A probabilistic model of weld penetration depth based on process parameters
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Abstract
In welded structures using robotized metal active gas (MAG) welding, unwanted variation in penetration depth is typically observed. This is due to uncertainties in the process parameters which cannot be fully controlled. In this work, an analytical probabilistic model is developed to predict the probability of satisfying a target penetration, in the presence of these uncertainties. The proposed probabilistic model incorporates both aleatory process parameter uncertainties and epistemic measurement uncertainties. The latter is evaluated using a novel digital tool for weld penetration measurement. The applicability of the model is demonstrated on fillet welds based on an experimental investigation. The studied input process parameters are voltage, current, travel speed, and torch travel angle. The uncertainties in these parameters are modelled using adequate probability distributions and a statistical correlation based on the volt-ampere characteristic of the power source. Using the proposed probabilistic model, it is shown that a traditional deterministic approach in setting the input process parameters typically results in only a 50% probability of satisfying a target penetration level. It is also shown that, using the proposed expressions, process parameter set-ups satisfying a desired probability level can be simply identified. Furthermore, the contribution of the input uncertainties to the variation of weld penetration is quantified. This work paves the way to make effective use of the robotic welding, by targeting a specified probability of satisfying a desired weld penetration depth as well as predicting its variation.
Keywords
Weld penetration depth Process variables MAG welding Aleatory uncertainty Epistemic uncertaintyNomenclature
- y
Weld penetration depth
- f_{Y}(y)
Probability density function of penetration depth
- F_{Y}(y)
Cumulative distribution function of penetration depth
- Prob[y > y_{0}]
Probability of satisfying y > y_{0}
- μ_{y}
Mean of penetration depth
- σ_{y}
Standard deviation of penetration depth
- λ
Skewness of penetration depth
- γ
Kurtosis of penetration depth
- x_{1}, x_{2}, x_{3}, x_{4}
Voltage, current, travel speed, and torch travel angle
- x
Vector of input process parameters
- f_{Xi}(x_{i})
Probability density function of process parameter
- F_{Xi}(x_{i})
Cumulative distribution function of process parameter
- μ_{xi}
Mean of process parameter
- μ_{x}
Vector of means of input process parameters
- σ_{xi}
Standard deviation of process parameter
- \(\mu _{xi}^{\text {eq}}\)
Equivalent mean of normalized variable
- \(\boldsymbol {\mu }^{\text {eq}}_{\mathbf {x}}\)
Vector of equivalent means
- \(\sigma _{xi}^{\text {eq}}\)
Equivalent standard deviation of normalized variable
- S^{eq}
Diagonal matrix of equivalent standard deviations
- \(\rho _{x_{1}x_{2} }\)
Correlation coefficient between voltage and current
- C
Matrix of correlation coefficients
- α
Slope of volt-ampere characteristic curve
1 Introduction
The robotic arc welding process involves complicated sensing and control techniques applied to various process parameters. These measures enhance the quality and improve repeatability of welds [1]. However, there remains limitations in making welded joints consistent and repeatability is still difficult to achieve. One of the major limitations is the inability to fully control input process parameters due to aleatory uncertainties. These stems from imperfect operating conditions such as measuring inaccuracies as well as limited capability of power sources to control and provide required dynamic and static characteristics [2]. The limited accuracy in measuring devices, repeatability in robots, and controllability in power sources results in variations in gun angle, torch travel angle, and robot trajectory [3] as well as current and voltage [4]. Reducing this process variation is crucial in reducing over-processing, saving cost, and increasing production capacity [5]. Thus, it is of utmost importance to understand the influence of welding process parameters on weld profile and produced quality, regarding both average values and variabilities.
In this work, a probabilistic model is proposed to estimate the probability of satisfying a desired penetration depth as well as to predict its variation. The uncertain process parameters are voltage, current, travel speed, and torch travel angle which were studied based on an experimental investigation. The weld penetration depth is evaluated from macrographs using a digital tool developed in MatLab [19]. The epistemic measurement uncertainty related to this evaluation is quantified and incorporated in the probabilistic model. The paper starts with an overview on welding parameters and sources of uncertainties, followed by the presentation of the studied experimental set-up. Thereafter, the proposed probabilistic model is thoroughly presented. Finally, in the result section, the applicability of the model and its advantage over the traditional deterministic approach currently used in the manufacturing process are demonstrated.
