tg Theexperimental parameters are listed inTable 1 The remaining parameters arefixed at Ig = 80 A Ib = 30 A Tp1 = 3 msIe = 120 A Te = 3 ms Tb = 30 ms Ifd =175 A Tfd = 5 ms

Damping Coefficient

According to Equation 13 thedamping coefficient can be calculatedas follows

+ + =m x b x k x F0 0 0 0

( )= + ω + ψ

ε = ω = minus ε

minusε

where

xFk

Ae sin t

b2m

km

0

0

t0

00

00

0

002

0

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Fig 13 mdash Droplet oscillation under different Te A mdash Te = 2 ms B mdash Te = 3 ms C mdash Te = 4 ms D mdash Te = 5 ms

A B

C D


(14)

where T1 is the oscillation period ofthe first oscillation cycle after the ex-citing pulse

As shown in Fig 6 A1 and A2 arethe oscillation amplitude of the firstand second oscillation cyclerespectively The droplet oscillation pe-riod T1 and the initial droplet mass m0are measured from the recorded high-speed image sequences UsingEquation 14 the damping coefficientin experiments 1ndash4 are calculated Theresults are shown in Fig 7 It can beseen that the damping coefficient isapproximately at the same level whenthe droplet mass is increasingThereby the damping coefficient isfixed at 00012 Nsm in this studywhich is the average of the measuredvalues from experiments 1ndash4

Spring Constant

Based on Equation 13 the dropletoscillation period of the firstoscillation cycle T1 is given by

(15)since the damping coefficient is only at10ndash3 Nmiddotsm Equation 15 can be simplified to

(16)

the initial droplet mass m0 and thedroplet oscillation period of the firstcycle T1 are measured fromexperiments 1ndash4 and then thecorresponding spring constant can becalculated as shown in Fig 8

It can be seen that the droplet springconstant is approximately linearlyincreasing with the droplet mass suchthat the spring constant calculationmodel can be established by linearly fit-ting the experimental values using theLeast Square method as follows

k = 774 + 12796129middotm (17)

Modeling Results andDiscussion

A simulation program based on themodel is developed in MatlabSimulink to compute the dynamicdroplet oscillation and detachment Theprogram structure is shown in Fig 9 Itcan be seen that the dynamic dropletmass radius displacement the springconstant and the total detaching forcecan all be obtained from this numericalmodel Forth-order Runge-Kuttaalgorithm is used to solve Equation 1The physical constants used are listed inTable 2 (Refs 13 17 18)

Optimal Exciting Phase Delay

In order to fully utilize the preoscil-lation before the exciting pulse the

optimal exciting phase delay undergiven growing parameters needs to bepredicted first According to the exper-imental study in the first part of thisinvestigation the optimal phase delaycorresponds to the reversing momentof the droplet oscillation (Ref 1)Given the current waveform parame-ters the dynamic dropletdisplacement and forces can be calcu-lated based on the above equations Byreading the time coordinate of the pre-oscillation reversing moment and theexciting end moment from the wave-form and droplet displacement curvesin Matlab the optimal exciting phasedelay Tp1 at given waveform parame-ters can be determined In order toguarantee the demonstration of thereversing moment relatively large Tp1needs to be used The calculation ofthe optimal exciting peak duration andoptimal detaching phase delay willproceed in similar ways to guarantee

=π

ω=

π

minusT

2 4 m

4k m b1

0

0

0 0 02

= πT 2 m k1 0 0

=b2mT

lnAA0

0

1

1

2

WELDING RESEARCH


Table 1 mdash Growing Duration in Experiments 1ndash4

No Tg (ms)

1 102 203 304 40

Table 2 mdash Physical Constant Used in the Model

Symbol Value Unit Description

C1 2885endash10 m3(A s) Melting constantC2 522endash10 m3(A W s) Melting constantrw 00004 m Wire radiusle 0006 m Wire extensionrr 07836 Wm Resistivity of Wirer 6800 kgm3 Density of liquid dropr 7860 kgm3 Density of solid wire

mo 125664endash6 kg mA2s2 Permeability of free spaceg 12 Nm Surface tension coefficient

Fig 14 mdash Te under different Tg Fig 16 mdash Effect of Tp2 on the droplet osshycillationdetachment

Fig 15 mdash Example for predicting T p2


the demonstration of the exciting peakmoment and reversing moment of themain excited oscillation

