## 4 Discussion
This is understandable for two reasons. First, these system inputs have not been used for parameter identification. Second, the quintic polynomial motion has higher acceleration values, which is why the system response also assumes higher values and therefore model inaccuracies are of greater importance. Nevertheless, this investigation shows that the derived models are well suited to simulate realistic system responses for arbitrary motion specifications that have not been used for parameter identification. Eventually, this means that the models can also be used for automated trajectory optimization in an expedient way, as initially required. Figure 10: Exemplarily validation results and \(SEOE\) value displayed for geometry 2 and an operating speed of \(n=45\,1/\min\), applying the Multiple-Operating-Speed Model (MOSM) strategy and model approach \(\mathcal{M}_{1}\). In order to get a better impression of the derived results and model parameters, they are shown in Figure 11 as a function of the varied fluid width \(w\). In addition, an approximation equation is given in the literature [7] for the resulting first undamped natural frequency, which is supplementary illustrated in Figure 11:  
\[\omega_{0}=\sqrt{\frac{\pi g}{w}\ \tanh\left(\frac{\pi h}{w}\right)}. \tag{20}\]  
For describing the damping ratio \(\zeta\), no approximation formula is known to the author's knowledge. The undamped natural frequency decreases by increasing the fluid's width. It can be seen that there is very good agreement between the derived model parameter from the experiments and the predicted parameters from Equation (20). This observation fits the vibration theory, since it is well known that, with increasing mass, which results from a larger volume, natural frequency decreases. Therefore, the results are plausible. With one exception, the damping ratio also decreases with the fluid's width. This coherence is evident by the decreasing influence of the dissipative frictional energy between the fluid and the container walls in relation to the total kinetic and potential energy, simultaneously with the fluid width. Whether the derived models are applicable for predicting the residual vibration had to be investigated. Therefore, the measure of \(\eta_{\text{eff}}\) introduced in Equation (17) was evaluated for the measured fluid edge, the line approximation, and the model response. An exemplarily result is displayed in \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Geometry** & \(\mathbf{\zeta}\) & \(\mathbf{\omega_{0}}\) & \(\mathbf{n_{\text{max}}}\) & **Identification: max\{_SEOE_\}** & **Validation: max\{_SEOE_\}** \\ \hline
1 & 0.042 & 22.498 rad/s & 481/min & 0.823 deg & 1.037 deg \\
2 & 0.050 & 27.347 rad/s & 571/min & 1.265 deg & 1.956 deg \\
3 & 0.036 & 21.001 rad/s & 45 1/min & 0.602 deg & 0.874 deg \\
4 & 0.042 & 25.351 rad/s & 541/min & 0.965 deg & 1.718 deg \\
5 & 0.029 & 19.622 rad/s & 42 1/min & 0.471 deg & 0.620 deg \\
6 & 0.039 & 23.729 rad/s & 51 1/min & 0.852 deg & 1.461 deg \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of investigations: for each geometry, the derived model parameters as well as the maximum operating speed are stated. Furthermore, the maximum _SEOE_ values are given for both the derived model applied to the identification motion profile and that applied to the validation motion profile. Figure 11: Depiction of derived model parameters \(\omega_{0}\) and \(\zeta\) dependent on the container’s width \(w\). Furthermore, the analytical expectation of the angular natural frequency is displayed. Figure 12. Furthermore, the difference measures introduced in Equations (18) and (19) were calculated and are additionally shown. Overall a good accordance of the different measures can be seen, whereby expectably the difference measure \(\Delta\eta_{\text{simul}}\) is a bit greater than \(\Delta\eta_{\text{appr}}\). In Table 5, these measures are shown for all investigated geometries. The evaluated differences are very small and hardly vary among the different geometries.