Control of penetration in gas-tungsten-arc welding : a puddle impedance approach

CONTROL OF PENETRATION IN GAS-TUNGSTEN-ARC WELDING -A PUDDLE IMPEDANCE APPROACH

BY

MIRIAM ZACKSENHOUSE / /

BSME, TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY (1980)

SUBMITTED IN.PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

JUNE 1982

@)

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Signature of Author

---

Signature Redacted

De par tm en t of Mechanical Engineering

Signature Redacted

Certified by . . . . _ , . ,

-Thesis Supervisor

Signature Redacted

Accepted by

-Chairman, Department Committee on Graduate Students

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MP.SSACHUSETTS iNSTITUTE

OF TECHNOLOGY

JUL 3

Q

1982

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CONTROL OF PENETRATION IN GAS-TUNGSTEN-ARC WELDING -A PUDDLE IMPEDANCE APPROACH

by

MIRIAM ZACKSENHOUSE

Submitted to the Department of Mechanical Engineering on June 22, 1982 in partial fulfillment of the requirements for the Degree of Master of Science in Mechanical Engineering.

ABSTRACT

This work is part of a three-year project sponsored by the Department Of Energy (DOE) with the purpose of improving the quality and reliability of pipe welding. A major determinant of weld quality is the extent of penetration, and the target of this work is to develop a real-time closed-loop control of the extent of full penetration in Gas-Tungsten-Arc Welding, with no access to the back side of the weld.

As full penetration is reached the impedance of the puddle is changed drasticly and is related to the size of the fully penetrated puddle, The puddle impedance can by characterized by either the gross deflection of the puddle or by the natural frequency of the puddle. The corresponding measurement techniques· are investigated for. the purpose of real-time quality control.

The theory of puddle impedance, and the relation between the puddle impedance and the puddle size are developed and checked experimentally. The ability to sense in-process the puddle impedance, based on the simple measurement of the a~c-voltage, is investigated.

The measurement technique, based on real-time natural frequency computation, is modeled and is used to calculate the overall closed-loop system response. The results of these simulations demonstrate the feasibility of this control strategy.

Thesis supervisor Title

David E. Hardt

Table of Contents

Page

Title Page 1

Abstract 2

Table of Content 3

List of Figures and Tables 6

Acknowledgement 14

Part 1 INTRODUCTION AND BACKGROUND 15

Chapter 1 Problem Statement 15

1.1 Motivation 15

1.2 Problem Statement 16

1.3 Thesis Structure 19

1.4 Penetration Measurement Techniques 19 1. 5 The Basic . Ideas in This Work 21 Chapter 2 Basic Experimental Apparatus 24 2.1 The Continuous Weld Experimental Apparatus 24 2.2 The Stationary Weld Experimental Apparatus 26 Chapter 3 Arc Charateristics- The Foundation of

an Impedance Measurement Technique 3.1 Electrical Features 3.2 Arc Force 30 30 40 --· Page

3

Table of Contents (cont'd)

Part 2 MEASUREMENT TECHNIQUE BASED ON PUDDLE IMPEDANCE

Chapter 4 Puddle Deflection and The Extent of Full Penetration

..

44

45

Chapter 5 Puddle Impedance- Theory 52

5.1 Modeling Assumptions 52

5.2 The Velocity Potential Problem 54

5.3 Lumped Parameter Model 63

5.4 Numerical Evaluation 66

5.5 The -System Transfer Function 69 "5.6 Radius Criterion and The Surface Tension 69

5.7 Puddle Excitation 70

5.8 Steady vs. Continuous Case 74

Chapter 6 Puddle Dynamic Impedance- Experiments 76

6.1 Arc-Volt_age Measurement 77

6.2 Laser Based Experimental Technique 92 6.3 Eliminating The Constant Arc-Resistance 112

Part 3 THE CONTROLLED

SYSTEM-BASED ON NATURAL FREQUENCY MEASUREMENT 129 Chapter 7 Measurement and Plant Model 131

7.1 Measurement Model 131

7.2 Plant Model 138

Table of Contents (cont'd)

Chapter 8 System Response 8.1 The Controller 8.2 The Overall System 8.3 Performance Study

Part 4 SUMMARY AND SUGGESTIONS Chapter 9 Summary and Suggestions

Appendix A On Line System Identification Using Cross Correlation References Page 141 141 142 147 .161 161 164 169 Page 5

Fig 1-1 1-2 2-1 2-2 3-1 3-2 3-3 3-4 3-5 3-6 List of Figures

Schematic of Quality Control Strategy Model of a Completely Melted Weld Bead The Continuous Weld Experimental Apparatus The Stationary Weld Experimental Apparatus V-I Curves; For 3/32" Dia. Electrode

V-I curves; For 0.040" Dia. Electrode Transfer Function; Arc on Copper DC=l20A Coherence and Phase; corresponding to Fig 3-3 Transfer Function; Arc on Copper DC=60A

Coherence and Phase; corresponding to Fig 3-5 3-7 Variation of Plasma Jet Momentum

With Current For An Argon Arc 4-1 Arc-Voltage at Melt Through

4-2 Arc-Voltage; Change In Heat Input Du~ing a Continuous Weld 4-3 Puddle Deflection

5-1 Model of Puddle at Full (a), and Partial (b) Penetration

5-2 Forces on an Isolated Ring Of A Cylindrical Puddle

6-1 Constant and Varying Arc-Length

Page 17 22 25 27 32 33 35 36 37 38 42 47 48

so

53 65 78 Page 6

List of Figures (cont'd)

Fig Page

6-2 Arc-Voltage: Partial To Full Penetration Transition On Thin Plate

6-3 Arc-Voltage: Partial To Full Penetration Transition On Thick Plate 6-4 Arc-Voltage: Partially Penetrated Puddle 6-5 Arc-Voltage: ·Step Change in Arc-Force,

On Copper DC=75A

6-6 Arc-Voltage: Step Change in Arc-Force, On Copper DC=lOOA

6-7 Arc-Voltage: Step Change in ARc-Force, On Copper DC=l25A

6-8 Auto Spectrum. of Arc-Voltage: DC•72A,

Fully Penetrated Puddle On Thin Plate 6-9 Auto Spectrum of Arc-Volatge: DC 126A,

80 80 82 82 83 83 85

Partially Penetrated Puddle On Thick Plate 86 6-10 Auto Spectrum of Arc-Voltage: DC 130A,

Fully Penetrated Puddle On Thick Plate 6-11 Frequency Characteristics of Arc at 10-250Hz,

Fully Penetrated Puddle on Thin Plate 6-12 Transfer Function and Coherence

Corresponding to Fig 6-11

87

89

90

List of Figures (cont'd)

Fig Page

6-13 Schematic of The Laser Based Measurement Technique 6-14 Photodiode Voltage: 2Hz Sinousoidal

Component in The Arc-Current,

Fully Penetrated Puddle on Thin Plate 6-15 Auto Spectrum of Arc-Current

Constructed by Method A 6-16 Auto Spectrum of Arc-Current

Constructed by Method B 6-17 Auto Spectrum of Photodiode

Voltage-Initial Stage of Fully Penetrated Puddle

94

96

98

100

(Current- Method A) 101

6-18 Auto Spectrum of Photodiode

Voltage-Developed Puddle (Current- Method A) 102 6-19 Auto Spectrum of Photodiode

Voltage-Very Developed Puddle (Current- Method A) 103 6-20 Transfer Function: Current-Input,

Photodiode Voltage-Output.