A simulation using Wave2 as theinput to the model is first performed asa preliminary verification on the modelwith the following current waveformparameters Ig = 80 A Tg = 20 ms Ib = 30A Tp1 = 5 ms Ie = 120 A Te = 3 ms Tp2= 3 ms Id = 140 A Td = 5 ms Tb = 20ms Ifd = 175 A and Tfd = 5 ms Thetime step for computation is 0001 msThe time cost for the computation on acommon Desktop PC is only a few sec-onds Figure 10A shows the droplet os-cillation and detachment under thegiven current waveform parametersThe corresponding dynamic detachingforce droplet gravitational force inertiaforce electromagnetic force and thedroplet radius are shown in Fig 10BndashFrespectively The calculated dynamicforces all correspond to the currentwaveform shown in Fig 10A One cansee from Fig 10C that the droplet gravi-tational force increases linearly in eachsubstage and a larger increasing rate as-sociates with higher current Since thewire diameter and the half angle arefixed the electromagnetic force is deter-mined by the current and droplet It canbe seen from Fig 10D that the electro-magnetic force is primarily determinedby the welding current Higher currentproduces larger electromagnetic forceWhile even the current is constant ineach substage the electromagnetic forcestill grows gradually because thedroplet radius is increasing Figure 10Eshows the dynamic inertia force Thestep changes in Fig 10E correspond tothe rising or falling edges of the pulsesin the current waveform which make

sudden changes on the electromagneticforce and thus sudden changes on thedroplet acceleration The inertia forcedoesnrsquot show direct correlation to thewelding current It is determined by thedroplet displacement and velocityBecause the detaching pulse is appliedwhen the droplet starts to move awayfrom the wire tip the droplet changesfrom being compressed to beingelongated in the detaching peak timeand the direction of the inertia force isthus also changed It can be seen thatthe excited droplet oscillation producesconsiderable inertia force which is syn-chronized with the electromagneticforce produced by the detaching pulseand the total detaching force is thus sig-nificantly increased as shown in Fig10B Hence the droplet is successfullydetached under only 140 A detachingcurrent The detached droplet radiusapproximately equals that measured inthe experiment using the same wave-form parameters

From Fig 10A one can see thatthe droplet is excited into a slightpre-oscillation under 80 A growingcurrent Tp1 is intentionally set at 5ms to demonstrate the reversing mo-ment of the preoscillation marked inFig 10A By reading the time of theexciting end moment and thepreoscillation reversing moment fromthe computed curves in Matlab thecorresponding T p1 can be obtainedIn the case as Fig 10A shows Tp1 isdetermined to be 28 ms Using thesame method Tp1 under differentdroplet mass (controlled by the grow-ing duration) can be obtained Themodeling and corresponding experi-mental results are shown in Fig 11 It

can be seen that the droplet growingtime changes in 10ndash40 ms thus thedroplet diameter is within 1ndash16 mmwide enough for discussing dropspray transfer characterized byrelatively small droplet size It can becalculated that the absolute modelingerror on Tp1 is only 007 ms whichindicates satisfactory modeling accu-racy When the initial droplet masschanges significantly in the widerange T p1 does not changesignificantly but are all within 2ndash4ms If the initial droplet size iscontrolled no larger than 125 mmcorresponding to 80 A20 ms growingpulse Tp1 can be roughly fixed at 2 or3 ms as a quick set while the utiliza-tion of the preoscillation is almostnot compromised

Optimal Exciting Peak Duration

Also using Wave2 as the input ofthe model the exciting peak durationis first set at 30 ms which is longenough to demonstrate the dropletdynamic response to the excitingpulse and gives a better understand-ing of its effect on the excited dropletoscillation The model calculation re-sult is shown in Fig 12

It can be seen that the droplet isfirst excited into a peak elongationand then goes into a forced oscillationduring the long exciting peak periodAs a contrast the droplet oscillationduring the base period after the excit-ing pulse can be considered as a freeoscillation since the electromagneticforce under the base current is negli-gible Figure 12 clearly demonstrates

WELDING RESEARCH


Fig 17 mdash Comparison between modeling and experimental result on Tp2

Fig 18 mdash Prediction of Tp2 using the analytical model

Xiao 9-14_Layout 1 82014 917 AM Page 328

that the droplet displacement and ve-locity at the end moment of the excit-ing pulse is exactly the initial condi-tion of the consequent freeoscillation Since the forcedoscillation during the exciting peakduration is damping the maximumamplitude of the free oscillation afterthe exciting pulse can only beachieved if the exciting pulse ends atthe elongation peak moment ie theoptimal exciting peak duration Tecorresponds to the elongation peakmoment Figure 13 demonstrates theeffect of the exciting peak durationon the magnitude of the free oscilla-tion The growing current is fixed at80 A with 20 ms duration The excit-ing peak current is fixed at 120 Awhile the exciting peak durationchanges from 2ndash5 ms It can be seenthat the droplet oscillationmagnitudes under 3 and 4 msexciting peak durations areapproximately even at the maximumlevel The droplet oscillation under 2ms exciting peak duration is a littleweaker while that under 5 ms excit-ing peak duration is significantlyweaker These results agree withthose of the experimentalexamination in Ref 8

