## 4 Discussion  
The question posed at the beginning of the present paper concerning the maximum possible operating speed up to which the assumption of a straight line for describing the fluid surface is valid can be answered with the help of the presented evaluation method. The determined deviations increase with increasing operating speed, since the waveform of the fluid surface assumes an increasing characteristic shape and thus deviates significantly from a straight line. The fact that the standard deviations of \(\overline{\epsilon}\) increase with the operating speed may indicate a higher sensitivity of the system to the initial state at higher operating speeds. Another conclusion that can be derived from the results concerns the relationship between container width and maximum operating speed. With increasing container width, the maximum possible operating speed decreases. This is due to the fact that as the container width increases, the fluid volume to be conveyed increases as well. Due to the associated greater inertia, the characteristic surface waveform is formed at a lower motion's profile velocity. It has been shown that model approaches \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\) both achieve good results, with the second approach having a free parameter in the form of a proportionality factor to reinforce the motion's acceleration profile. Apparently, the factor \(\omega_{0}^{2}/g\) given by the linearized pendulum equation is already a very good specification, which is represented almost identically by the free proportionality factor \(p\). Therefore, no additional value is achieved in the model quality, as proposed in [23; 24]. The second question formulated in the beginning concerns the applicability of the different model approaches for the whole range of valid operating speeds. Therefore, the two different strategies SOSM and MOSM were investigated. The SOSM strategy provides slightly better results, but those  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Geometry** & \multicolumn{3}{c}{**SOSM**} & \multicolumn{3}{c}{**MOSM**} \\ \hline  & \(\mathcal{M}_{1}\) & \multicolumn{3}{c}{\(\mathcal{M}_{2}\)} & \multicolumn{3}{c}{\(\mathcal{M}_{1}\)} & \multicolumn{3}{c}{\(\mathcal{M}_{2}\)} \\ \hline  & **max\{\(SEOE\)**} & **\#\{P\}** & **max\{\(SEOE\)**} & **\#\{P\}** & **max\{\(SEOE\)**} & **\#\{P\}** & **max\{\(SEOE\)**} & **\#\{P\}** \\ \hline
1 & 0.785 deg & 26 & 0.748 deg & 39 & 0.823 deg & 2 & 0.811 deg & 3 \\
2 & 1.118 deg & 32 & 0.991 deg & 48 & 1.265 deg & 2 &  
are hardly significant. From this it can be concluded that the model parameters can be regarded as constant within the valid operating speed range and that a model for the respective range is therefore sufficient for the considered application. Finally, in the sense of a model validation, it should be discussed whether the models of approach \(\mathcal{M}_{1}\) in combination with the MOSM strategy of the different geometries derived from the acceleration pulse motion profile also provide similar simulation results for the validation quintic polynomial motion profile. To examine this, repeat tests are performed with \(K_{2}=5\), and the measurements are compared with the corresponding simulation results. The \(SEOE\) value is used again for quantification. In Figure 10, exemplarily simulation results and a comparison to measurements are shown for geometry 2 and operating speed \(n=45\,1/\min\). In principle, it can be stated that the dynamic behavior of the system is also well reproduced for this modified input. Furthermore, the \(SEOE\) value is similar to the \(SEOE\) derived from the parameter identification measurements (see Figure 8). For a better comparison of all considered geometries and a corresponding assessment regarding the model's applicability, Table 4 shows the results of the maximum \(SEOE\) values recorded for the parameter identification motion profile and the validation measurements. The respective values are proportional to each other, whereby the results for the validation evaluation are higher.