Fully Penetrated Puddle (Current- Method A) 104 6-21 Auto Spectrum of Photodiode

Voltage-Initial Stage of Fully Penetrated Puddle

(Current- Method B) 105

List of Figures (cont'd)

Fig Page

6-22 Auto Spectrum of Photodiode

Voltage-Developed Puddle (Current- Method B) 6-23 Auto Spectrum of Photodiode

Voltage-106

More Developed Puddle (Current- Method B) 107 6-24 Auto Speatrum of Photodiode

Voltage-Most Developed Puddle (Current- Method B) 108 6-25 Auto Spectrum of Photodiode

Voltage-Developed Puddle (Current- Method B) 6-26 lOOHz Component in Arc-Current and

Arc-Voltage, On Copper DC=l25A 6-27

v

8 - The Signals in Fig 6-26 Are

109

115

Subtracted by The Oscilloscope Amplifier 115 6-28 Arc Voltage (lower), and 5V₀

-Corresponding to Fig 6-26 116

6.:..29 lOOHz Component in Arc-Current and

Arc-Voltage, On Copper DC=63A 116 6-30

v

_{8 -} The Signals in Fig 6-29 Are

Subtracted by The Osciloscope Amplifier 117 6-31 Arc-Voltage (lower), and 5V₈

-Corresponding to Fig 6-29 6-32 Transfer Function: Current-Input,

v

8-output, On Copper DC•l25A

117

118

List of Figures (cont'd)

Fig

6-33 Coherence and

Phase-Corresponding to Fig 6-32 6-34 Transfer Function: Current-Input,

Ve-Output, On Copper DC•60A 6-35 Coherence and

Phase-Corresponding to Fig 6-34 6-36 Transfer Function: Current-Input,

Ve -Output, Fully Penetrated Puddle

On Thin Plate 6-37 Auto Spectrum of

Arc-Current-Fully Penetrated Puddle On Thin Plate Electrode Dia. 0.040"

6-38 Auto Spectrum of

Ve-Corresponding to Fig 6-37 6-39 Transfer Function: Current-Input,

\ -Output, Corresponding to Fig 6-37 6-40 Auto Spectrum of The Photodiode Voltage (1)

and

v

₈ (2). Fully Penetrated Puddle

On Thin PLate 7-1

7-2 7-3

The Closed Loop System

Schematic of The Measurement System Delay Time 119 120 121 123 124 125 126 128 130 133 137 Page 10

List of Figures (cont'd)

8-lA The Overall System:

Modeled With Continuous Disturbances 8-lB The Overall System:

Modeled With Sampled Disturbances 8-2 Z-plane Root Locus For Eq 8-17

Zero at z=. 7, T='p

I

10, TT=1> 8-3 Z-plane Root Locus For Eq 8-17

Zero at z=.845, T=1p/10, TT~Tp 8-4 Z-plane Root Locus For Eq 8-17

..

Zero at z=.905, T=1p/10, TT=Tp Zero Cancells The Plant Pole 8-5 Z-plane Root Locus For Eq 8-17

Zero at z=.955, T=Tp/10, TT=Tp 8-6 Step Response of The System in Fig 8-4,

K•0.0055

8-7 Step Response of The System in Fig 8-4, K•0.01

8-8 Step Response of The System in Fig 8-4, K•0.0145 Page 11 143 145 150 151 152 153 154 156 157

List of Figures (cont'd)

Fig Page

8-9 Z-Plane Root Locus For Eq 8-16 With n=3,

T=Tp/10, TT=Tp

Zero Cancells Plant Pole at z=0.905 8-10 Step Response of The System in Fig 8-9,

K•0.02

A-1 On Line System Identification: Method (a),

Is Not Affected by Reference Signal (b), or Noise in The Output (c)

159

160

165

List of Tables

Table

5-1 Nomenclature List

5-2 Natural Frequencies of Partially and Fully Penetrated Puddle

Page

55

68

Page 14 ACKNOWLEDGEMENT

The author of this Thesis is grateful to her supervisor, Professor Dave E. Hardt, for his advise and inspiration, and to Professor William Unkel for his useful consultation.

Thanks to the technicians at the Laboratory for Manufacturing and Productivity for their help and technical advise, and to Dr. Jose Convert! for introducing to me the facility available at the Energy and Chemical Dynamics Laboratory and the concepts in his Doctoral Thesis.

This research was funded by the Department Of Energy under Contract No. DE-AC02-79ER10474.AOOO.

Page

15

Part 1 INTRODUCTION AND BACKGROUND

Chapter 1 Problem Statement

1.1 Motivation

The wide purpose of the work described in this thesis is to develop a real-time closed-loop control of the quality of the weld. The work is sponsored by the Departement Of Energy with the aim of improving the quality of pipe welding for nuclear plants. This is only one example of a need that can be filled by achieving the above purpose •

•

Most of the high quality welding operations are done by skillfull welders, but those welders are rare and expensive and there is always room for human errors. The percentage of defective welds is high and the cost of repair is a significant portion of any high quality welding project. In addition, implementing automatic control of the weld quality will allow reduction of fit-up tolerances and an increase in torch velocities; resulting in significant reduction in the total time to weld.

The concept of quality control should be differentiated from two other kinds of automatic welding machines already in use. The first kind is the welder control machines that monitor the conditions at the interface between the welder and the work-piece (e.g. heat output, torch to workpiece distance, wire feed rate, and torch velocity). Those parameters are adjusted to fixed values based upon experience for the particular weld

Page 16 being done. It results in an open-loop control of the quality of the weld; a control method that is especially unfit to deal with the wide range of disturbances that affect the welding process. In the quality control approach the welder control is used as an inner part of an overall closed-loop controller. (See Fig 1-1 ) •

The second kind is the automatic seam tracking machines that guide the torch so as to maintain a fixed position and orientation with respect to the seam. Seam tracking is a necessary condition for automatic high quality welding, but clearly an insufficient one.