Through the model based on Equa-tion 1 Te can be obtained byintentionally setting relatively long ex-citing peak duration Figure 14 shows acomparison between the experiment-tested and model-predicted resultsunder different droplet masses(controlled by the growing duration)The figure demonstrates that the exper-imental results are all slightly largerthan the corresponding model-predicted ones The maximum absoluteerror is up to 017 ms This is caused bythe limited rising speed of the realwelding current when it is changedfrom the base to peak In general therising time is approximately 1 ms suchthat the oscillation peak time becomesa little longer than that calculated fromthe model However such a level of dif-ference on the exciting peak durationwill not significantly affect theamplitude of the free oscillation afterthe exciting pulse Here the excitingpeak duration can also be fixed at 3 msas a quick set as long as the growingparameters are properly selected tocontrol the droplet size not exceeding125 mm

Optimal Detaching Phase Delay

The detaching phase delay is themost important parameter because itdetermines the synchronizationbetween the detaching pulse anddroplet downward momentum Thefirst part of this investigation (Ref 1)has confirmed that the featuredetaching phase delay corresponds tothe reversing moment of the excitedfree droplet oscillation is the optimaldetaching phase delay This conclusionis verified here based on the modelusing Ig = 80 A Tg = 20 ms Ib = 30 A Tp1= 3 ms Ie = 120 A and Te = 3 ms Figure15 shows the model-predicted T p2 is308 ms Using Wave2 as the input ofthe model a group of Tp2 around thisfeature value are used to perform simu-lations to verify the optimality of thefeature detaching phase delay Here Id isset at 110 A Thus it will elongate thedroplet but the elongation is not strongenough to detach the droplet Hencethe peak droplet displacement duringthe detaching pulse under different Tp2denoted as xp can be collected to evalu-ate the effect of Tp2 The results areshown in Fig 16 It can be seen that themaximum xp is achieved when Tp2equals the predicted Tp2 308 msOverall the simulation results supportthat the feature phase delay correspon-ding to the oscillation reversingmoment is the optimal for maximumenhancement on the dropletdetachment

Figure 17 shows a comparison be-tween the modeling and theexperimental results of Tp2 underdifferent initial droplet massescontrolled the growing duration Itcan be calculated that the maximumerror is only 01 ms and the averageerror calculated from the foursamples is only 0058 ms Such a lowlevel of prediction error indicates sat-isfactory accuracy of the model HereTp2 under different growing parame-ters cannot be simply fixed at acertain value as a quick set becausethe utilization of the dropletdownward momentum is sensitive tothe selection of Tp2 The tolerancerange of Tp2 for maximum utilizationof the downward momentum was ex-perimentally estimated to be [Tp2 ndash02 Tp2 + 04] ms in the first part ofthis investigation (Ref 1)

Analytical Model onDroplet Oscillation

Model Derivation

From the above results and analy-sis one can see that the establishedmodel gives a comprehensiveunderstanding on the dynamic dropletoscillation and detachment in the en-hanced active metal transfer controlThe dynamic droplet sizemassdroplet motion and the forces exertedon the droplet can be computed andthe computation time is only severalseconds However if possible asimpler analytical model with accept-able accuracy would be more appreci-ated Since the numerical modeling re-sults imply that Tp1 and Te can bothbe quickly set as long as the initialdroplet size is controlled within 1ndash13mm the analytical model will only aimat the prediction of Tp2which deter-mines the optimal synchronization ofthe detaching pulse and the dropletdownward momentum To this endthe following approximations are ap-plied to simplify the original modeland thus to avoid numerical computa-tions as follows

1 Set the damping coefficient b tozero Since the viscous damping indroplet oscillation was found to have anegligible effect on calculating thedroplet oscillation frequencycompared with other factors such assurface tension and gravity (Ref 9)the damping coefficient can be set tozero in predicting Tp2

2 Use a constant droplet mass toreplace the time-varying droplet massduring a short period As the same as-sumption used for the coefficient cal-ibration the droplet mass during theexciting peak period and the first freeoscillation cycle is considered to beconstant and equals the value meas-ured at the end of exciting pulse de-noted as m0 Based on Equations 3and 4 m0 can be calculated by

(18)

m

C I C l I T

C I C l I T

C I C l I T

C I C l I T

0

1 g 2 r e g2

g

1 b1 2 r e b12

b1

1 e 2 r e e2

e

1 d 2 r e d2

re

( )( )( )( )

= ρ

+ ρ

+ + ρ

+ + ρ

+ + ρ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

WELDING RESEARCH



where Tre represents the residual peakduration of the forced detaching pulseFor 5 ms forced detaching peak dura-tion Tre equals 05 ms approximately

