## 2 Materials and Methods
### FEM-Simulated Natural Vibrations and Basic Data Insights
#### 2.3.2 Geometry-Induced Eigenfrequency Shifts versus Defect Impacts  
Understanding the relationship of the specific structure of a part towards its eigenfrequencies is fundamental for reaching a reliable defect detection. Based on FEM, Table 1 gives an estimation of the connecting rod eigenfrequencies depending on the geometric effects, together with an optional through-hole defect of 2 mm diameter in the center of the bar (central 2 mm defect). Besides the geometrically mean part, different batches each containing 1000 virtual parts from the basic population with or without defect were used to generate the table data. Table 1 contains the following mode-specific information:  
* _Frequency without defect:_ This column contains the eigenfrequencies of the geometrically mean part without any defect. The averaged frequencies of the defect-free parts derived from the basic population are roughly the same;
* _Geometry-induced standard deviation:_ These columns contain the standard deviations of the eigenfrequencies that can be obtained for the normally distributed basic population (the frequencies are also nearly normally distributed). The values refer to the defect-free parts, but approx. identical standard deviations can be observed for the parts with a central 2 mm defect. The values are given in absolute and relative terms;
* _Frequency shift due to a central 2 mm defect:_ These columns present the eigenfrequency shifts or defect impacts induced by a central 2 mm defect. The shifts refer to the geometrically mean part. On average, similar shifts can be observed for the defective parts from the basic population, all featuring an identical, central 2 mm defect. However, the individual defects' impacts depend significantly on the specific random part geometry. The values are given in an absolute and relative manner. Figure 4: FEM-calculated eigenmodes of a connecting rod up to a frequency of around 30 kHz taking the example of the geometrically mean part; italic font: mode type (bending in \(y\)- or \(z\)-direction, torsional or longitudinal character); compare the previous publication [34]. Referring to the geometrically mean part, the defect impact with respect to the defect position and size was analyzed with FEM. Figure 5 visualizes the impact of a 2 mm defect at various positions in the bar for some eigenmodes. Such nonlinear and mode-specific behavior depending on the defect position is well known [29; 36]. However, a useful defect detection requires the consideration that a defect can influence an eigenfrequency with an individual maximum or has nearly no effect, which strongly depends on the defect position. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Mode no. **} & **Frequency** & **Geometry-Induced** & **Frequency Shift due to a** \\  & **without Defect** & **Standard Deviation** & **Central 2 mm Defect** \\  & **(in Hz)** & **(in Hz)** & **(in \%)** & **(in Hz)** & **(in \%)** \\ \hline
1 & 1760 & 45.3 & 2.58 & \(-\)5.9 & \(-\)0.34 \\
2 & 3619 & 93.9 & 2.59  
Since a linear growth in the defect diameter results in a quadratic rise in the defect area and volume, a quadratic dependence of the defect-induced frequency shift from the defect diameter was expected. Once again, using the fixed defect position in the center of the bar, such behavior could approximately be confirmed for the moderate defect sizes up to 3 mm or 4 mm, which is sufficient for our range of interest. Nevertheless, there are more complicated or interactional effects in the case of a larger defect, or in the case of a variable defect position in combination with geometric variations. For example, a large defect has an impact on the geometry-induced eigenfrequency standard deviations of a population, even if the position is fixed. Of course, when the defect position is also variable, this has an additional impact on the statistical characteristics.