## 3 Results and Discussion
### Force Control  
The force was measured via load cells LC1 and LC2 during closed-loop force control (Figure 12). The maximum compressive force during welding was expected for the rigid clamping situation where the movement of sheet 2 in the y-direction was completely prevented, hence the maximum strain occurred. Therefore, the setpoint values of the forces were selected to be smaller than the value of the maximum compressive force in rigid clamping which was determined to be approximately 100 N (see Section 3.1). The linear actuators adjusted sheet 2 to reach the specified setpoint value of the compression force at the start of the welding process. If a load cell detected deviations, it was compensated by the related linear actuator. Hence, the force was controlled individually for both positions of the linear actuators by assigning load cell LC1 to linear actuator LA1 and load cell LC2 to linear actuator LA2. Figure 13 shows the resulting gap over time for the different inductive probes IP1-3 based on different setpoint forces \(F_{set}\) compared to an unhindered gap formation. The unhindered gap formation (uncontrolled process, no rigid clamping) in Figure 13a was already discussed in Section 3.1 but is used as a reference for describing the effects based on the force control. The time interval \(\phi\) is introduced to compare the different times needed for the joint gap to increase in size for different forces adjusted by the closed-loop control. Figure 13b shows a setpoint value \(\mathrm{F_{set}}\) of 0 N and depicts the resulting joint gaps and the measured forces over time. An influence on the gap formation can already be seen for nominal zero forces, i.e., the gap size measured at IP1 did not increase significantly compared to the unhindered process in Figure 13a. The negative gap dimension obtained can be explained by the contraction of the sheets after the welding process, i.e., the outer dimension of the welded sheets in the y-direction was reduced compared to the initial value and the inductive probes measure against the outer edge. The maximum value of the joint gap at IP3 was significantly reduced by 0.09 mm to 0.51 mm. The set force at the begin of the sheet was reached rapidly and remains rather constant. However, for LC1 a peak was determined in each case for a setpoint value of 0 N indicating a strong reaction to the occurring compressive forces at the begin of the welding process. A slightly increased compression force (max. \(\Delta\mathrm{F}=-8.6\) N) than specified was measured at LC2 because of the chosen control strategy for the first half of the welding time. The proportional gain of the controller could be increased at this point, but there is a risk of the system overshooting and it should be considered that each load cell operated at the lower limit of its resolution. It should be noted that the small peaks of the force curves that occur continuously are due to the proportional controller and do not represent signal noise. The smaller deviations in the controlled force values did not have a noticeable effect on the joint gap, so any negative effect on the position of sheet 2 could be ruled out. A comparison to the uncontrolled rigid clamping (see Section 3.1) shows the difference in characteristics clearly. Figure 12: Closed-loop force control. The effect of a further increase in the set compression force \(F_{set}\) to \(-30\) N is shown in Figure 13c. The time \(\Phi\) until the gap size is increasing was significantly extended. It can be assumed that the gap opening started after the process force is greater than the setpoint compression force. It should be noted that the linear actuator started retraction before \(\Phi\) was reached, i.e., the force control delayed the gap formation. A further decrease in gap size down to 0.33 mm as maximum value at IP3 was reached while the control kept the forces on a quite constant level resulting in an average weld seam length of 260 mm.