Finally the model depicting thedropletrsquos dynamic response to the ex-citing pulse can be expressed as

(19)

(20)

(21)

From Equation 21 Tp2 under givengrowing and exciting parameters is de-rived to be

(22)

The comparison between the analyti-cal modeling and experimental results isshown in Fig 18 It can be seen that theanalytical model also shows satisfactoryaccuracy in predicting Tp2 since themaximum absolute error is only 01 ms

Conclusions1 A theoretical model on the

dynamic droplet oscillation and detach-ment in the enhanced active metaltransfer control is established based onthe mass-spring system The critical co-efficients ie the spring constant anddamping coefficient are experimentallycalibrated It is found that the dampingcoefficient is approximately independ-ent of the droplet mass but the springconstant increases with the dropletmass linearly

2 The model is numericallycomputed The effects of the criticalwaveform parameters on the droplet

oscillationdetachment are analyzedbased on the model The optimal excit-ing detaching phase delaycorresponds to the reversing momentof the droplet oscillation The optimalexciting peak duration corresponds tothe elongation peak moment Theseresults agree with those from theexperimental study in the first part ofthis paper

3 The numerical model enables oneto predict the critical waveformparameters at adequate speed and ac-curacy and can be used to effectivelydetermine the waveform parametersfor the enhanced active metal transfercontrol The exciting phase delay andexciting peak duration can both befixed as quick set as long as the grow-ing parameters are properly selectedsuch that the droplet size be relativelysmall as desired with the needed dropspray transfer

4 An analytical model on the exciteddroplet oscillation has also been estab-lished through acceptableapproximations such that the most im-portant parameter ie the optimal de-taching phase delay can be analyticallycalculated with adequate accuracy

This work is financially supported bythe State Key Laboratory of AdvancedWelding and Joining Harbin Instituteof Technology Harbin China and theNational Science Foundation undergrant CMMI-0825956 J Xiao greatlyappreciates the scholarship from ChinaScholarship Council (CSC) that fundedhis visit to the University of Kentuckyto conduct this research

1 Xiao J Zhang G J Zhang W Jand Zhang Y M 2014 Active metal trans-fer control by utilizing enhanced dropletoscillation Part 1 Experimental studyWelding Journal 93(8) 282-s to 291-s

2 Thomsen J S 2006 Control ofpulsed gas metal arc welding InternationalJournal of Modelling Identification and Con-trol 1(2) 115ndash125

3 Kim Y S and Eagar T W 1993Metal transfer in pulsed current gas metalarc welding Welding Journal 72(7) 279-sto 287-s

4 Amin M 1983 Pulse current param-eters for arc stability and controlled metaltransfer in arc welding Metal Construction15 272ndash278

5 Jacobsen N 1992 Monopulse inves-tigation of droplet detachment in pulsedgas metal arc welding Journal of Physics DApplied Physics 25 783ndash797

6 Zhang Y M Liguo E and Kovace-vic R 1998 Active metal transfer controlby monitoring excited droplet oscillationWelding Journal 77(9) 388-s to 395-s

7 Zhang Y M and Liguo E 1999Method and system for gas metal arc weld-ing US Patent 6008470

8 Xiao J Zhang G J Zhang Y M etal 2013 Active droplet oscillation excitedby optimized waveform Welding Journal92(7) 205s to 217-s

9 Choi J H Lee J and Yoo C D2001 Dynamic force balance model formetal transfer analysis in arc weldingJournal of Physics D Applied Physics 342658ndash2664

10 Jones L A Eagar T W and LangJ H 1998 A dynamic model of drops de-taching from a gas metal arc welding elec-trode Journal of Physics D Applied Physics31 107ndash123

11 Wu C S Chen M A and Li S K2004 Analysis of excited dropletoscillation and detachment in active con-trol of metal transfer Computational Mate-rials Science 31(1-2) 147ndash154

12 Chen M A Wu C S Li S K andZhang Y M 2007 Analysis of active con-trol of metal transfer in modified pulsedGMAW Science and Technology of Weldingand Joining 12(1) 0ndash14

13 Lesnewich A 1958 Control ofmelting rate and metal transfer in gasshielded metal arc welding Part 1 Controlof electrode melting rate Welding Journal37(9) 343-s to 353-s

14 Amson J C 1965 Lorentz force inthe molten tip of an arc electrode BritishJournal of Applied Physics 16 1169ndash1179

15 Kim Y S and Eagar T W 1993Analysis of metal transfer in gas metal arcwelding Welding Journal 72(6) 269-s to277-s

16 Huang Y Shao Y and Zhang YM 2012 Nonlinear modeling of dynamicmetal transfer in laser-enhanced GMAWWelding Journal 91(5) 140-s to 148-s