1.2. Problem Statement

The purpose statement in the beginning of the chapter uses the term quality control. This is a broad concept and for this work has been narrowed to full penetration control. The work concentrates on the root pass(es) using Gas-Tungsten-Arc Welding (GTAW) and not on the filling process using Gas-Metal-Arc Welding (GMAW ). Although the filling occupies considerable time in welding a thick plate, the root pass requires a higher level of control and precision, and therefore can benefit more from application of control techniques.

The main problems with GTA root pass weld are incomplete melting and burn-through. A full penetration control will command the weld device to melt only enough material to completely fuse the joint without risking a burn through, and then to maintain a specific backbead width.

DESIRED WELD

CHARACTERISTICS

..

'.

WELD

It.

CONTROLLER

. ~

HEJ\T.

SOURCE

WELDER

_.

I

_I

~

...

_.

INNER

LOOP

OUTER LOOP

-

.

WORl

PIECE

Fig 1-1 Schematic of Quality

Control Strategy

Page

17

_WORK

PIECE

-. -- -

--Page 18 Thus, the purpose of this work is to develop a measurement technique and a controller scheme for on-line closed-loop control of full·penetration in GTAW. Furtheremore, the measurement technique should not rely on access to the back side of the plate (or the inner side of the pipe).

As stated before the long-range purpose is to improve the quality and reliability of pipe welding. This purpose is kept in mind while developing the measurement technique for the simpler case of plate welding (primarily on steel plates). To be expandable to pipe welding, the measurement technique should not rely on a particular gravity orientation relative to the weld, and the sensor should be easily portable. (Although pipe welding is sometimes done by rotating the pipe while the torch is stationary and its orientation constant).

Having restricted ourselves to full penetration control it is important to view the place of full penetration control in improving weld quality. In controlling the quality of the welds the purpose is to achieve certain strength, thoughness and fatigue properties in the weld and the Heat Affected Zone (RAZ). These quality parameters are related to the geometry of the weld and there is a clear correlation between them and the width of the back bead. This is clear intuitively because the extent of penetration determines the cross sectional area of the weld. With incomplete penetration the area available to support the loads is less than it could be, while extensive penetration may result in burning through and unnecessary heating of the weldment. In addition, incomplete penetration may result in very high stress consetration at the weld-plate-air interface. Nevertheless, there are other factors that affect the quality

Page 19 of the weld and they cause a significant scattering in the correlation between the back bead width and the strengh of the resulting weld. Those factors include the thermal history in the weld and in the RAZ, the size of the HAZ, porosity, slages inclusions and undercut, to mention the most important ones.

In the beginning of this three-year project an extensive literature survey was carried out. A list of references on welding automation, with all its aspects, is given in Ref 1.

1.3 Thesis Structure

The Thesis is organized into four parts and one appendix. Part 1 has begun with the description of the problem and the value of its solution. From here on the foundations for the main work are provided. Part 2 describes the theoretical and experimental work done to develop the measurement technique. Part 3 analyzes the overall controlled system and its response to disturbances. Part 4 summarizes the work done and outlines future work along this path. The appendix covers the method used in this

work to compute the frequency domain characteristics of a system.

1.4 Penetration Measurement Techniques

The fundamental step toward developing a weld geometry control is to conceive the physical phenomenon that relates the weld geometry to a measurable parameter. 'nlis work establishes the relation between the back bead radius and the puddle impedance based on puddle dynamics. Before describing this concept it is worthwhile to review penetration measurement techniques that have been investigated by other researchers. The physical

Page 20 phenomena that have been used as a basis for developing penetration measurement techniques are: (1) Til.e density and viscosity change at the liquid-metal interface, and (2) Til.e radiation from the hot puddle. Here at M.I.T., Joel Katz (Ref 2 ) is developing a method to sense the extent of penetration and even the puddle shape, using ultrasonic pulse echo techniques. Til.e technique is based on the fact that the density and the ability to sustain shear is changed at the metal-liquid interface. Til.e technique is in development stage, and so far he was able to experimentally relate the ultrasonic reflactions to the size of the puddle in a rod geometry (where the puddle is made in one end of the rod and the tranducers are in the ather end).

Several attemps have been done to develop a penetration measurement technique based on puddle radiation. Vroman and Brandt (Ref 3) used a line scan camera to measure the width of the top side of a weld. Til.e width to depth ratio of a weld remains relatively constant under fixed welding conditions and as long as the puddle penetration is small compared to the. thickness of the workpiece. Under those conditions, controlling the top weld width results in a regulated penetration. 'nle results reported by Vroman and Brandt were not conclusive. It will be even more complex to use top width control to regulate full penetration unless an additional measurement is done.

Another approaches involved sensing the back bead radiation. Nomura et. al. (Ref 4) used photodetectors to detect the infrared radiation from the back side of the weld for partial penetration submerged welding. 'nlis radiation increases drasticly when full penetration is reached and at

Page 21

M.I.T., Dave Garlow (Ref 5) has used an infrared phototransistor to measure the back bead width of full penetration welds. He has used this measurement technique along with a computer based controller to regulate the back bead width.

1.5 'nle Basic Ideas in This Work

'nlis work concentrates on, two ideas: using puddle mechanical impedance measurement to sense full penetration, and using the arc-voltage to measure the puddle impedance. The first idea is fundamental while the other idea, although very attractive, is not the only way to sense the puddle motion.

Both ~he static and the dynamic impedance of the puddle can be used·to distinguish between full and partial penetration. As full penetration is reached there is no metal base to support the puddle and the weight of the puddle is supported mostly by surface tension. As the puddle moves vertically the angle of those forces changes providing a position-dependent vertical force or, in other words, a spring (a non linear spring), (see Fig

,

1-2). The static impedance or stiffness of the puddle determines the steady state depression of the puddle under gravity and external forces. The dynamic impedance determines the response of the puddle to periodic forces, and can be charaterized by the natural frequency of the puddle. Both the steady state depression and the natural frequency of the puddle are related to the size of the puddle, and can be used to sense indirectly the back bead width.

N M

~Page 22

PLASMA FORCE

N (Surface

Tension)

p Vg

(Gravity)

(Vertical

Non-Linear

Spring)

Fig 1-2

Model of a Completely Melted

Weld Bead

Page 23 Using the static impedance phenomenon the distance between the torch and the upper surface of the puddle, as well as the torch to workpiece distance, should be regulated to maintain the desired back bead width. The measurement technique based on the dynamic impedance of the puddle involves exciting the puddle with a vertical force (orginating, for example, from the plasma jet momentum), measuring the puddle motion, and then calculating and regulating the natural frequency.

