## 4 Demonstration and Validation of the Simulation-Based Generation of ART Training Data  
As a final step, our novel approach of a simulation-based generation of representative and valid ART training data and its application are demonstrated in practical use. For this, the 24 machine-made validation parts, which were not considered for any previous modeling, were used within a blind test. Our application example aims at extracting the actual geometry of those produced validation parts. For each part, the 18 individual eigenfrequencies up to approx. 30 kHz measured by ART served as input data for geometry evaluation. To determine the geometry information from the eigenfrequencies of a connecting rod, we postulated the easy relationship given in the following equation. \[y\approx a_{0}+a_{1}\cdot f_{1}+a_{2}\cdot f_{2}+\cdots+a_{18}\cdot f_{18}\]  
This formula has to be regarded in a mathematical and not in a physical sense. It describes any geometric dimension, which is denoted symbolically by \(y\) and which should be calculated approximately from a constant \(a_{0}\) as well as the products from the coefficients \(a_{1},a_{2},\ldots,a_{18}\) and the eigenfrequencies \(f_{1},f_{2},\ldots,f_{18}\). For each separate geometric parameter, we ultimately used the formula as a linear regression model. The constant and coefficients were calculated using the synthetic training data mentioned in the following paragraph as input with the help of the method of least squares. For specific implementation of the fit, we used _MATLAB_ and its function _fitlm_ (fit linear regression model). However, many software provide similar functionality for performing such regression model fitting. Such an idea of an eigenfrequency-based geometry estimation has already been discussed by the exclusive use of simulated data [34; 35]. However, for the first time within a manuscript, this method is transferred to experimentally measured eigenfrequencies of real existing parts. The generation of synthetic ART training data was first carried out using random numbers and FEM simulations. The eigenfrequencies of in total 5000 virtual training parts were calculated as a function of random geometric variations assuming 8 independent geometric parameters. A homogeneous and isotropic material according to literature as well as the simplified assumptions such as disregarding damping and support loads, as discussed in the previous chapters, were considered. For random number generation, the procedure described in Section 2 and the scatter values given in Table 1 were used, whereby uniformly distributed random numbers were used to describe the specific dimensions of the parts and the scatter values were each scaled upwards by a factor of 1.5. In this way, for each geometric parameter of the virtual training parts, a value range increased by 50% in comparison to the unscaled range or to the range of the machine-made connecting rods is described. This scaling is intended to ensure that training parts (and thus the training data) should always cover the geometric properties of the machine-made connecting rods, which vary normally or uniformly in a smaller  
In a second training and validation step, the determination of the regression functions was repeated after the simulated training data set was adapted to measurement using the correction factors. The revised results using the eigenfrequencies of the validation parts are shown in Figure 10. They once again refer to geometric parameter \(d_{\text{R1}}\). Beneath the results derived by including measured eigenfrequencies within the updated regression function, results using simulation data are shown as reference (for the reference results, it is irrelevant whether the training data as well as the validation data both are left unchanged or both are adapted multiplicatively using the same correction factors). After updating, a very good agreement between the values estimated by the regression function based on measured eigenfrequencies and the actual dimensions can be seen.