17 Choi S Kim Y S and Yoo C D1999 Dimensional analysis of metal trans-fer in GMA welding Journal of Physics DApplied Physics 32 326ndash334

18 Naidu D S Moore K L YenderR and Tyler J 1997 Gas metal arc weld-ing control Part 1 mdash Modeling and analy-sis Nonlinear Analysis Methods and Appli-cations 30(5) 3101ndash 3111

m x k x F t0 0 ( )+ =

( ) ( )( ) =

le le

ge

⎧⎨⎪

⎩⎪F t

F I - F I 0 t T

0 t T

Define w k m thus the analytical

solution of Equation 19 is derived to be

em e em b e

e

0 0 0

[ ]

( ) ( )

( ) ( )

( )

( ) =

minus

minus ω lt le

minus

ω minus minus ω⎡⎣ ⎤⎦ ge

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x t

F I F I

k

1 cos t 0 t T

F I F I

k

cos t T cos t t T

em e em b

0

0 e

em e em b

0

0 e 0 e

T34

T12

T

T 2 m k

p2 1 e

1 0 0

= minus

= π

lowast

WELDING RESEARCH


References

Acknowledgments


detaching current to be further reducedThe lower limit of the detaching currentis determined to be not only muchlower than the spray transition currentbut also significantly lower than that ofthe original active metal transfercontrol The experimental study also in-dicates that the enhanced active metaltransfer control process is sufficientlyrobust (Ref 1)

Figure 3 demonstrates a typicaldroplet excitation and detachment inthe enhanced active metal transfercontrol where Wave2 is used Ig = 80 ATg = 20 ms Ib = 30 A Tp1 = 2 ms Ie =120 A Te = 3 ms Tp2 = 32 ms Id = 125A and Td = 4 ms Tb = 20 ms Ifd = 175A Tfd = 5 ms It can be seen that thedroplet is first elongated by the excit-ing pulse (Frames 2ndash4) and goes intooscillation in the base period called de-taching phase delay (Frames 5ndash7) and

then the droplet is accelerated and de-tached by the detaching pulse of only125 A4 ms (Frames 8ndash13)

To better depict the excited dropletoscillation the following conceptsdefined in the experimental study arerepeated here 1) The moment at whichthe excited droplet reaches itsmaximum elongation is referred to asthe elongation peak moment 2) Themoment at which the droplet changesits moving direction from upward(toward the wire) into downward (awayfrom the wire) during the preoscillationor the main oscillation is referred to asthe oscillation reversing moment Ascan be seen from Fig 2 the followingcurrent waveform parameters need tobe properly selected to first maximizethe droplet oscillation amplitude andthen maximize the consequentenhancement on the droplet

detachment in the enhanced activemetal transfer control

1 Exciting phase delay Tp1 ie thebase duration between the growingand exciting pulse As has beenverified in Ref 8 the droplet may beexcited to a preoscillation during Tp1as long as the growing current is highenough to preelongate the droplet Ifthe exciting pulse is synchronized withthe downward momentum during thepreoscillation the main oscillationafter the exciting pulse can be furtherenhanced If the growing current issufficiently low for example80 A used in the experimental study

the selection of Tp1 will not affect themain droplet oscillation significantlyIn this sense this parameter was notdiscussed in the experimental studybut it will be analyzed in this theoreti-cal study

WELDING RESEARCH


Fig 1 mdash Current waveform for enhanced droplet oscillation Fig 2 mdash Current waveform for enhanced active metal transfercontrol

Fig 3 mdash Typical metal transfer of enhanced active control 1 ms per frame ER70Sshy608shymm wire 15Lmin argon gas flow 6shymm arc length 6shymm wire extension beadshyonshyplate welding

Fig 4 mdash Illustration of massshyspring model ofdroplet oscillation













WELDING RESEARCH









(1)

F(t) = Fem + Fd + Fg (2)








WELDING RESEARCH



A B

C D

E F







(6)

(7)




(8)

(9)





Fs = 2rw g (11)






( ) ( )

( )

=μ

π

+ λ⎡

⎣⎢

⎤

⎦⎥

F tI t4

lnr t

r

em0

2

d

w

( )

( )


+minus θ + θ

1n sin14

11 cos

2

1 cosln

21 cos2

F12


A r rp d2

w2( )= π minus

WELDING RESEARCH





b = 3m Vx2


Calibration Method



(12)


(13)




Damping Coefficient


+ + =m x b x k x F0 0 0 0

( )= + ω + ψ

ε = ω = minus ε

minusε

where

xFk

Ae sin t

b2m

km

0

0

t0

00

00

0

002

0

WELDING RESEARCH



A B

C D


(14)