2 Welding parameters and sources of uncertainty
In this section, the welding parameters and sources of uncertainties are described. These parameters are set-up and measured experimentally and the described uncertainties are based on industrial recommendations.
2.1 Weld penetration depth
2.2 Travel speed and torch travel angle
The travel speed is the linear rate with which the torch arc moves along the weld bead. A schematic illustration of the torch arc movement is shown in Fig. 2 b. Conservatively, the travel speed may vary by ± 10% of the pre-set value due to undesirable vibration or backlash. The torch travel angles, defined in Fig. 2 b, is set using a digital angle gauge. Various factors results in variation in the torch travel angle, such as limited repeatability in robots and inaccuracies of measuring devices. The assumption is that the uncertainty is ± 3 degrees of the setting value. It should be observed that this variation is assumed to be the same regardless of the pre-set value.
2.3 Voltage and current
The voltage is pre-set at the power source which has a constant voltage characteristic. However, the dynamic characteristic of the welding arc, under the influence of electric and thermal conductivity of arc and arc length [4], results in variations in both voltage and current. This variation is assumed to be within ± 20% of their average values. It should be noted, however, that this uncertainty could be larger than what is assumed in this work [21].
3 Experimental investigation
3.1 Experimental set-up
Cruciform joints were produced from SSAB Domex 650 MC steel sheets with dimensions 700 × 700 × 10 mm. The steel sheets were fixed in zero gap and tack welded with approximately 20-mm weld length at the ends on each side. A single pass weld with a weld leg length of 7 mm was produced from robotic MAG welding in PA welding position [23], where the gun angle between the plates was 45 degrees. There were four welds with selected welding process parameters on each cruciform joint, see Fig. 1 a. The steel sheets were fixed by an external fixture to prevent welding distortion. After welding the first pass, the specimen was allowed to cool down to room temperature and then rotated to weld another pass with the same welding direction. The welding equipment consists of a Motoman HP20-B00 robot equipped with a Motoman CWK- 400 welding torch, MT1-250 positioner, and EWM Phoenix 521 Progress Pulse coldArc power source to which a Phoenix Progress Drive coldArc wire feed unit was connected. An ESAB AristoRod 13.29 solid wire having 1.2-mm diameter was used as filling material and the shielding gas was Mison-18 gas (20 l/min). During welding, the current was measured using a LEM HT 500-SBD current transducer and the voltage was measured between the wire feeder and the grounding point of the positioner. After welding, the cross sections were taken at the steady state part of the welds. The macrographs were made, with grinding and polishing to 1.0-μ m diamond size followed by etching with 2% Nital.
3.2 Design of experiment
Design of Experiment, pre-set values and measured mean values
Travel angle | Cont. to work dist. | Measured current (A) / voltage (V) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
29.6 V^{a} | 30.6 V^{a} | 31.6 V^{a} | 34.1 V^{b} | 35.1 V^{b} | 36.1 V^{b} | 38.4 V^{c} | 39.4 V^{c} | 40.4 V^{c} | ||
20^{∘} | 20 mm | 308/28.1 | 258/29.2 | 306/30.2 | 357/32.6 | 302/33.7 | 348/34.6 | 406/36.7 | 345/37.7 | 382/38.8 |
20^{∘} | 25 mm | 238/28.7 | 239/29.8 | 244/30.7 | 269/33.3 | 279/34.4 | 351/35.2 | 385/37.7 | 392/38.8 | 398/39.5 |
20^{∘} | 30 mm | 194/29.4 | 207/30.2 | 289/31.2 | 239/34.1 | 318/34.9 | 310/36 | 285/38.7 | 336/39.5 | 344/40.5 |
25^{∘} | 20 mm | 270/28.1 | 263/28.9 | 265/29.9 | 330/32.3 | 382/33.2 | 328/34.2 | 371/36.5 | 422/37.1 | 382/38.2 |
25^{∘} | 25 mm | 310/29 | 303/29.9 | 239/30.9 | 342/33.6 | 352/34.5 | 283/35.4 | 389/40 | 386/39 | 390/40 |
25^{∘} | 30 mm | 298/29.4 | 295/30.4 | 215/31.3 | 323/34.3 | 335/35.2 | 320/36.1 | 356/39 | 350/40 | 357/40.7 |