Both methods require a sensor that is sensitive to the vertical position of the top surface of the weld. An attractive method to do so uses the fact that the puddle motion changes the arc length which in turn changes the arc potential. Consequently, arc voltage measurement is sufficient and no additional instrumentation is needed. Interpreting the arc voltage is very difficult, as will be demonstrated in the next part, but the advantage promised by this technique should provide the incentives to push forward in this direction. Nevertheless the fact that other sensors can be developed should not be ignored. Optical sensing of the top surface of the puddle that can pick up elevation changes (based, for example, on interference) is one way to go. In this work only the voltage measurement technique has been investigated for in-process uses, while an additional technique has been developed for research purposes only.

Page 24 Chapter 2 Basic Experimental Apparatus

Two different types of experimental apparatus have been used. The first provides the means to control the velocity of the workpiece and to make a continuous weld, while in the other experimental apparatus both the torch and the workpiece are stationary. A sophisticated current regulator has been used with the stationary weld experimental apparatus, and although portable, the current regulator has not been transfered to the continuous weld experimental set. The continuous weld experimental set has been built gradually at the Laboratory for Manufacturing and Productivity and is shared with other students working on different projects. 'nle stationary weld experimental set has been built for this pr~ject (the three-year project sponsored by the DOE) by Yehuda Dror and Jose Converti who have worked on this project during the first two years, and is located at the Energy and Chemical Dynamics Laboratory.

2.1 'nle Continuous Weld Experimental Apparatus

The basic facility is shown schematically in Fig 2-1 (see Ref 6). An AIRCO 200 amp power supply is used in conjunction with a gas cooled torch and tungsten electrodes. 'nle torch is attached to the quill of a vertical milling machine having computer controlled table positioning capability. The workpiece can be moved, at a controlled velocity, past the stationary torch. The entire torch-workpiece area is shielded by an aluminum box with viewing ports in the sides.

QUILL

Page

25

GAS

~POWER

TORCH SHIELD

NOT SHOWN

COMPUTER COMMANDED

0. C. SERVO POSITIONING SYSTEM

Fig 2-1 The Continuous Weld Experimental Apparatus

Page 26

The table velocity, and therefore the welding velocity, is controlled by a DC servo system commanded by a microcomputer (A Cromemco Z2-D operating at 4MHz, executing FORTRAN). This computer is also interfaced to the arc voltage through a 12 bit, high accuracy A/D system (based upon the BURR-BROWN data acquisition module).

Most of the experiments were done under the standard conditions described here. The experiments were done on 11-gauge (0.1" thick ) HR steel cut into either 3X5" or 6Xl0" coupons. A 3/32" diameter tungsten, 2% toriated, electrode was used, 0. l" above the workpiece. Argon gas was used for shielding at 20GPH. Those conditions will be referred to as standard conditions in relation to experiments done at the continuous weld experimental apparatus.

2.2 'nle Stationary Weld Experimental Apparatus

The basic facility is shown in Fig 2-2 (see Ref 7). 'nle DC power supply is a 3-phase rectified supply capable of providing a maximum voltage of 70 volts and a maximum arc power of approximately 4-SKw. A GTAW water cooled torch (Airco Heliweld H20-C) is used with Argon shielding gas. A traverse mechanism is used to position the torch above the workpiece. Cooling water is supplied to the various components from a common manifold through a water flow meter. 'nle facility provides maximum access to the weld and new epuipment can be easily added; a capability that has been used as described later on.

WATER

TO COOL

ELECTRODE

WATER TO DRAIN

M O BRICKS

g

COOLED COPPER PLATE

Fig 2-2 The Stationary Weld

Experimental Apparatus

(Ref

7)

1-d Pl (Jl:j (D N ----.)

Page 28 The arc-current is measured by the voltage

placed between the positive side of the

drop across a resistor, current

workpiece. The voltage is measured directly between the and the workpiece (grounded through the water pipe).

regulator and the torch (negative)

In most of the experiments done in this facility the ability to regulate the profile of the current has been essential. This capability has been provided by a power transistor current regulator made by Alexander Kusko, Inc. to specifications written by Jose Converti (see Ref 8 for an extensive description of the current regulator). The reference signal to the current regulator is constructed according to the specific need at each experiment, and in general, although not during this work, can be supplied directly from a computer.

(The grounding of the current regulator has caused a lot of troubles. While the positive terminal of the current regulator is connected to the workpiece and to the ground (through the water pipe), the ground of the. reference signal is connected to the negative terminal. Thus, the signal reference must be kept floating with respect to ground, otherwise the two terminals are shorted. This problem has caused additional inconvenience: the signal reference and the arc-current or arc-voltage cannot be measured simultaneously on the same instrument (i.e. the same Oscilloscope or Tape recorder). Furthermore, the instruments that have been used to construct the reference signal have been floating with respect to ground at as much as -50V (when the power supply is on but the arc is off).)

Page 2 9

It is convenient to describe at this stage the standard conditions applied in most of the experiments done at this experimental apparatus. A 3/32" diameter, 2% toriated tungsten electrode machined to a point was used at .2" above the workpiece. The standard workpiece was an 11-gauge (0.1" thick) HR steel cut into 3X5" coupons (will be referred to as thin plates). Argon shielding gas at 20GPH, and water cooling at 15-20psi have been used.

All the puddles have been formed on plate with no filling material.

Two other kinds of workpieces have been used besides the thin plates: A 1/4" thick HR steel cut into 3X5" coupons (referred to as thick plates) and copper plates.

Page 30 Chapter 3 Arc Characteristics

-The Foundation of an Impedance Measurement Technique

In this chapter, two characteristics of the arc are described. The electrical voltage-current curves and the Arc-force phenomenon. While the concept of puddle impedance and its relation to puddle penetration has been described in previous chapters; this chapter concentrates on those arc characteristics which are important

measurement technique.

3.1 Electrical Features

in implementing an impedance

The voltage drop across the arc is divided into three regions: The cathode fall space, the plasma column, and the anode fall space. The major voltage drop is acrossed the anode and cathode terminals (see Ref 9).

The Voltage-Current Characteristics Curves: The total potential across the arc initially falls with increasing current and then rises with a further increase in current. The initial decrease in the total Arc-voltage with in~reasing current can be attributed

thermal ionization and thermally induced electron emission

to a growth of at the arc cathode. As current is increased the temperature rises the thermal mechanisms for electron emission becomes stronger and a smaller electrical field is needed. At higher currents this pbemonenon is saturated and the impedance characteristic of the arc is apparent.

Page 31

Typical characteristic curves are shown in Fig 3-1 and Fig 3-2 (see Ref 7). 'nlose curves were obtained by Yehuda Dror, during his Master research, from experiments done on the stationary puddle experimental apparatus (Section 2 .2). The

voltage-current characteristics for

curves represent the steady-state an Argon arc maintained between a tungsten electrode and a water-cooled copper plate. Arc lenghs of .2" and

.42" and electrode tips of .094" (3 /32" )and .040" were studied.