Spring Constant



(16)



k = 774 + 12796129middotm (17)






=π

ω=

π

minusT

2 4 m

4k m b1

0

0

0 0 02

= πT 2 m k1 0 0

=b2mT

lnAA0

0

1

1

2

WELDING RESEARCH



No Tg (ms)

1 102 203 304 40
















WELDING RESEARCH











Model Derivation




(18)

m

C I C l I T

C I C l I T

C I C l I T

C I C l I T

0

1 g 2 r e g2

g

1 b1 2 r e b12

b1

1 e 2 r e e2

e

1 d 2 r e d2

re

( )( )( )( )

= ρ

+ ρ

+ + ρ

+ + ρ

+ + ρ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

WELDING RESEARCH





(19)

(20)

(21)


(22)



























m x k x F t0 0 ( )+ =

( ) ( )( ) =

le le

ge

⎧⎨⎪

⎩⎪F t

F I - F I 0 t T

0 t T



em e em b e

e

0 0 0

[ ]

( ) ( )

( ) ( )

( )

( ) =

minus

minus ω lt le

minus


⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x t

F I F I

k

1 cos t 0 t T

F I F I

k

cos t T cos t t T

em e em b

0

0 e

em e em b

0

0 e 0 e

T34

T12

T

T 2 m k

p2 1 e

1 0 0

= minus

= π

lowast

WELDING RESEARCH


References

Acknowledgments













WELDING RESEARCH






Xiao 9-14_Layout 1 81514 344 PM 23



(1)

F(t) = Fem + Fd + Fg (2)








WELDING RESEARCH



A B

C D

E F







(6)

(7)




(8)

(9)





Fs = 2rw g (11)






( ) ( )

( )

=μ

π

+ λ⎡

⎣⎢

⎤

⎦⎥

F tI t4

lnr t

r

em0

2

d

w

( )

( )


+minus θ + θ

1n sin14

11 cos

2

1 cosln

21 cos2

F12


A r rp d2

w2( )= π minus

WELDING RESEARCH





b = 3m Vx2


Calibration Method



(12)


(13)




Damping Coefficient


+ + =m x b x k x F0 0 0 0

( )= + ω + ψ

ε = ω = minus ε

minusε

where

xFk

Ae sin t

b2m

km

0

0

t0

00

00

0

002

0

WELDING RESEARCH



A B

C D


(14)



Spring Constant



(16)



k = 774 + 12796129middotm (17)






=π

ω=

π

minusT

2 4 m

4k m b1

0

0

0 0 02

= πT 2 m k1 0 0

=b2mT

lnAA0

0

1

1

2

WELDING RESEARCH



No Tg (ms)

1 102 203 304 40
















WELDING RESEARCH











Model Derivation




(18)

m

C I C l I T

C I C l I T

C I C l I T

C I C l I T

0

1 g 2 r e g2

g

1 b1 2 r e b12

b1

1 e 2 r e e2

e

1 d 2 r e d2

re

( )( )( )( )

= ρ

+ ρ

+ + ρ

+ + ρ

+ + ρ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

WELDING RESEARCH





(19)

(20)

(21)


(22)



























m x k x F t0 0 ( )+ =

( ) ( )( ) =

le le

ge

⎧⎨⎪

⎩⎪F t

F I - F I 0 t T

0 t T



em e em b e

e

0 0 0

[ ]

( ) ( )

( ) ( )

( )

( ) =

minus

minus ω lt le

minus


⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x t

F I F I

k

1 cos t 0 t T

F I F I

k

cos t T cos t t T

em e em b

0

0 e

em e em b

0

0 e 0 e

T34

T12

T

T 2 m k

p2 1 e

1 0 0

= minus

= π

lowast

WELDING RESEARCH


References

Acknowledgments




(1)

F(t) = Fem + Fd + Fg (2)








WELDING RESEARCH



A B

C D

E F







(6)

(7)




(8)

(9)





Fs = 2rw g (11)






( ) ( )

( )

=μ

π

+ λ⎡

⎣⎢

⎤

⎦⎥

F tI t4

lnr t

r

em0

2

d

w

( )

( )


+minus θ + θ

1n sin14

11 cos

2

1 cosln

21 cos2

F12


A r rp d2

w2( )= π minus

WELDING RESEARCH





b = 3m Vx2


Calibration Method



(12)


(13)




Damping Coefficient


+ + =m x b x k x F0 0 0 0

( )= + ω + ψ

ε = ω = minus ε

minusε

where

xFk

Ae sin t

b2m

km

0

0

t0

00

00

0

002

0

WELDING RESEARCH



A B

C D


(14)



Spring Constant



(16)



k = 774 + 12796129middotm (17)






