### 1. Introduction

^{1}

^{-}

^{4)}.

^{5}

^{,}

^{6)}. Also heat sources that are guided spirally around the pipe have been investigated

^{7)}.

^{7}

^{-}

^{9)}. Also the evaluation of effects from laser beam power, welding speed and beam angle, such as the width and depth of molten pool at T-joints is predicted

^{10)}.

^{11)}. Such an approach could also be used to adapt welding parameters for changing boundary conditions.

### 2. Used welding setup

### 3. Modelling

Reduced weld root welding speed, relatively to the speed on the surface.

Smaller volume, relatively to linear sections in which heat can dissipate.

Weld path movement into stronger preheated areas, located perpendicular to linear weld path.

Possible short term reduction of robot speed in the radius.

^{7}

^{,}

^{9}

^{,}

^{10)}. Used thermal conductivity

*k*, specific heat capacity

*c*

_{P}and global heat transfer coefficient

*h*are shown in Fig. 1. Density

*ρ*of aluminum materials is assumed to be 2.69 g/cm³. Heat exchange between bodies and the surrounding stationary medium air takes place at the sheet surface. Ambient temperature is set to

*T*= 20 °C. Forced convection due to gas exchange in the near field of the torch and thermal radiation were neglected in the model. Required computing time could thus be reduced.

*q*of the simulation corresponds to that of the real weld. Seam width and heat flux

*q*are constant over the weld seam path.

*v*

_{S}= 40 mm/s.

*r*= 20 mm,

*r*= 35 mm, and

*r*= 50 mm. The Fig displays that there is an elevated temperature in the area of the radii. The smaller the radius, the higher the temperature increases.

### 4. Power reduction method

*r*occurs during welding with constant power, which may lead to defects in the weld seam. In the following the increase of the temperature will be explained on the basis of the volumetric heat flux density

*q*

_{V}.

*q*

_{V}is shown in eq. (1) as a function of the heat input and output

*Q*, the related volume

*V*, and the time increment Δ

*t*. With or without radius, due to constant welding speed, the time increment Δ

*t*is constant. Also there is no change in heat input

*Q*, due to constant Power

*P*

_{Gesamt}.

*q*

_{V}increases. According to the energy balance flow eq. (2) from Anders

^{3)}the increase of the volumetric heat flux density

*q*

_{V}means an increase in temperature

*T*. This applies for a volume element with side length

*dx*,

*dy*, and

*dz*, the specific heat capacity

*c*, and density

*ρ*.

*Q*

_{Adjust}, which is based on the power

*P*

_{Adjust}. According to eq. (3) the amount of adjustment of the power

*P*

_{Adjust}depends on the specific heat capacity

*c*, the mass

*m*, and the temperature rise Δ

*T*

_{Elevate}in the area of the radius.

*c*depends on the temperature, increasing degressively with temperature rise. For the range in consideration, the change of the factor is a relatively small compared to the temperature multiplier. The bending of the sheets lead to a constant mass

*m*. Due to the relatively constant heat capacity

*c*, the necessary power adjustment

*P*

_{Adjust}is, as shown in eq. (4), approximately proportional to the raise in temperature Δ

*T*

_{Elevate}.

*r*= 20 mm,

*r*= 35 mm, and

*r*= 50 mm. These were simulated with constant and with adjusted power in the radius. Total input power corresponds to the functions ① and ②. The temperatures resulting from these powers are shown in the functions ③ and ④.

### 5. Evaluation and Conclusion

*T*

_{Radius}are shown relatively to temperatures

*T*

_{Linear}, which are present in linear sections. There are clear excesses of almost 30 %.

*q*

_{V}is still effective. This means necessary expansion of the adjustment beyond the start and end point of the radius. Due to the heat flow in front and the resulting heat accumulation, the smaller the radius, the larger the expansion must be.

*r*= 50 mm a well working prediction. The intended root connection is reached. For radii with

*r*= 35 mm one of the two welded specimen show a partial missing root connection. And for the smallest radii of

*r*= 20 mm, there is a deviation from the desired welding result.

*v*

_{S}in the radii. The smaller the radius, the faster the robot must move with its welding head. A limitation of the welding speed could counteract this. The smaller the radius, the smaller the maximum permitted speed would have to be. Another alternative would be to reduce the height of the laser-MIG hybrid welding head.

Illustration of 2-D welds of convex radii in the simulation and via real trials.

Function of a virtual sensor, which follows the heat source in the simulation, for the determination of the temperature rise in convex 2-D radius, compared to linear 1-D welds.

Establishment of a power adjustment for the radii, based on a physical model and its real verification.

Display of necessity for further research for a complete automation of the parameter choice for 2-D welds. A radius-specific ramping in and out of the power adjustment would be conceivable.