The curves for the smaller diameter electrode reach the minimum at lower currents and show a larger slope in the region of positive slope. 'nle data for the two arc-lengths can be used to extrapolate to the limit of zero arc-length and the resulting curves are indicated in the figures. The slope of the resulting voltage curve at the region - of positive slope represents the resistive losses in the tungsten tip. 'nle resistance of the tungsten tip is about .040 ohms for the small diameter tip, and can be neglected for the larger electrode.

Arc Impedance: The major elements that contribute to arc impedance are the tip resistance, the arc resistance, and the arc inductance. To study the overall arc-impedance, the transfer function of the arc was calculated ('nle method used to calculate the transfer function is based on a noise signal in the input, as described in Appendix A). 'nle arc, about .2" long, was maintained between a 3/32" tungsten electrode and a water-cooled copper plate (1/2" thick). 'nle experiments were done in the stationary weld set (Section 2.2) at standard conditions as mentioned there. In this arrangement no puddle is formed on the copper thus, the arc dynamics is decoupled from the puddle dynamics.

40

30

(~LT~

20

10

10

Fig

3-1

V-I Curves· For

3/32"

Diameter Electrode

(Ref?)

L,

=

0.42"

L2

=

0. 2

11

•

' ,

--...

______

\

______ _

____ _

.

L3

= 0

( EXTRAPOLAtED)

20

30

40

50

60

70

80

I

(AMPS)

--90

---~ Pl (J'q (D \..,J l\)

30

(VvoLTs)

20

10

L

_{1 ::}

0.42"

L

₁ =

0

(EXTRAPOLATED)

""--

-10

20

30

40

50

60

1

(AMPS)

Fig

3-2

V-I curves For 0.040" Diameter Electrode

(Ref

7)

70

---

---

--

--

----80

90

'"d

~

_(D \.,.) \.,.) "'"""!I

Page 34

The DC current was either 60A or 120A. The AC component (about 12A peak-to-peak), had a constant power at the range of frequencies of interest: between lOHz and 250Hz. The input signal was achieved by passing the output of a Random

law-pass-filter before adding· it to circuit ). The current and voltage

Noise Generator through a 250Hz the DC voltage (from a battery were measured with the arrangement described at section 2.2. The current signal was amplified by a factor of 100. Both the current and voltage signals were FM recorded on tape on a scale of -.5V to +.5V at a speed of 3 3/4 ipm. The data was used as an input to the HP Structural Dynamics Analyser (described in Appendix A) to calculate the arc transfer function. (Current input, Voltage output ).

The transfer function, both phase and log magnitude, and the coherence (which indicates the causality between the input current and the ouput arc-voltage), are shown in Fig 3-3 through Fig 3-6. The 120A DC arc has the following characteristics : The log magnitude of the transfer function reaches a constant value (-1.38 db ) at 60Hz and is less than 2db below this value between 20Hz and 60Hz. The coherence is very high, between 9.50m and 1000m (the maximum), and the phase is constant at 5 degrees. Comparing the arc characteristics at 60A DC current with those at 120A: The coherence is much lower, between 400m and 1000m, the phase is larger, (about 17 degrees). The log magnitude curve has much more variance and the corresponding constant value is smaller at -2.8db.

The transfer function indicates that the arc-impedance below 250 Hz is mainly a resistor. Any significant inductance would have resulted in a

TRANS

(Transfer Function)

,,., '20 _o;poo

4.o · .,

2. 0 ·

o.o

-1.4

_{I .}

~ l ~ ! \ , J ~

~ ¥

-6.

O •

-8.0

-10.

o.J_ _ _ _ _

-,---.---,-,---r--,---,----,--,-

_{1 ~ ~}

-20

₄₀

_{LG HZ}

₁₀₀

Fig

3-3

Transfer Function (Current-Inputp Voltage-Output) of Arc on Copper DC=120A

"'Cl Pl Otl (I) \..,.) \J\

colQ (Coherence)

990

m

REAL

959

m,...

20 I I ..

l""

! '

I \ /

I I l \ '

/.

~.i

)

,,,, 2Q . I

.

I :

.

. {

,.

~ .• ~ I • • 11 " A t 1, , I ' / I /1 t;l'\j! • II ''"", ,/ \"": l • A , A , l A • 1

" J,

₁

···._/·.J ,,

;i , 1.:.d. , ,

,1.

:

1,:

:'1/

1 ; }· 1

:\r~!

_{._)!.-J.·:, 1r}

'.:'i

i

.,1

/ \ • \ \ • 1 ! V •

lJ

! '.' . ·1, ,,: Jf ~ . ./, 1 ! . I /i \t J • :, IJI.

I

r

~ j •I ' ' V , I ; /· • 1 . I I ., 1 't l ' It}'~ i/; • ·1:-1· I

'\; ,. ·\ i! .1. i. J t I ·f.t'i;:

1·1·:

!.'~ I

i

.

.

'. . .

~ I \ I • ,·. . ' .; ·,,., • , / •I 'f \;';i I •:: > t•,J • I. I. :1 .. 1 I , . :' If J I I • , 1 ·: ': 1 · 1 I .

\ :i

_.,

40

l

. ')

:

·. • .

~ f " •1 . '.! LGHZ

100

TRANS

(Transfer· Function)

_{IA. 28 EXPAND}

180.0

J

i

... : -; PJtASE

_!

l _j i • ~ ' -180.88-j 28.000

₄₀

LG HZ 100

Fig 3-4 Coherence and Phase Corresponding

to Pig

J-J

•

1-d Pl OQ (D \..v ()\

1RANS (

Transfer Functiqn)

I, Q ~. t:f'. 2.0 0. 0

-LGMAG

DB-·,·

-6.o

-8.0

\t .

"" 21 -10.

o

,.1---r---.--~---r--r----r-r-r---'

FigJ-5

20.800

₄₀

LG HZ

₁₀₀

I

Transfer Function (Current-Input, Voltage-Output) of Arc on Copper-DC=60A

! i

...

l"(j Pl (}t} CD \.,J --.J

COffER

(Coherence)

,,.. 20

902

'j

I

/1

~

,. ·. r..f\/\

in

jl

·N'1

·

._,

• .

' . \ A

I

\

'

IV \

i \

fl/

i

/i

A

A

~

. . ' V

i

1

'

ii

!1 ,'

I

1~!1

- -;;;--r---r-:-:;-r~~--~'·

"

u

I

..

