## 2 Materials and Methods
### Evaluating the Model's Assumptions  
The main assumption of the derivated model above is the approximation of the fluid's surface as a straight line. In a first step, this assumption has to be investigated, because in reality the fluid surface has a characteristic waveform. For doing so, for each geometry \(\mathcal{G}_{\
u}\), operating speed \(n_{u}\), repetition \(k\), and captured frame \(j\) corresponding to a discrete time value \(\hat{t}_{j}\), an approximation in the form of a line of the measured fluid's surface has to be derived. Therefore, the measured fluid's surface \(\eta_{\mathcal{G}_{\
u},n_{u}}(t,x)\) is approximated as a linear function \(\tilde{\eta}_{\mathcal{G}_{\
u},n_{u}}(t,x)=\theta_{\mathcal{G}_{\
u},n_{u}}( t)\cdot x\) with time-dependent slope \(\theta_{\mathcal{G}_{\
u},n_{u}}(t)\). This slope has to be determined as part of a parameter identification. Thus, the residuals between the measured surface edge in the discrete form and the approximated line in the discrete form surface are quadratically summed:  
\[r\left(\theta_{\mathcal{G}_{\
u},n_{u},k}\left(\hat{t}_{j}\right)\right):=\sum _{i=1}^{I}\left(\eta_{\mathcal{G}_{\
u},n_{u},k}(\hat{t}_{j},\hat{x}_{i})- \tilde{\eta}_{\mathcal{G}_{\
u},n_{u},k}(\hat{t}_{j},\hat{x}_{i})\right)^{2} =\sum_{i=1}^{I}\left(\eta_{\mathcal{G}_{\
u},n_{u},k}(\hat{t}_{j},\hat{x}_{i}) -\theta_{\mathcal{G}_{\
u},n_{u},k}\left(\hat{t}_{j}\right)\cdot\hat{x}_{i} \right)^{2}. \tag{11}\]  
The value \(\theta_{\mathcal{G}_{\
u},n_{u}}(t_{j})\) that minimizes this sum of residuals is the value to be identified:  
\[\theta_{\mathcal{G}_{\
u},n_{u}}(t_{j})=\arg\min r\left(\theta_{\mathcal{G}_{ \
u},n_{u}}(t_{j})\right). \tag{12}\]  
The gradient-based algorithm of [PERSON]([PERSON], 1993) is applied for performing the optimization, due to the convexity of the problem. Hence, short calculation times can be achieved, which are not realizable with global optimization strategies. In Figure 6, an exemplarily sketch of a surface with an approximated line is shown, whereas the difference \(\Delta\eta\) is displayed over the horizontal coordinate \(x\).  
The remaining residuals are a measure for the approximation's goodness and are therefore used to score the goodness of line approximation over all discrete time points. Due to the effect of in-phase interception discussed in [19], cancellation of residual vibration may occur at some operating speeds of the motion profile. To enable comparability, the dwell phase is not considered evaluating the goodness of line approximation, which is why only the captured frames of the motion phase \(J_{b}\) are considered:  
\[\bar{\epsilon}_{\mathcal{G}_{\
u},n_{u},k}:=\sqrt{\frac{1}{J_{b}\cdot I}\sum_ {j=1}^{J_{b}}\sum_{i=1}^{I}\left(\eta_{\mathcal{G}_{\
u},n_{u},k}\left(\hat{t }_{j},\hat{x}_{i}\right)-\tilde{\eta}_{\mathcal{G}_{\
u},n_{u},k}\left(\hat{t }_{j},\hat{x}_{i}\right)\right)^{2}}. \tag{13}\]  
Figure 6: Results of investigating the linearity of the detected fluid surface.  
Due to the repetitions of the experiments, the mean values \(\overline{\tilde{\epsilon}}\) as well as the corresponding standard deviations \(\Delta\overline{\tilde{\epsilon}}\) are examined to compare the obtained results [27]:  
\[\overline{\tilde{\epsilon}}_{\mathcal{G}_{\
u},n_{u}} :=\frac{1}{K_{1}}\sum_{k=1}^{K_{1}}\overline{\epsilon}_{\mathcal{G }_{\
u},n_{u},k} \tag{14}\] \[\Delta\overline{\epsilon}_{\mathcal{G}_{\
u},n_{u}} :=\sqrt{\frac{1}{K_{1}-1}\sum_{k=1}^{K_{1}}\left(\overline{ \epsilon}_{\mathcal{G}_{\
u},n_{u},k}-\overline{\tilde{\epsilon}}_{\mathcal{G }_{\
u},n_{u}}\right)^{2}}. \tag{15}\]  
Setting a tolerable value for \(\overline{\tilde{\epsilon}}\) of \(0.5\,\mathrm{mm}\), this evaluation allows the determination of a maximum operating speed \(n_{\mathrm{max}}\) up to which the assumption of a fluid's surface as a straight line is valid. This enables the answering of the first formulated question in the introduction.