Spot welding system and spot welding method and nugget diameter estimation system and nugget diameter estimation method

• US 6,043,449 A
• Filed: 08/26/1998
• Issued: 03/28/2000
• Est. Priority Date: 08/27/1997
• Status: Expired due to Term

• Current Assignee: Nissan Motor Co., Ltd.
• Original Assignee: Nissan Motor Co., Ltd.
• Inventors: Kanjo, Mitsunori
• Primary Examiner(s): MILLS, GREGORY L
• Application Number:
  US09/140,316
• Time in Patent Office: 580 Days
• Field of Search: 219/86.41, 219/86.51, 219/91.1, 219/86.1, 219/108, 219/109, 219/110, 364/477.06, 700/212
• US Class Current: 219/109
• CPC Class Codes: B23K 11/25 Monitoring devices
• CPC Class Codes: B23K 11/253 the measured parameter being a displacement or a position
• CPC Class Codes: B23K 11/255 the measured parameter being a force B23K11/253 takes precedence
• CPC Class Codes: B23K 31/12 relating to investigating the properties, e.g. the weldability, of materials

Abstract

A welding spot is held between electrodes either biased with a controlled pressure, having a controlled voltage applied thereacross, and variations of an inter-electrode distance are detected for a welding quality control, in which the distance variations are time-integrated, a nugget size is estimated on a basis of the integration, and a decision is made of a conformity of welding quality based on an estimated nugget size.

Claims

1. A spot welding system comprising:

a first and a second electrode adaptive for a relative advancement to each other to hold therebetween a welding spot of a work and for a relative retreat form each other to set free the welding spot;

a mechanism for biasing either electrode of the first and the second electrode with a controlled pressure in a direction in which the either electrode makes the relative advancement;

a power supply for supplying a controlled voltage between the first and the second electrode;

a detector for detecting a variation of a distance between the first and the second electrode; and

a controller for controlling a welding quality of the welding spot on a basis of a detected variation of the distance, the controller comprising;

an extrenum holder for holding an extrenum of the detected variation before a spattering at the welding spot;

a base setter for setting a base to be the detected variation upon a restart of a thermal expansion of the welding spot after the spattering with a continued welding current conducted through the welding spot;

an adder for adding a difference between the base and the detected variation to the extrenum to provide a corrected variation;

an integrator for determining a time integration of the corrected variation;

an estimator for estimating a size of a nugget of the welding spot on a basis of the time integration; and

a decision-maker for making a decision on a conformity of the welding quality on a basis of an estimated size of the nugget.

2. The spot welding system as claimed in claim 1, wherein the integrator makes the time integration over an integration interval covering a time interval in which the thermal expansion proceeds, as the welding spot is heated with the continued welding current.

3. The spot welding system as claimed in claim 1, wherein the integrator makes the time integration over an integration interval covering a time interval in which the welding spot shrinks with the continued welding current interrupted after the thermal expansion.

4. The spot welding system as claimed in claim 1, wherein the integrator makes the time integration over an integration interval covering an arbitrary one of plural time intervals preset to a corresponding one of plural combinations of welding conditions of the welding spot.

5. The spot welding system as claimed in claim 1, wherein the estimator has a linear first model representing the size of the nugget, the first model including the time integration as a model element thereof.

6. The spot welding system as claimed in claim 5, wherein the estimator further has a second model for evaluating an error of the first model.

7. The spot welding system as claimed in claim 5, wherein the first model is representative by a recurrence formula for a multivariate analysis of the size of the nugget.

8. The spot welding system as claimed in claim 7, wherein the recurrence formula includes as explanation variables thereof an amplitude and a conduction time of the continued welding current, and the controlled pressure.

9. The spot welding system as claimed in claim 7, wherein the welding spot comprises a pair of plate members, and the recurrence formula includes as explanation variables thereof a plate thickness, a material, a joining pattern and a surface preparation of the plate members.

10. The spot welding system as claimed in claim 5, wherein the model element has a correction factor in consideration of one of a relationship between the thermal expansion and a contraction of the welding spot and an effect of the spattering.

11. The spot welding system as claimed in claim 1, further comprising processor elements for

having a first value of the time integration along the thermal expansion with the continued welding current conducted and a second value of the time integration along a thermal shrinkage with the continued welding current interrupted,

calculating a correction coefficient equivalent to the second value divided by the first value, and

multiplying values of the time integration along the thermal expansion by the correction coefficient to correct an attenuation of the inter-electrode displacement due to a tip contact area of the electrodes.