I I -- I - I l

I

LG HZ 100 20

₄₀

TR/INS (

Transfer Function)

_{fAt 29 EXP/IND}

198.11.1,..., _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

"

PH"SE

-180. 11 .J

28. fflJ8

40

LG HZ

100

Fig

J-6

Coherence and Phase Corresponding to Fig

J-5 ·

p, "'d Ol:l

(I)

\.,.)

Page 39 non constant phase.

The comparsion made above between the arc characteristics at 60A and 120A shows that at 120A the variance is smaller and the coherence (or causality) is larger. This phenomenon can be attributed to the fact that at 60A the voltage/current curve is very non linear, and it becomes more linear at higher currents. See Fig 3-1. Any measurement technique that relies on interpreting the ar~-voltage will be easier to implement if the arc is maintained at high DC currents (e.g.

currents (e.g. 60A ).

120A) instead of low DC

Arc Voltage dependance on Arc Length: Of particular important to any attempt to detect the puddle motion by arc-voltage measurement is the change in the arc voltage due to a change in arc-length. For a 3 /32" diameter electrode this parameter can be estimated from the data available at Fig 3-1. At 60A the corresponding voltage values are:

V(L=.2") = 10(1+3/3 .5)V==l8.6V V(L=.42 '')= 10(1+1. 5/3. 5 )V=l4 .3V Renee: V(L==.2 ")-V(L•.42 ") Eq 3-1 - = D,.L • 42 "- .2" • 20V/in (•.8 V/mm)

For a 1mm. change in plate to torch distance the steady state change in arc voltage is .8V (800 mV ). 'nlis parameter is appropriate to describe the steady-state relation between the initial depression of the puddle and the corresponding change in arc voltage. To find the dynamic response of the arc-voltage to the puddle motion (at 10-250Rz ),the dynamic version of this parameter had to be calculated. Jose Converti during his Ph.D.

Page 40 'nlesis work has investigated the arc-dynamics (see Ref 10). Based on his work, the arc time constant is less than .OOlsec. Hence, in the range of frequencies of interest (below 250Hz ), the above steady state relation, Eq 3-1, can be used for the dynamic case as well.

3 .2 Arc Force

In this section, the arc-force and the plasma jet momentum phenomena are summarized. 'nlese phenomena will enable us to excite the puddle by manipulating the current signal while keeping the average current or heat input constant. 'nle work described here was done by Jose Converti (see Ref 10 ). Plasma jet formation in a welding arc results from the expansion of the current path from the imposed constriction at the torch tip to a larger area. 'nle main (axial) welding current induces a magnetic field oriented in the azimuthal direction, the axial and radial current components interact with this self-induced field to cause motion of the fluid. 'nle radial pressure results from the interaction of the axial current and the azimuthal magnetic field. The total pressure force on the plate (or puddle) is independent of the distribution of current and is given by: Eq 3-2 F(from the radial pressure)= (r/87r)I2

Where

r

is the permeasibility of the molten metal and is approximately same as the permeasibility of the air, and I is the arc-current.

the

The interaction of the radial current with the azimuthal magnetic field results in an axially directed body force. Since the radial curent is highest near the tip, i.e. where the main current expansion occurs, the momentum is imparted to the fluid in the tip region. 'nle flux of momentum is converted to a force on the plate, since the flow direction is altered

Page 41 by the plate. For a fixed, flat plate the axial force is:

Eq 3-3 F(axial)

=

(f/8iT)i2

(21n

~/R

i)

where~ and

Rz

are the radii at the tip and after expansion, respectively.

The total force is the sum of the two components described above: Eq 3-4

In his work, Converti determined the momentum plasma jet by measuring the deflection of the arc caused by the application of an external magnetic field. The variation of momentum flux with current for a torch with a tip angle of 32 degre (shown in Fig 3-7) follows the expected square law d~pendence. For comparsion the force from the radial pressure distribution is shown, and can be seen to be approximately a factor of 3 times lower, at least for the tip angle of these experiments. The variation of momentum flux with tip angle has also been measured during that work and shows the expected trend of increased momentum at sharper angles (where R

₁

is expected to be smaller for the same tip surface area).

The expected square law dependence is noteworthy because it indicates that the force increases faster than the heat input, which increases roughly linearily with current. Furthermore, the arc force can be varied while keeping the average heat input and puddle size constant. A sinousoidal component in the current at frequency f (that is higher than the inverse of the thermal response time of the puddle), will result in sinousoidal components in the arc force, while the average heat input and the resulting weld width will remain constant. This is the physical basis for the technique of exciting the puddle by arc-current manipulation.

TIP ANGLE

=

32.8°

Plasma Jet Moment~

_.,,....

,..

/

Page 42

-

-

--

--

- -

~

Force From

Radial Pressure

0

20

40

60

80

100

f20

140

160

!(Amps.)

Fig

J-7

Variation of Plasma Jet Momentum

With Current For An Argon

Arc

Page 43 More quantitatively, the arc force resulting from a sinousoidal component in the current at frequency w is (according to Eq 3-4):

2

2

2

·

Eq 3-5 F = (r/2rT)(Io Tla t2+2IoiaS1n(wt) +Ia /2 Cos(2wt))

Where Io and Ia are the DC and AC components of the current, respectively. 'nle force is a non linear function of the AC amplitude, and has components at both wand 2w. During the worK done to develop an impedance measurement it has been important, when the puddle transfer function has been investigated, to keep a linear relation between the current and the arc force. For Ia<<Io the relation between the AC component of the current and the AC component of the arc force can be linearized. 'nle AC component of the force at frequency 2w is neglected and the component at frequency w is: Eq 3-6

•·

When important, the AC component of the current has been kept small compared to the DC current (about 10%). 'nle system, in this case, can be approximately described by the linear relation, Eq 3-6.

Page

44

Part 2 MEASUREMENT TECHNIQUES BASED ON PUDDLE IMPEDANCE

Tilis part develops the concepts of the static and the dynamic impedance of the puddle that were introduced in section 1.5. In chapter 4, the ability to sense the puddle depression by measuring the DC level of the arc-voltage is investigated. Chapter 5 develops the theory of the puddle impedance and states the relation between the natural frequency of the puddle and the puddle size. Chapter 6 checks this theory experimentally and describes the efforts to sense the puddle impedance by measuring the arc-voltage.

Page 45 Chapter 4 Puddle Deflection and The Extent of Full Penetration

The basic concept: As full penetration is reached there is no metal base beneath the puddle, and the puddle must rely upon the surface tension forces to remain in place. Since these forces are elastic in nature, the puddle must deflect until a force equilibrium is established. This new steady state equilibrium position will be lower than the non-fully penetrated case and thus indicate the transition. Further melting beyond this point will cause greater static deflection as the puddle mass increases faster than the surface tension. (The mass is proportional to the puddle volume whereas the surface tension spring constant is proportional to the length of the liquid-air-metal interface.)