=π

ω=

π

minusT

2 4 m

4k m b1

0

0

0 0 02

= πT 2 m k1 0 0

=b2mT

lnAA0

0

1

1

2

WELDING RESEARCH



No Tg (ms)

1 102 203 304 40
















WELDING RESEARCH











Model Derivation




(18)

m

C I C l I T

C I C l I T

C I C l I T

C I C l I T

0

1 g 2 r e g2

g

1 b1 2 r e b12

b1

1 e 2 r e e2

e

1 d 2 r e d2

re

( )( )( )( )

= ρ

+ ρ

+ + ρ

+ + ρ

+ + ρ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

WELDING RESEARCH





(19)

(20)

(21)


(22)



























m x k x F t0 0 ( )+ =

( ) ( )( ) =

le le

ge

⎧⎨⎪

⎩⎪F t

F I - F I 0 t T

0 t T



em e em b e

e

0 0 0

[ ]

( ) ( )

( ) ( )

( )

( ) =

minus

minus ω lt le

minus


⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x t

F I F I

k

1 cos t 0 t T

F I F I

k

cos t T cos t t T

em e em b

0

0 e

em e em b

0

0 e 0 e

T34

T12

T

T 2 m k

p2 1 e

1 0 0

= minus

= π

lowast

WELDING RESEARCH


References

Acknowledgments







(6)

(7)




(8)

(9)





Fs = 2rw g (11)






( ) ( )

( )

=μ

π

+ λ⎡

⎣⎢

⎤

⎦⎥

F tI t4

lnr t

r

em0

2

d

w

( )

( )


+minus θ + θ

1n sin14

11 cos

2

1 cosln

21 cos2

F12


A r rp d2

w2( )= π minus

WELDING RESEARCH





b = 3m Vx2


Calibration Method



(12)


(13)




Damping Coefficient


+ + =m x b x k x F0 0 0 0

( )= + ω + ψ

ε = ω = minus ε

minusε

where

xFk

Ae sin t

b2m

km

0

0

t0

00

00

0

002

0

WELDING RESEARCH



A B

C D


(14)



Spring Constant



(16)



k = 774 + 12796129middotm (17)






=π

ω=

π

minusT

2 4 m

4k m b1

0

0

0 0 02

= πT 2 m k1 0 0

=b2mT

lnAA0

0

1

1

2

WELDING RESEARCH



No Tg (ms)

1 102 203 304 40
















WELDING RESEARCH











Model Derivation




(18)

m

C I C l I T

C I C l I T

C I C l I T

C I C l I T

0

1 g 2 r e g2

g

1 b1 2 r e b12

b1

1 e 2 r e e2

e

1 d 2 r e d2

re

( )( )( )( )

= ρ

+ ρ

+ + ρ

+ + ρ

+ + ρ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

WELDING RESEARCH





(19)

(20)

(21)


(22)



























m x k x F t0 0 ( )+ =

( ) ( )( ) =

le le

ge

⎧⎨⎪

⎩⎪F t

F I - F I 0 t T

0 t T



em e em b e

e

0 0 0

[ ]

( ) ( )

( ) ( )

( )

( ) =

minus

minus ω lt le

minus


⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x t

F I F I

k

1 cos t 0 t T

F I F I

k

cos t T cos t t T

em e em b

0

0 e

em e em b

0

0 e 0 e

T34

T12

T

T 2 m k

p2 1 e

1 0 0

= minus

= π

lowast

WELDING RESEARCH


References

Acknowledgments



b = 3m Vx2


Calibration Method



(12)


(13)




Damping Coefficient


+ + =m x b x k x F0 0 0 0

( )= + ω + ψ

ε = ω = minus ε

minusε

where

xFk

Ae sin t

b2m

km

0

0

t0

00

00

0

002

0

WELDING RESEARCH



A B

C D


(14)



Spring Constant



(16)



k = 774 + 12796129middotm (17)






=π

ω=

π

minusT

2 4 m

4k m b1

0

0

0 0 02

= πT 2 m k1 0 0

=b2mT

lnAA0

0

1

1

2

WELDING RESEARCH



No Tg (ms)

1 102 203 304 40
















WELDING RESEARCH











Model Derivation




(18)

m

C I C l I T

C I C l I T

C I C l I T

C I C l I T

0

1 g 2 r e g2

g

1 b1 2 r e b12

b1

1 e 2 r e e2

e

1 d 2 r e d2

re

( )( )( )( )

= ρ

+ ρ

+ + ρ

+ + ρ

+ + ρ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

WELDING RESEARCH





(19)

(20)

(21)


(22)



























m x k x F t0 0 ( )+ =

( ) ( )( ) =

le le

ge

⎧⎨⎪

⎩⎪F t

F I - F I 0 t T

0 t T



em e em b e

e

0 0 0

[ ]