12. The spot welding system as claimed in claim 1, wherein the time integration starts when the distance between the electrodes has a predetermined value or with a lapse of a predetermined time after a start of the continued welding current.

13. The spot welding system as claimed in claim 1, wherein the time integration ends when the distance between the electrodes has a predetermined value or with a lapse of a predetermined time after a start of the continued welding current.

14. The spot welding system as claimed in claim 1, wherein the controller processes the variation of the distance between the electrodes to be insignificant along the spattering.

15. A spot welding system comprising:

a first and a second electrode adaptive for a relative advancement to each other to hold therebetween a welding spot of a work and for a relative retreat from each other to set free the welding spot;

a mechanism for biasing either electrode of the first and the second electrode with a controlled pressure in a direction in which the either electrode makes the relative advancement;

a power supply for supplying a controlled voltage between the first and the second electrode;

a detector for detecting a variation of a distance between the first and the second electrode; and

controlling means for controlling a welding quality of the welding spot on a basis of the detected variation of the distance, the controlling means comprising;

extrenum holding means for holding an extrenum of the detected variation before a spattering at the welding spot;

base setting means for setting a base to be the detected variation upon a restart of the thermal expansion of the welding spot after the spattering with a continued welding current conducted through the welding spot;

adding means for adding a difference between the base and the detected variation to the extrenum to provide a corrected variation;

integrating means for determining a time integration of the corrected variation;

estimating means for estimating a size of a nugget of the welding spot on a basis of the time integration; and

deciding means for making a decision on conformity of the welding quality on a basis of an estimated size of the nugget.

16. The spot welding system as claimed in claim 15, further comprising processing means for

having a first value of the time integration along the thermal expansion with the continued welding current conducted and a second value of the time integration along a thermal shrinkage with the continued welding current interrupted,

calculating a correction coefficient equivalent to the second value divided by the first value, and

multiplying values of the time integration along the thermal expansion by the correction coefficient to correct an attenuation of the inter-electrode displacement due to a tip contact area of the electrodes.

17. A spot welding method comprising the steps of:

having a first and a second electrode making a relative advancement to each other to hold therebetween a welding spot of a work;

biasing either electrode of the first and the second electrode with a controlled pressure in a direction in which the either electrode makes the relative advancement;

supplying a controlled voltage between the first and the second electrode;

detecting a variation of a distance between the first an the second electrode; and

controlling a welding quality of the welding spot on a basis of the detected variation of the distance, the controlling step comprising;

holding an extrenum of the detected variation before a spattering at the welding spot;

setting a base to be the detected variation upon a restart of the thermal expansion of the welding spot after the spattering with a continued welding current conducted through the welding spot;

adding a difference between the base and the detected variation to the extrenum to provide a corrected variation;

determining a time integration of the corrected variation;

estimating a size of a nugget of the welding spot on a basis of the time integration; and

making a decision on a conformity of the welding quality on a basis of an estimated size of the nugget.

18. The spot welding method as claimed in claim 17, further comprising the steps of:

having a first value of the time integration along the thermal expansion with the continued welding current conducted and a second value of the time integration along a thermal shrinkage with the continued welding current interrupted;

calculating a correction coefficient equivalent to the second value divided by the first value; and

multiplying values of the time integration along the thermal expansion by the correction coefficient to correct an attenuation of the inter-electrode displacement due to a tip contact area of the electrodes.

19. A nugget size estimation system comprising:

an extrenum holder for holding an extrenum of a detected variation of a distance between electrode that have a welding spot held therebetween under pressure, as the extrenum has occurred before a spattering at the welding spot;

a base setter for setting a base to be the detected variation upon a restart of a thermal expansion of the welding spot after the spattering with a continued welding current conducted through the welding spot;

an adder for adding a difference between the base and the detected variation to the extrenum to provide a corrected variation;

an integrator for determining a time integration of the corrected variation; and

an estimator for estimating a size of a nugget of the welding spot on a basis of the time integration.