Measurement of gross changes in the steady state puddle height can be used, therefore, to detect the point of complete penetration and to further indicate the extent of melting beyond that point. The subject of puddle depression has been discussed in the literature (see Ref 11 and Ref 12 ) as a way to enhance weld penetration but not as a way to control full penetration.

Experiments: The deflection of the puddle, as expected to occur at full penetration, changes the arc length. As discussed in section 3.1, a change in the arc length is characterized by a proportional change in the arc-voltage. Thus, the arc-voltage can be used to sense the puddle deflection and to regulate the extent of full penetration. A fundamental experiment to demonstrate this phenomenon was done in a simple way: The arc-voltage between a tungsten electrode and a stationary puddle was recorded. At the instant of complete penetration (detected by observation)

Page 46 the voltage rapidly increased. During the experiment shown in Fig 4-1, for example, the voltage increased from about 11.8 volts to approximately 12.4 volts.

In the next step it was important to demonstrate the ability to sense changes in the puddle depression while welding continuously. The experiments were done at the continuous weld experimental apparatus (described in section 2.1) and the necessary software to control the table velocity was written. As before the arc voltage has been recorded while the arc was started and the table was moving at a constant initial velocity. During the course of the experiments the velocity was sudenly changed resulting in a step change in the heat input per length of the weld, since the welding current was held constant.

A typical record of the arc-voltage (through a lHz second order digital Low Pass Filter) is shown in Fig 4-2. The current was held constant at 100A. The initial velocity was 5.4 ipm, and the resulting weld was partially penetrated. When the velocity was reduced to 4.5 ipm (at the vertical line) full p

₇

netration was reached and the puddle depressed. As

expected the magnitude of the arc-voltage increased, by about .2 volts (200mV ), with a characteristic time constant. (In Fig 4-2 the negative arc-voltage is shown, and the increase in the magnitude of the arc-voltage is shown as a decrease in the negative value of the arc-voltage.)

More experiments, like the one described, were done by Ken Pasch, and the repeatability of those results was demonstrated. His experimental work showed that for the same change in the welding velocity, and under the same welding current, the resulting change in the arc-voltage depends on the

-1°4

rMe

l

t-through

Arc

_-12

~

Voltage

(volts)

· -10

Time

Fig 4-1 Arc-Voltage at Melt Through

Stationary Puddle

Page

47

100mV

~

Arc-Voltage

{Negative)

Step Change in Welding Velocity

_.at t=O

1

Time (sec)

2 ₃

4

₅

6

₇

8

Fig 4-2 Arc-Voltage Respons·e To· 'Change In Heat Input

During a Continuous Weld.

{Step Change in Welding Velocity at The Horizontal Line)

1-'d S}) ~ co .(::-(X)

Page 49

initial velocity, or in other words, on the initial extent of penetration.

Control strategy based on sensing the puddle deflection: Once full penetration is reached it can be maintained by controlling the welding velocity to keep a constant arc-voltage. (Such a control . strategy is discussed in Ref 6). The main problem with such a strategy is that the arc-voltage is used extensively in welding machines to position the torch

"

at an approximately constant distance above the workpiece to maintain constant heat input. An additional use of the arc-voltage for penetration control will interfere with the purpose of this system.

To control the extent of penetration, the amount of the puddle deflection has to be controlled. Or al ternat.i vel y, both the torch to workpiece distance (d), and the distance between the torch and the upper surface of the puddle (1) have to be maintained constant. See a schematic in Fig 4-3. For this purpose the arc-voltage can be used to measure the distance 1 (instead of the distanced as is done in seam tracking machines) while an additional mechanism should be developed to measure the torch to workpiece disance (d). Jon Weinert is currently developing this approach at MIT and will publish his results under his Master Thesis.

Another control strategy based on puddle deflection measurement can be developed without interfering with the use of the arc-voltage signal for seam-tracking. This control strategy relies on the fact that a constant change in the welding velocity will result in different arc-voltage changes according to the initial extent of penetration (as demonstrated by the experiments done by Ken Pasch). The approach involves changing the welding velocity in order to measure and regulate the corresponding change in the

J~

_ . . . _ __ .

Puddle

~

Fig

4-3

Puddle Deflection

d- Torch to Workpiece Distance

1- Arc Length, Distance Between

Torch and Upper Surface of Puddle

Page 51 arc-voltage. These controlled changes in the welding velocity should be small to result in a smooth and nearly constant back bead width. More experiments, with a small change in the welding velocity should be done to determine the feasibility of this control scheme.

Advantages, disadvantages and further suggestions: The advantage of this approach is that the extent of full penetration can be controlled by the simple mea~urement of the arc-voltage. But since the arc-voltage signal is often used for the purpose of keeping a constant torch to workpiece distance, a more sophisticated approach should be developed. The main disadvantage of this approach is that it relies heavily on gravity orientation relative to the weld, and is not expandable to all types of pipe welding.

One problem in implementing this approach is that the .arc-voltage is affected by other factors. As discussed in section 3.1 the arc-voltage drop is mainly across the electrode space, while this method is concerned with the arc-voltage drop across the plasma arc only. It is wise to decouple the voltage drop across the arc from the voltage drop across the electrodes. One way to do so is to include a high frequency sinousoidal component (e.g. lOOHz) in the arc-current. The corresponding AC component of the arc-voltage will depend on the electrode tip resistance and on the arc-impedance (which is proportional to the arc-length) but not on the large voltage drop across the electrodes. The experience gained during this work about the arc-voltage behaviour (see sections 3.1 and 6.3) can be used to check the improvement from adding an AC component to the arc-current at the cost of the more sophisticated current regulator.

Page 52

Chapter 5 Puddle Impedance - Tileory

Tilis section provides the theoretical analysis of the puddle motion. The purpose is to demonstrate the difference between the modes of motion at full vs. partial penetration, and to develop the expressions for the corresponding natural frequencies. In doing so, the goal is to identify the main parameters that influence the puddle motion, and to get an estimate of the values of the natural frequencies. Tile analysis is carried out for a simplified case to gain insight into the problem, and absolute accuracy is not important.

The surface motion is investigated in two ways • Using fluid

.

mechanics, the surface motion is developed as part of a velocity-potential problem. 'nlis method clarifies the theoretical distinction between the partial and full pentration cases. Since only the surface motion is of interest (and not the value of the velocity- potential function throughout the puddle) a lumped parameter approach is also developed. Tilis approach requires even more assumptions but is more intuitive.