( ) ( )

( ) ( )

( )

( ) =

minus

minus ω lt le

minus


⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x t

F I F I

k

1 cos t 0 t T

F I F I

k

cos t T cos t t T

em e em b

0

0 e

em e em b

0

0 e 0 e

T34

T12

T

T 2 m k

p2 1 e

1 0 0

= minus

= π

lowast

WELDING RESEARCH


References

Acknowledgments


(14)



Spring Constant



(16)



k = 774 + 12796129middotm (17)






=π

ω=

π

minusT

2 4 m

4k m b1

0

0

0 0 02

= πT 2 m k1 0 0

=b2mT

lnAA0

0

1

1

2

WELDING RESEARCH



No Tg (ms)

1 102 203 304 40
















WELDING RESEARCH











Model Derivation




(18)

m

C I C l I T

C I C l I T

C I C l I T

C I C l I T

0

1 g 2 r e g2

g

1 b1 2 r e b12

b1

1 e 2 r e e2

e

1 d 2 r e d2

re

( )( )( )( )

= ρ

+ ρ

+ + ρ

+ + ρ

+ + ρ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

WELDING RESEARCH





(19)

(20)

(21)


(22)



























m x k x F t0 0 ( )+ =

( ) ( )( ) =

le le

ge

⎧⎨⎪

⎩⎪F t

F I - F I 0 t T

0 t T



em e em b e

e

0 0 0

[ ]

( ) ( )

( ) ( )

( )

( ) =

minus

minus ω lt le

minus


⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x t

F I F I

k

1 cos t 0 t T

F I F I

k

cos t T cos t t T

em e em b

0

0 e

em e em b

0

0 e 0 e

T34

T12

T

T 2 m k

p2 1 e

1 0 0

= minus

= π

lowast

WELDING RESEARCH


References

Acknowledgments










WELDING RESEARCH











Model Derivation




(18)

m

C I C l I T

C I C l I T

C I C l I T

C I C l I T

0

1 g 2 r e g2

g

1 b1 2 r e b12

b1

1 e 2 r e e2

e

1 d 2 r e d2

re

( )( )( )( )

= ρ

+ ρ

+ + ρ

+ + ρ

+ + ρ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

WELDING RESEARCH





(19)

(20)

(21)


(22)



























m x k x F t0 0 ( )+ =

( ) ( )( ) =

le le

ge

⎧⎨⎪

⎩⎪F t

F I - F I 0 t T

0 t T



em e em b e

e

0 0 0

[ ]

( ) ( )

( ) ( )

( )

( ) =

minus

minus ω lt le

minus


⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x t

F I F I

k

1 cos t 0 t T

F I F I

k

cos t T cos t t T

em e em b

0

0 e

em e em b

0

0 e 0 e

T34

T12

T

T 2 m k

p2 1 e

1 0 0

= minus

= π

lowast

WELDING RESEARCH


References

Acknowledgments








Model Derivation




(18)

m

C I C l I T

C I C l I T

C I C l I T

C I C l I T

0

1 g 2 r e g2

g

1 b1 2 r e b12

b1

1 e 2 r e e2

e

1 d 2 r e d2

re

( )( )( )( )

= ρ

+ ρ

+ + ρ

+ + ρ

+ + ρ

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

WELDING RESEARCH





(19)

(20)

(21)


(22)



























m x k x F t0 0 ( )+ =

( ) ( )( ) =

le le

ge

⎧⎨⎪

⎩⎪F t

F I - F I 0 t T

0 t T



em e em b e

e

0 0 0

[ ]

( ) ( )

( ) ( )

( )

( ) =

minus

minus ω lt le

minus


⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x t

F I F I

k

1 cos t 0 t T

F I F I

k

cos t T cos t t T

em e em b

0

0 e

em e em b

0

0 e 0 e

T34

T12

T

T 2 m k

p2 1 e

1 0 0

= minus

= π

lowast

WELDING RESEARCH


References

Acknowledgments




(19)

(20)

(21)


(22)



























m x k x F t0 0 ( )+ =

( ) ( )( ) =

le le

ge

⎧⎨⎪

⎩⎪F t

F I - F I 0 t T

0 t T



em e em b e

e

0 0 0

[ ]

( ) ( )

( ) ( )

( )

( ) =

minus

minus ω lt le

minus


⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

x t

F I F I

k

1 cos t 0 t T

F I F I

k

cos t T cos t t T

em e em b

0

0 e

em e em b

0

0 e 0 e

T34

T12

T

T 2 m k

p2 1 e

1 0 0

= minus

= π

lowast

WELDING RESEARCH


References

Acknowledgments


active metal transfer control by utilizing enhanced droplet … · 2016-01-13 · enhanced active...

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