30^{∘} | 20 mm | 326/28 | 268/29 | 322/30.5 | 354/32.7 | 343/33.2 | 365/35.1 | 389/37.3 | 420/37.2 | 391/39.4 |
30^{∘} | 25 mm | 250/28.4 | 260/31 | 254/30.5 | 318/32.8 | 310/34 | 325/34.8 | 380/37.1 | 368/38.5 | 389/39 |
30^{∘} | 30 mm | 234/28.9 | 226/30.1 | 234/31 | 278/33.8 | 282/34.8 | 285/35.7 | 331/38.3 | 334/39.3 | 339/40.2 |
3.3 Penetration measurements
The uncertainty of the measured weld penetration is dependent on the uncertainty from measuring the scale, the uncertainty from estimating the intersection of the plates—i.e., from where the penetration should be measured, and the uncertainty from estimating the weld penetration maxima. A quantification of the contribution of these uncertainties is performed using Monte Carlo (MC) simulation, where the location of the defined points A–F is assumed normally distributed, see Fig. 5 b. For the points A and B, the standard deviation is set to 1.1 pixels. For the rest of the points the standard deviation is set to 1.35 pixels. These input uncertainties in the location of the points (i.e., 1.1 pixels and 1.35 pixels) are computed using ten (10) manual measurements of the penetration on one randomly selected macrograph.
As can be seen from Fig. 5 c, the resulting standard deviation of the measured penetration is 0.11 mm, or equivalently ± 0.21 mm at 95% confidence interval. Since the uncertainties in the input points (locations A–H) are independent of the calculated lengths, it is clear that the uncertainty in the penetrations depth does not depend on its mean value. From Fig. 5 d, it is noted that the largest contribution to the uncertainty is the extrapolated line, EF, which corresponds to the blue point cloud in Fig. 5 b.
4 Probabilistic model
4.1 Quadratic deterministic model
4.2 Model of input uncertainties
Description of random variables
Parameter | Notation | Distribution | Mean | Standard deviation | |
---|---|---|---|---|---|
Case 1 | Case 2 | ||||
Voltage | x_{1} | Normal | Lognormal | μ_{x1} | σ_{x1} = 0.1μ_{x1} |
Current | x_{2} | Normal | Lognormal | μ_{x2} | σ_{x2} = 0.1μ_{x2} |
Travel speed | x_{3} | Normal | Lognormal | μ_{x3} | σ_{x3} = 0.05μ_{x3} |
Torch angle | x_{4} | Normal | Normal | μ_{x4} | σ_{x4} = 1.5^{o} |
Measurement error | 𝜃 | Normal | Normal | μ_{𝜃} = 0 | σ_{𝜃} = 0.11mm |
For the voltage and current, most of the variation is within ± 20% of the mean value, see Section 2, i.e., ± 0.2μ_{xi}. If normal distribution and 95% confidence is assumed, this uncertainty range is equal to ± 2σ_{xi}, resulting in a coefficient of variation of σ_{xi}/μ_{xi} = 0.1. The uncertainty in the travel speed is assumed to be ± 10% of its mean value, i.e., a coefficient of variation of 0.05. For the torch travel angle, the standard deviation σ_{x4} = 1.5^{o}, corresponding to ± 3^{o} with 95% confidence, is assumed independent of the mean value.
The error 𝜃 in the measured penetration, see Table 2, is assumed to follow a normal distribution \(\mathcal {N} \left (0,\sigma _{\theta } {~}^{2} \right )\), with mean value zero and standard deviation σ_{𝜃} = 0.11mm estimated from repeated measurement, see Section 3.3. Therefore, two types of uncertainties are considered: the aleatory uncertainty in process parameters x and the epistemic measurement uncertainty \(\theta \sim \mathcal {N} \left (0,\sigma _{\theta } {~}^{2} \right )\) in the penetration y.
4.3 Model of uncertainty in weld penetration
4.3.1 Model including aleatory uncertainties
The expressions for the PDF, CDF, and the reliability Prob[y > y_{0}] according to Eqs. 16, 17, and 18, respectively, can be applied directly given \(\mathbf {{A}^{\prime }}\), \(\mathbf {{k}^{\prime }}\), and \(c^{\prime }\). As can be seen from Eq. 11, these are functions of the deterministic fitting parameters (A, k, and c), the correlation between the process parameters (through T and D) as well as μ_{x,eq} and S_{eq} which in turn are functions of the distributions of the process parameters as is seen from Eq. 9.