20. The nugget size estimation system as claimed in claim 19, further comprising processor elements for

having a first value of the time integration along the thermal expansion with the continued welding current conducted and a second value of the time integration along a thermal shrinkage with the continued welding current interrupted,

calculating a correction coefficient equivalent to the second value divided by the first value, and

multiplying values of the time integration along the thermal expansion by the correction coefficient to correct an attenuation of the inter-electrode displacement due to a tip contact area of the electrodes.

21. A nugget size estimation system comprising:

extrenum holding means for holding an extrenum of a detected variation of a distance between electrodes that have a welding spot held therebetween under pressure, as the extrenum has occurred before a spattering at the welding spot;

base setting means for setting a base to be the detected variation upon a restart of a thermal expansion of the welding spot after the spattering with a continued welding current conducted through the welding spot;

adding means adding a difference between the base and the detected variation to the extrenum to provide a corrected variation;

integrating means for determining a time integration of the corrected variation; and

estimating means for estimating a size of a nugget of the welding spot on a basis of the time integration.

22. The nugget size estimation system as claimed in claim 21, further comprising processing means for

having a first value of the time integration along the thermal expansion with the continued welding current conducted and a second value of the time integration along a thermal shrinkage with the continued welding current interrupted,

calculating a correction coefficient equivalent to the second value divided by the first value, and

multiplying values of the time integration along the thermal expansion by the correction coefficient to correct an attenuation of the inter-electrode displacement due to a tip contact area of the electrodes.

23. A nugget size estimation method comprising the steps of:

holding an extrenum of a detected variation of a distance between electrodes that have a welding spot held therebetween under pressure, as the extrenum has occurred before spattering at the welding spot;

setting a base to be the detected variation upon a restart of a thermal expansion of the welding spot after the spattering with a continued welding current conducted through the welding spot;

adding a difference between the base and the detected variation to the extrenum to provide a corrected variation;

determining a time integration of the corrected variation; and

estimating a size of a nugget of the welding spot on a basis of the time integration.

24. The nugget size estimation method as claimed in claim 23, further comprising the steps of:

having a first value of the time integration along the thermal expansion with the continued welding current conducted and a second value of the time integration along a thermal shrinkage with the continued welding current interrupted;

calculating a correction coefficient equivalent to the second value divided by the first value; and

multiplying values of the time integration along the thermal expansion by the correction coefficient to correct an attenuation of the inter-electrode displacement due to a tip contact area of the electrodes.

25. A method of estimating a nugget diameter in a spot welding, the method comprising the steps of:

holding a welding spot between electrodes under pressure to detect an inter-electrode displacement therebetween;

conducting a welding current through the electrodes and the welding spot, causing a thermal expansion of the welding spot;

detecting a spattering occurred at the welding spot during the thermal expansion;

identifying an extrenum of detected values of the inter-electrode displacement before the spattering;

processing detected values of the inter-electrode displacement after the spattering to determine a growth of the thermal expansion after the spattering;

adding the growth to the extrenum to provide substitute values for the detected values after the spattering;

integrating detected values of the inter-electrode displacement, using the substitute values, to make a time integration of the inter-electrode displacement; and

estimating a nugget diameter of the welding spot on the basis of the time integration.

26. The method as claimed in claim 25, further comprising the steps of:

having a first value of the time integration along the thermal expansion with the welding current conducted and a second value of the time integration along a thermal shrinkage with the welding current interrupted;

calculated a correction coefficient equivalent to the second value divided by the first value; and

multiplying values of the time integration along the thermal expansion by the correction coefficient to correct an attenuation of the inter-electrode displacement due to a tip contact area of the electrodes.

27. A method of estimating a nugget diameter in a spot of welding, the method comprising the steps of:

holding a welding spot between electrodes under pressure, to detect an inter-electrode displacement therebetween;

detecting an initial value of the inter-electrode displacement before conducting a welding current through the electrodes and the welding spot,

continuing a conduction of the welding current, causing a thermal expansion of the welding spot;

detecting a spattering occurred at the welding spot during the thermal expansion;

interrupting the conduction of the welding current, causing a thermal shrinkage of the welding spot;

detecting a final value of the inter-electrode displacement associated with the thermal shrinkage;

adding a difference between the initial and final values to detected values of the inter-electrode displacement after the spattering, to provide substitute values for the detected values;

identifying an extrenum of the detected values of the inter-electrode displacement before the spattering;

integrating detected values of the inter-electrode displacement using the substitute values, to make a time integration of the inter-electrode displacement; and

estimating a nugget diameter of the welding spot on a basis of the time integration.

