## 2 Materials and Methods
### FEM-Simulated Natural Vibrations and Basic Data Insights
#### 2.3.3 Theoretical Limits of a Simple Eigenfrequency Defect Detection  
One single eigenfrequency is generally not suitable for an ART defect detection, because a frequency may not be significantly affected or sensitive to a defect, depending on its position. Therefore, an approach based on a single frequency can only work for special defect positions. In addition, the defect influence may be much less compared to the geometry-induced effects on the eigenfrequencies. Therefore, geometric scatter can superpose the effects from the structural defects. FEM data were used for an estimation of the defect detection limits for the connecting rods using a single eigenfrequency. A central through-hole defect in the middle of the bar was assumed as an example. Figure 6 is derived from Table 1. It refers to the parts from the basic population and visualizes the relative sensitivity of the connecting rod eigenfrequencies to a 2 mm defect. Every sensitivity value is the mode-specific ratio between the frequency shift caused by the defect on average compared to the eigenfrequency standard deviation caused by the geometric variations. As can be seen, Mode-16 has the highest sensitivity towards such a central 2 mm defect.  
Figure 7 visualizes the distribution of the eigenfrequencies regarding Mode-16 for large batches of defect-free parts and for parts with a central through-hole. The figure shows density curves derived for the basic population parts, as well as for the parts from a scaled population with only 10% geometric scatter in cross-comparison. There are curves for the different defect sizes. In the case of the basic population, there is no chance of a reliable separation between the damaged parts containing a defect with a diameter from 1 mm to 5 mm and the good parts, due to a significant overlap of the frequency ranges. A separation may only be possible for large defects with a size of 10 mm or more. This, however, is an academic finding and not a demonstration of a satisfactory performance. In  
Figure 6: Relative defect sensitivity to a central 2 mm defect in a connecting rod (the relative defect sensitivity is the ratio of the defect impact and the geometry-induced standard deviation).  
contrast, for the small geometric variations, the defects with a size of around 3 mm or larger should, in theory, be detectable with the help of that single eigenfrequency. Nevertheless, the utility for a defect detection is still limited to special positions and a 3 mm defect is quite large, taking into account the low geometric variations. Furthermore, these results are based on simulation data and practice-relevant effects may result in a reduced detection power.  
The previous explanations illustrate that a simple ART defect detection approach based on a single eigenfrequency is extremely limited. This is particularly valid in the case of large variations in the geometry from part to part or when a possible defect is not limited to a small and known region of a part. To counteract the observed challenges, we describe a much more powerful approach within this paper, based on a more efficient data evaluation method as well as on extensive synthetic ART training data.