4.3.2 Model including epistemic uncertainty
It should be noted that, by adding the measurement uncertainty, the mean μ_{y} of the penetration is unchanged. However, the standard deviation σ_{y} increases, which results in a decrease of the skewness λ and kurtosis γ of the distribution according to Eqs. 14 and 15, respectively.
5 Results
5.1 Deterministic model
Based on the measured penetration values at the 81 data points, see Fig. 4, the deterministic model according to Eq. 2 is fitted. This results in:
5.2 Probabilistic model
6 Discussion
A deterministic hyper-parabolic model is applied to express the weld penetration as a function of process parameters. The model is simple for practicing engineers, with 2n + 1 fitting coefficients for n process parameters. For the 4 studied process parameters in this work, the model captures both the trend and non-linearity of weld penetration. Based on the computed deterministic sensitivities, it has been shown that the largest influence is attributed to the current. This is in accordance with results presented in previous work [15].
A probabilistic model based on the fitting coefficients of the quadratic deterministic model, the joint probability distribution of process parameters as well as the epistemic measurement uncertainty is proposed. The presented probability formulas for both the PDF and CDF are closed-form analytical expressions. Probabilistic sensitivities, defined as the influence of the means of process parameters on the probability of satisfying a desired penetration depth, can be efficiently computed. Although the proposed probabilistic approach has major advantages compared with traditional deterministic methods, further experimental validation of the computed probabilities is necessary. This would however necessitate a substantially larger experimental dataset than the 81 experiments used in this work, depending on the reliability level to be validated. Furthermore, in order to extend the usage of the proposed probability model, it should also cover a wide range of materials, plate thickness, welding positions, welding robots, and power sources.
It has been shown that the predicted penetration depth using the deterministic model is 3 mm for the arbitrarily chosen set-up, 30 V, 300 A, 30 cm/min, and 25^{∘}. However, the probability of satisfying a 3-mm penetration depth, computed for the same set-up using the proposed probability model, is only 58%. This result is not specific for the studied set-up. It can be expected that deterministic predictions generally yield reliability levels between 40 and 60% depending on the skewness of the PDF of the weld penetration.
The aleatory process parameter uncertainty is modelled using either normal or log-normal distributions. It is observed that the probability of satisfying the predicted penetration is similar in both cases. Although the formulation of the probabilistic model is simpler using normal variables, a log-normal distribution is more appropriate to describe positive quantities such as current, voltage and welding speed.
The epistemic measurement uncertainty is modelled using a normal distribution with a standard deviation 0.1 mm. However, the contribution of welding distortion of the specimen is not taken into account in the proposed estimation. Therefore, the epistemic measurement uncertainty is likely to be higher than the estimated valued, since distortion from welding is likely to increase the error of approximating the plates by a line. It should however be noted that this error is likely to be small, since the gap size is fixed to zero and the welding distortion is prevented by external fixture (see Section 3.1). It should also be noted that, an epistemic model uncertainty, i.e., uncertainties in the model itself, can be added to the proposed formulation through the law of total probability [29].
7 Conclusion
When a reliability level of 90% is required, the proposed probabilistic model yields process parameters set-ups that differs significantly compared with a traditional deterministic approach. This is due to the fact that deterministic approaches yield reliability levels close to 50% regardless of the input uncertainties.
Using the proposed probabilistic model, it is concluded that the uncertainty in welding current shows the largest contribution to the variation in the weld penetration depth. Therefore, in order to limit this variation, the capability of power sources to control and provide required characteristics has to be enhanced.
A digital image tool is developed to measure the penetration depth and quantify the epistemic measurement uncertainty. It is shown that this source of uncertainty has a substantial contribution to the overall uncertainty in penetration depth.
The proposed probability model can be used to formulate guidelines for process parameters set-ups that satisfy a desired reliability level. The approach paves the way for optimization under uncertainty, since both probabilities and sensitivities can be efficiently computed.
Notes
Acknowledgments
Open access funding provided by Royal Institute of Technology. This research was financially supported by Sweden’s Innovation Agency (Vinnova) programme for Strategic vehicle research and innovation (FFI), contract number 2016-03363. The experimental data was provided by Swerim AB. The support is gratefully acknowledged.
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