28. The method as claimed in claim 27, wherein the time integration is made within an integration interval between a start point and an end point, wherein

the start point is defined as one of:

a point of time when the current conduction is interrupted;

a point of time after the current conduction when the inter-electrode displacement reaches or exceeds a predetermined level; and

a point of time when a predetermined period has elapsed after the current conduction, and

the end point is defined as one of;

a point of time depending on a welding condition; and

a point of time when a predetermined period has elapsed after the current conduction.

29. A method estimating a nugget diameter in spot welding, the method comprising the step of:

estimating an actual weld nugget diameter by an integration of an inter-electrode displacement due to a thermal expansion during a conduction of a welding current, wherein

the integration of the inter-electrode displacement due to the thermal expansion is performed within an integration interval defined by a combination of an integration start point and an integration end point,

the integration start point is one of;

an integration start point determined in advance on basis of welding conditions;

an integration start point determined as a point during a current conduction, where the inter-electrode displacement reaches or exceeds a predetermined level; and

an integration start point determined as a point where a predetermined period of time has elapsed from a start of a current conduction, and

the integration end point is one of;

an integration end point determined in advance on basis of the welding conditions;

an integration end point determined as a point during or after a current conduction, where the inter-electrode displacement as it has once exceeded a maximum value, decreases to or exceeds a predetermined level;

integration end point determined as a point where a predetermined period of time has elapsed from the integration start point;

an integration end point determined as a point where a thermal expansion rate based on the inter-electrode displacement during a current conduction becomes a 0; and

an integration end point determined as a point where a predetermined period of time has elapsed from a start of a current conduction, and

a regression expression is computed for determining an estimated nugget diameter on a basis of an inter-electrode displacement integration value calculated by integrating the inter-electrode displacement due to the thermal expansion during a conduction of a welding current, a welding current value, a current conducting time and a welding pressure, while searching a statistical model on a basis of multivariate data to estimate the actual weld nugget diameter, such that

space="preserve" listing-type="equation">y=b0+b1x1+b2x2+b3x3+b4x4 and

space="preserve" listing-type="equation">Yi=a+byi+e,

where y;

estimated nugget diameter

x1;

inter-electrode displacement integration value

x2;

welding current

x3;

current conducting time

x4;

welding pressure

Yi;

i-th measured nugget diameter

yi;

i-th estimated nugget diameter

a;

intercept

b;

b0 to b4;

regression coefficients

e;

error.

30. The method as claimed in claim 29, further comprising the steps of:

holding a maximum displacement just before an occurrence of a spattering, as the spattering has occurred during a welding;

setting a base point when welding plates restart thermally expanding with a continued welding current;

adding a relative displacement from the base point to the maximum displacement so that an inter-electrode displacement after the occurrence of the spattering is corrected;

calculating an inter-electrode displacement integration based on the corrected inter-electrode displacement; and

computing the regression expression by using the inter-electrode displacement integration.

31. A method of estimating a nugget diameter in a spot welding, the method comprising the step of:

estimating an actual weld nugget diameter by integrating an inter-electrode displacement due to a thermal shrinkage after a finish of a conduction of a welding current, wherein

the integration process of the inter-electrode displacement due to the thermal shrinkage is performed within an integration interval defined by a combination of an integration start point and an integration end point,

the integration start point is one of;

an integration start point determined as a finish point of a current conduction;

an integration start point determined as a point after a finish of a current conduction, where the inter-electrode displacement reaches or exceeds a predetermined lever; and

an integration start point determined as a point where a predetermined period of time has elapsed from a finish of a current conduction, and

the integration end point is one of;

an integration end point determined in advance on basis of the welding conditions;

an integration end point determined as a point where a predetermined period of time has elapsed from the integration start point;

an integration end point determined as a point where a thermal expansion rate based on the inter-electrode displacement during a current conduction becomes a 0; and

an integration end point determined as a point where a predetermined period of time has elapsed from a finish of a current conduction, and

a regression expression is computed for determining an estimated nugget diameter on a basis of an inter-electrode displacement integration value calculated by integrating the inter-electrode displacement due to the thermal shrinkage after a finish of conduction of a welding current, a welding current, a current conducting time and a welding pressure, while searching a statistical model on a basis of multivariate data to estimate the actual weld nugget diameter, such that

space="preserve" listing-type="equation">y=b0+b1x1+b2x2+b3x3+b4x4 and

space="preserve" listing-type="equation">Yi=a+byi+e,

where y;

estimated nugget diameter

x1;

inter-electrode displacement integration value

x2;

welding current

x3;

current conducting time

x4;

welding pressure

Yi;

i-th measured nugget diameter

yi;

i-th estimated nugget diameter

a;

intercept

b;

b0 to b4;

regression coefficients

e;

error.

