## 3 Results
### Determining the Model's Parameters and Their Goodness  
Knowing the maximum operating speeds for all investigated geometries for which assuming the fluid surface as a straight line is valid, the discussed model approaches are derived, and the related \(SEOE\) values are calculated as a measure for the model's goodness. Therefore, for each geometry and a valid operating speed, an SOSM approach is calculated. This leads to two models, each for one operating speed, with two and three parameters, respectively. Furthermore, for each geometry, an MOSM approach is calculated considering model equations \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\), resulting overall in two models with two and three parameters, respectively.  
Figure 8 shows exemplarily the measured and simulated results for an operating speed of \(45\,\text{1/min}\) and the second geometry, as well as the related \(SEOE\) values for the SOSM and MOSM approaches. The simulated courses appear to fit very well to the measured courses, and the differences between the different model approaches are very small.  
In order to get a better impression of the \(SEOE\) values determined for the different approaches and strategies for different operating speeds and geometries, Figure 9 shows the results for two exemplarily geometries. The results of the other geometries behave analogously and are therefore not shown.  
Figure 7: Mean errors of fluid’s surface line approximations for different geometries and operating speeds as well as the correspondent standard deviations. Assuming a tolerable error of \(0.5\,\mathrm{mm}\), a maximum operating speed \(n_{\mathrm{max}}\) is derivable. Up to this operating speed, the considered fluid’s surface motion can be described as a line, and thus can be used for the pendulum modeling.  
Figure 8: Exemplarily results of measured and simulated courses for all different model approaches of Geometry 2 and \(n=45\,1/\min\).  
Figure 9: Determined standard error of estimate (\(SEOE\)) values for the two different model approaches and strategies for all considered operating speeds and the two exemplarily geometries 1 and 2.  
It can be seen that the \(SEOE\) values for model approaches 1 and 2 are essentially identical, since the data points lie approximately on the line \(y=x\). Furthermore, it can be noted that the MOSM strategy provides slightly higher values than the SOSM strategy. For a comparison of all results, Table 3 summarizes the maximum \(SEOE\) values for all approaches and geometries. In addition to that, the amount of necessary model parameters \(\#\{P\}\) is stated. These quantities are identical for all geometries applying the MOSM strategy. Applying the SOSM strategy, the amount of necessary parameters depends on the maximum valid operating speed, which in turn depends on the considered geometry.  
Hence, it can be summarized that, expectably, due to a higher amount of parameters, the SOSM strategy leads to better results than the MOSM strategy, and model approach \(\mathcal{M}_{2}\) leads to better results than \(\mathcal{M}_{1}\). Thereby, better results are understood as smaller \(SEOE\) values. However, it has to be stated that the differences are nevertheless very small. Due to the very good ratio of minor model errors and a small number of model parameters, model approach \(\mathcal{M}_{1}\) in combination with the MOSM strategy will be treated in the following as the preferred solution and investigated during a model validation more closely.