## 2 Materials and Methods
### Determining the Model Parameters and Evaluating the Model's Goodness  
For answering the second formulated question, it is to be examined whether the simulated values \(\theta_{\mathrm{sim}}\) from the model approaches \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\) are reproducing the approximated values of \(\theta\) within the ascertained range of operating speeds in a suitable way. Therefore, two approaches are investigated. First, for each operating speed \(n_{u}\) and geometry \(\mathcal{G}_{\
u}\), \(\mathcal{M}_{1,\mathcal{G}_{\
u},n_{u}}\) and \(\mathcal{M}_{2,\mathcal{G}_{\
u},n_{u}}\) models are examined by solving a least-squares problem, which considers all repetitions at once. Thereby, the least-square-optimization problem is solved with the Levenberg-Marquardt algorithm again. For calculating the objective function to be minimized, possible parameter candidates as well as the motion's acceleration profile \(a(t)\) are inserted into the model equations. The resulting differential equations are solved numerically using the Runge-Kutta method in order to finally calculate the residuals between the measured and simulated values of \(\theta\). Hence, a set of models are build to describe the measured behavior for its valid range of operating speeds. This strategy should be named the _Single-Operating-Speed Model_ (SOSM). Second, for each geometry \(\mathcal{G}_{\
u}\) and all valid operating speeds, the models \(\mathcal{M}_{1,\mathcal{G}_{\
u}}\), and \(\mathcal{M}_{2,\mathcal{G}_{\
u}}\) are examined in the same way considering all repetitions at once. Hence, for one geometry, only one model for approach \(\mathcal{M}_{1}\) and one for approach \(\mathcal{M}_{2}\) are derived. This strategy should be named the _Multiple-Operating-Speed Model_ (MOSM). Finally, the goodness of these different strategies and model approaches has to be examined. Therefore, the standard error of estimate (\(SEOE\)) is calculated by  
\[SEOE_{\mathcal{G}_{\
u},n_{u}}:=\sqrt{\frac{1}{K\cdot J}\sum_{k=1}^{K}\sum_{j =1}^{I}\big{(}\theta\left(\hat{t}_{j}\right)-\theta_{\mathrm{sim}}\left(\hat {t}_{j}\right)\big{)}^{2}}. \tag{16}\]  
This value is a measure of how much the simulated value of \(\theta_{\mathrm{sim}}\) differs on average from all approximated values of \(\theta\) that result from the measurements. This allows the comparison of the two model approaches \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\) as well as the two strategies SOSM and MOSM in the correspondent valid range of operating speeds. Finally, it must be checked whether the identified models can make a meaningful statement regarding the process-relevant residual sloshing behavior. First of all, an appropriate quality criterion is necessary. One possibility is the effective value of the fluid surface during the dwell phase, whose definition generally has the following form:  
\[\eta_{\mathrm{eff},k}:=\sqrt{\frac{1}{(1-b)\cdot T_{\mathrm{cycle}}\cdot w} \int\limits_{b\cdot T_{\mathrm{cycle}}}^{T_{\mathrm{cycle}}}\int\limits_{-w/2}^ {w/2}\eta_{k}^{2}(t,x)\,\mathrm{d}x\,\mathrm{d}t}. \tag{17}\]  
Due to the discretization of all treated quantities, the integrals are calculated numerically using the trapezoidal method. The investigation of this compacted size has the advantage that it can be applied to the actually measured edge courses of the fluid as well as to the approximated straight line and the simulated courses resulting from applying the mechanical model. Thus, this parameter is universally applicable and enables an evaluation of the process success, especially with regard to the investigation of the success of optimal trajectories. Furthermore, it is possible to compare the difference between real edge progression, straight line approximation, and model prediction.