32. The method as claimed in claim 31, further comprising the steps of:

sampling an inter-electrode displacement after an occurrence of a spattering during a welding, until a changing rate of the inter-electrode displacement due to the thermal shrinkage after a finish of a current conduction becomes a zero;

adding an absolute value of a relative displacement between an inter-electrode displacement when a changing rate of the inter-electrode displacement due to the thermal shrinkage after the finish of the current conduction has become the zero and a reference point of an inter-electrode displacement before a start of the current conduction to the data sampled after the occurrence of the spattering so that the inter-electrode displacement after the occurrence of the spattering is corrected;

calculating an inter-electrode displacement integration based on the corrected inter-electrode displacement; and

computing the regression expression by using the inter-electrode displacement integration.

33. A method of estimating a nugget diameter in a spot welding, the method comprising the step of

estimating an actual weld nugget diameter by an integration of an inter-electrode displacement due to a thermal expansion during a conduction of a welding current, wherein

the integration of the inter-electrode displacement due to the thermal expansion is performed within an integration interval defined by a combination of an integration start point and an integration end point,

the integration start point is one of:

an integration start point determined in advance on basis of welding conditions;

an integration start point determined as a point during a current conduction, where the inter-electrode displacement reaches or exceeds a predetermined level; and

an integration start point determined as a point where a predetermined period of time has elapsed from a start of a current conduction, and

the integration end point is one of;

an integration end point determined in advance on basis of the welding conditions;

an integration end point determined as a point during or after a current conduction, where the inter-electrode displacement, as it has once exceeded a maximum value, decreases to or exceeds a predetermined level;

an integration end point determined as a point where a predetermined period of time has elapsed from the integration start point;

an integration end point determined as a point where a thermal expansion rate based on the inter-electrode displacement during a current conduction becomes a 0; and

an integration end point determined as a point where a predetermined period of time has elapsed from a start of a current conduction, and

a regression expression is computed for determining an estimated nugget diameter on a basis of an inter-electrode displacement integration value calculated by integrating the inter-electrode displacement due to the thermal expansion during a conduction of a welding current, and a thickness, a material, a joining type and a surface preparation of welding plates, while searching a statistical model on a basis of multivariate data to estimate the actual weld nugget diameter; and

the regression expression, such that

space="preserve" listing-type="equation">y=b0+b1x1+b2x2+b3x3+b4x4+b5x5 and

space="preserve" listing-type="equation">Yi=a+byi+e,

where y;

estimated nugget diameter

x1;

inter-electrode displacement integration value

x2;

welding plate thickness

x3;

welding plate material

x4;

welding plate joining type

x5;

welding plate surface preparation

Yi;

i-th measured nugget diameter

yi;

i-th estimated nugget diameter

a;

intercept

b;

b0 to b5;

regression coefficients

e;

error.

34. A method of estimating a nugget diameter in a spot welding, the method comprising the step of:

estimating an actual weld nugget diameter by an integration of an inter-electrode displacement due to a thermal expansion during a conduction of a welding current, wherein

the integration of the inter-electrode displacement due to the thermal expansion is performed within an integration interval defined by a combination of an integration start point and an integration end point,

the integration start point is one of;

an integration start point determined in advance on basis of welding conditions;

an integration start point determined as a point during a current conduction, where the inter-electrode displacement reaches or exceeds a predetermined level; and

an integration start point determined as a point where a predetermined period of time has elapsed from a start of a current conduction, and the integration start point is one of;

an integration end point determined in advance on basis of the welding conditions;

an integration end point determined as a point during or after a current conduction, where the inter-electrode displacement as it has once exceeded a maximum value, decreases to or exceeds a predetermined level;

an integration end point determined as a point where a predetermined period of time has elapsed from the integration start point;

