A B C D 4). It was observed that the welding speed ranged from 1.5 to 0.5 mm/s. Consequently, the weld pool width, length, and convexity covered the ranges 4 to 7 mm, 4 to 9 mm, and 0.05 to 0.18 mm, respectively. The range in which the inputs reside implies that the resultant model can be used to estimate the weld pool parameters if the welding speed falls within the range defined by this distribution. System Modeling The system modeling was performed using the acquired welding speed and outputs (weld pool width, length, and convexity) data. The following two criteria are proposed to evaluate the performance of the developed models. The model average error is defined by (1) where n is the number of data points, the measured weld pool characteristic parameters is at instant k, and y(k) is the model estimated pool parameters. The root mean square error (RMSE) of the model is calculated by (2) n 2 The heat input of the arc in a unit interval along the travel direction can be written as a nonlinear function of I2 and 1/S (3) 2 1 Roughly speaking, one can assume that the volume of the weld pool is approximately proportional to the heat input. A linear model structure thus may be expressed as follows: (4) where g’s are linear functions of the current I and square root of the reciprocal of welding speed 1/√S. Given a constant welding current, the input of the system is defined as u = 1/√S. In the following subsections, MA and ARMA models are proposed and the model orders are carefully selected to form linear models of the system outputs (i.e., weld pool parameters). Moving Average (MA) Modeling The MA model is commonly used in linear system modeling and control applications. In this subsection, MA models are first considered with the following model structure: WELDING RESEARCH (5) N y k iu k j r where y(k) represents the weld pool parameters at instant k, β i and r are MA model parameters, u(k–j) is the system input, and N is the MA model order. Modeling trials suggest that for our system, MA model (with only the input information) is not sufficient to capture the dynamic response of the weld pool parameters, even with large model orders. As an example, Fig. 6 shows the MA model fitting result of the weld pool width with model order N = 10. As can be observed, there are consistant fitting errors. To better model the weld pool parameters, ARMA models with previous weld pool measurements are considered below. AutoRegressive Moving Average (ARMA) Modeling The linear models for the width, length, and convexity can be expressed as the following ARMA models (6) = Σ ( ) − ( ) ( = ) = E n y k y k k n 1 ave ˆ , 1,..., k n 1 Σ( ( ) − ( )) = = RMSE y k y k n ˆ k 1 Δ ∝ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ H f I S , ∝ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ W L C g I S , , , 1 ( ) =Σβ ( − ) + = j 1 ( ) W k NP Σ Σ ( ) ( ) ( ) ( ) = − = W 1 NI + − = a jW k j b ju k j j W j 1 W W APRIL 2015 / WELDING JOURNAL 129-s Fig. 9 — Verification experiment results. ^ ^
Welding Journal | April 2015
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