,

5.1 Modeling Assumptions

The analysis is done for a stationary puddle that has a cylindrical shape. (See Fig. 5-1). The cylindrical shape is somewhat similar to the actual shape of a stationary puddle at extensive full penetration. The shape, however does not accurately describe either a stationary puddle at partial or slighly full penetration (when the back bead radius is small compared to the upper bead radius), or the puddle shape under a moving torch.

h

..

h

Outside Pressure Pi

2R

Outside ~essure~z=W

P2

Outside Pressure P1

2R

••

I (b)

Fig 5-1 Model of Puddle at Full (a)p and Partial (b)· Penetration

\ ltj ~ (Jtj CD \J\ \...,J

Page 54

The molten metal is nearly incompressible and is assumed to be inviscid. Furthermore the molten metal is assumed to be homogeneous the surface tension (N) and the liquid density (q) are constant and independent of the temperature. (It is reasonable to assume that the fluid flow inside the puddle will result in a small gradient of temperature).

'nle surface tension at the interface of the weld puddle with the surrounding atmosphere supports the molten metal at full pentration and prevents burn-through. Surface tension at the metal-liquid interface is not considered. (For the assumptions about the molten metal see Ref 11). For convenience a complete nomenclature list for the variables in this chapter is provided in the next page.

5.2 The Velocity Potential Problem

The motion of the weld pool as a result of the Lorentz force has been investigated in the literature by Woods and Milner (Ref 13), and Scott (Ref 14). The effect of the surface tension on the molten metal motion inside the weld was considered by Ishizaki (Ref 12), and Bradstreet (Ref 16). The purpose of these works is to understand qualitatively the mechanism of weld formation, while in this section the ability to quantitatively correlate the extent of penetration with the upper surface motion for measurement purpose is investigated.

The Governing Equations: neglecting the liquid-metal

Assuming the molten metal is inviscid and viscous effects, the fluid flow can be described by the velocity-potential function ~. For an incompressible medium the mass conservation law results in laplace equation

Variabie

aoo' boo

-e F g h

Ia,

Ia

r

₀

(er)

Jm(kr)

k K(R) m N p p1' p2 p C q

r

R

s

t

wn

w

z

u

Page

55

Meaning

The first non-zero solution of

J

0

(a)=O,

J

_{0 (b)=O}

Parameter of Bessel function equal to (qg/N) 1/ 2

Arc-force

Gravity constant

Height of the puddle-cylinder

DC, AC Arc-current

Associated Bessel function, order O parametere

Bessel function of order m and parameter k

Parameter of Bessel function

Spring constant associated with surface tension

of puddle with radius R

Puddle mass

Surface tension of molten metal

Pressure

Outside pressure on the upper, lower surface

Constant Pressure

Density of molten metal

Radial coordinate

Radius of the puddle-cylinder

The shape of the upper surface of the puddle

as function of r,

(Q),

and t

Time

Na~_ural frequency of the puddle

The shape of the lower surface of the fully

penetrated puddle as function of r,

(Q),

and t

Vertical coordinate

Gross, steady state, puddle deflection

Amplitude of the puddle vibration

Velocity potential function

Azimutal coordinate

Permeasibility of molten metal

Page 56

'°v2(0)=0

In cylindrical coordinates the three space variables are r, z, and 8, and the mass conservation law has the form:

1 d d0 1 d2

0

d20 Eq 5-1 - - ( r - ) + - - + ~ = 0

r dr dr r d82 dz

The momentum equation for inviscid flow, after integration, gives:

d(/) 1 p

Eq 5-2

-

+

-'\/</J'v</J

+

-(r,8,t)

+

gz = F(t)

dt 2 q

/

for every point in the potential flow. Pis the fluid pressure, q constant density, and z the vertical position.

the

On the free surface(s) the fluid pressure equals the outside pressure plus (for the lower free surface at full penetration) or minus (for the upper free surface) the surface tension component:

2

P(r,8,t)=P

₁

(r,8,t)-NV (S) on the upper surface

2

the lower surface P(r,8,t)=Pz(r,8,t)+N'\i (W) on

See Fig 5-1 where S(r) and W(r) are the shapes (z as a function of r) of the upper and lower surface, respectively, and N is the surface tension coefficient in N/m (Newton/meter).

Using these expressions in Eq 5-2 give:

d(/) p _{(r,8,t) 1 N}

Eq 5-2a - +

+-WV</J

+ gS

:. =-vs

= F(t)

dt q 2 q

on the upper surface. And in the case of full penetration, d(/) P (r,8,t) 1 N--2

Eq 5-2b -

+

+ -V</JV</J +

gW

+

-v

VI = F(t)

Page 57

on the lower surface.

The fluid velocity at the free surface in the z-direction is related to the motion of the free surface by the kinematic condition

where D/Dt is the complete derivative with respect to time. In cylindrical coordinates this condition gives:

dS d0 dS 1 d0 dS

Eq S-3a

+

+ -

=

dt dr dr r d0 d9 dz z=S on the upper surface.

And in the case of full penetration:

dW d0 dW 1 d0 dW d0

Eq 5-3b

+

~

+ -

=

-dt dr dr r d0 d9 dz

z=W

on the lower surface.

D(z-S)/Dt=O

At partial penetration there can be no flow perpendicular to the z=O, metal-liquid interface. The boundary condition here is:

Eq 5-3c = 0

dz z::Q

Finally, the boundary condition at r=R, the metal-liquid interface, has to be specified. There are two reasonable conditions that contradicts each other. On one hand, there can be no flow perpendicular to the r=R, metal-liquid interface, which means that:

Eq 5-4a = 0

dr

Page 58

On the other hand, the boundary of the puddle surface should be fixed.

Eq 5-4b S(r=R)=h=constant (and W(r=R)=O)

The contradiction between these conditions, its cause and its consequences will be discussed later. The analysis is carried on as far as possible before this problematic boundary condition is needed.

(e.g.

Fluid flow problems that consider different parts of this situation under simple geometry or when gravity can be neglected) are described in Ref 16.

The Solution: In stating the problem, it has been shown that under the assumptions made, the situation is governed by Eq 5-1 and Eq 5-2 under the boundary conditions described by Eq 5-3 and Eq 5-4. Additional assumptions will be made in this section to get an analytic expression for the normal modes and natural frequencies.

To calculate the,~ormal modes and natural frequencies the unforced situation is investigated. i.e.

pressure does not vary with time. meaning~ and the term F(t) can

precisely, a new potential function From Eq 5-2a the resulting momentum

d9)

p. (r,8,t)=P. (r,8) i=l,2, the outside

J. J.

Only the gradient of~ has a physical be absorbed in the term D(~)/Dt (more ~·-~ J(t)dt has to be defined). equation is: N 2 Eq 5-5

- +

1i

(r,8) 1

+-'19lV</J +

gS - ·--'ils '""0 dt q 2 q z=S on the upper surface.