an integration end point determined as a point where a thermal expansion rate based on the inter-electrode displacement during a current conduction becomes a 0; and

an integration end point determined as a point where a predetermined period of time has elapsed from a start of a current conduction, and

a regression expression is computed for determining an estimated nugget diameter on a basis of an inter-electrode displacement integration value calculated by integrating the inter-electrode displacement due to the thermal shrinkage after a finish of conduction of a welding current, and a thickness, a material, a joining type and a surface preparation of welding plates, while searching a statistical model on a basis of multivariate data to estimate the actual weld nugget diameter; and

the regression expression is, such that

space="preserve" listing-type="equation">y=b0+b1x1+b2x2+b3x3+b4x4+b5x5 and

space="preserve" listing-type="equation">Yi=a+byi+e,

where y;

estimated nugget diameter

x1;

inter-electrode displacement integration value

x2;

welding plate thickness

x3;

welding plate material

x4;

welding plate joining type

x5;

welding plate surface preparation

Yi;

i-th measured nugget diameter

yi;

i-th estimated nugget diameter

a;

intercept

b, b0 to b5;

regression coefficients

e;

error.

35. A method of estimating a nugget diameter in a spot welding, the method comprising the step of:

estimating an actual weld nugget diameter by an integration of an inter-electrode displacement due to a thermal expansion during a conduction of a welding current, wherein

the integration of the inter-electrode displacement due to the thermal expansion is performed within an integration interval defined by a combination of an integration start point and an integration end point,

the integration start point is one of;

an integration start point determined in advance on basis of welding conditions;

an integration start point determined as a point during a current conduction, where the inter-electrode displacement reaches or exceeds a predetermined level; and

an integration start point determined as a point where a predetermined period of time has elapsed from a start of a current conduction, and the integration end point is one of;

an integration end point determined in advance on basis of the welding conditions;

an integration end point determined as a point during or after a current conduction, where the inter-electrode displacement, as it has once exceeded a maximum value, decreases to or exceeds a predetermined level;

an integration end point determined as a point where a predetermined period of time has elapsed from the integration start point;

an integration end point determined as a point where a thermal expansion rate based on the inter-electrode displacement during a current conduction becomes a 0; and

an integration end point determined as a point where a predetermined period of time has elapsed from a start of a current conduction, and

a correction coefficient is calculated from an inter-electrode displacement integration value associated with the thermal expansion during a conduction of a welding current and an inter-electrode displacement integration value associated with the thermal shrinkage after a finish of conduction of a welding current, such that

the correction coefficient =(the inter-electrode displacement integration value associated with the thermal shrinkage)/(the inter-electrode displacement integration value associated with the thermal expansion), and

the inter-electrode displacement integration value associated with the thermal expansion is multiplied by the correction coefficient to correct an attenuation of an inter-electrode displacement due to an electrode tip contact area.

36. A system for estimating a nugget size of a welding spot, the system comprising:

a spot welder for holding a welding spot between electrodes under pressure, detecting an inter-electrode displacement therebetween, and conducting a welding current through the welding spot;

a spattering detector for detecting a spattering occurred at the welding spot;

a first selector for selecting a first value of the inter-electrode displacement detected before the spattering;

a second selector for selecting a second value of the inter-electrode displacement detected after the spattering, the second value being different from the first value;

a corrector for correcting detected values of the inter-electrode displacement after the spattering, depending on a difference between the first and second values;

an integrator for integrating detected values of the inter-electrode displacement, using corrected values thereof as applicable, to make a time integration of the inter-electrode displacement; and

an estimator for estimating a nugget size of the welding spot on a basis of the time integration.

37. A method of estimating a nugget size in a spot welding, the method comprising:

holding a welding spot between electrodes under pressure, detecting an inter-electrode displacement therebetween;

conducting a welding current through the welding spot;

detecting a spattering occurred at the welding spot;

selecting a first value of the inter-electrode displacement detected before the spattering;

selecting a second value of the inter-electrode displacement detected after the spattering, the second value being different from the first value;

correcting detected values of the inter-electrode displacement after the spattering, depending on a difference between the first and second values;

integrating detected values of the inter-electrode displacement, using corrected values thereof as applicable, to make a time integration of the inter-electrode displacement; and

estimating a nugget size of the welding spot on a basis of the time integration.