## 2 Materials and Methods
### Modeling Liquid Slosst  
In order to be able to use the model to be created for motion optimization, as few degrees of freedom as possible should be used in order to avoid unnecessary computing time. Therefore, the already mentioned mechanical pendulum model is investigated in this paper. The real system, shown in Figure 5, can be abstracted to a model consisting of a horizontally movable mass \(m_{\text{M}}\) displaying the linear drive and an attached damped pendulum representing the fluid's surface displacement with length \(l\), mass \(m_{\text{P}}\), and damping constant \(d\). For describing the system's dynamic behavior, two generalized coordinates, namely \(x_{\text{M}}(t)\) as well as \(\theta(t)\), are introduced, which are summarized in the vector \(\mathbf{q}=(x_{\text{M}},\theta)^{\top}\). The system is controlled by the time-depending force \(F(t)\) affecting the mass \(m_{\text{M}}\). In contrast to [16], the model equations are motivated with respect to the applied drive. \begin{table}
\begin{tabular}{c c c} \hline
**Variable Experiment Element** & **Specification** & **Remarks** \\ \hline Container geometries & \(\mathcal{G}_{1}\): \(w=60\,\text{mm}\) & For all geometries: \\  & \(\mathcal{G}_{2}\): \(w=40\,\text{mm}\) & Fluid: water \\  & \(\mathcal{G}_{3}\): \(w=70\,\text{mm}\) & Filling level: 67 mm \\  & \(\mathcal{G}_{4}\): \(w=47\,\text{mm}\) & \\  & \(\mathcal{G}_{5}\): \(w=79\,\text{mm}\) & \\  & \(\mathcal{G}_{6}\): \(w=54\,\text{mm}\) & \\ \hline Motion profiles & Acceleration pulse motion & Used for parameter identification (\(K_{1}=10\)) \\  & Quintic polynomial motion & Used for validation (\(K_{2}=5\)) \\ \hline Set of operating speeds & \(n_{1}=12\,1/\text{min}\) & \\  & \(n_{2}=15\,1/\text{min}\) & \\  & \(\vdots\) & \\  & \(n_{17}=60\,\,1/\text{min}\) & \\ \hline \end{tabular}
\end{table} 
Table 2: Specification of experimental design for creating parameter identification and validation data. Figure 5: Mechanical motivation for model building due to abstracting the real process. The direct linear drive is described by a force actuated mass \(m_{\text{M}}\) on which a damped pendulum with length \(l\), mass \(m_{\text{P}}\), and damping coefficient \(d\) is mounted, which describes the fluid’s surface. According to [22], the Lagrange formalism is used for determining the equations of motion for the abstracted system. The kinetic energy \(T\), the potential energy \(U\) of the system, and the dissipation energy \(D\) due to the damper element affecting on the pendulum are given by  
\[T =\frac{1}{2}(m_{\text{M}}+m_{\text{P}})\cdot\dot{x}_{\text{M}}^{2}+ \frac{1}{2}m_{\text{P}}l^{2}\cdot\dot{\theta}^{2}-m_{\text{P}}l\cos(\theta) \cdot\dot{\theta}\cdot\dot{x}, \tag{1}\] \[U =-m_{\text{P}}gl\cdot\cos(\theta),\] (2) \[D =\frac{1}{2}dl^{2}\cdot\cos^{2}(\theta)\cdot\dot{\theta}^{2}. \tag{3}\]  
Subsequently, the equations of motion can be derived with the actuating force \(\mathbf{Q}=[F(t),\mathbf{0}]^{\top}\) from evaluating  
\[\frac{\text{d}}{\text{d}t}\left(\frac{\partial(T-U)}{\partial\dot{q}_{o}} \right)-\frac{\partial(T-U)}{\partial\dot{q}_{o}}+\frac{\partial D}{\partial \dot{q}_{o}}=\mathcal{Q}_{o}. \tag{4}\]  
This leads to a system of two nonlinear differential equations:  
\[\ddot{x} =\frac{F+m_{\text{P}}l\left(\cos(\theta)\cdot\ddot{\theta}-\sin( \theta)\cdot\dot{\theta}^{2}\right)}{m_{\text{P}}+m_{\text{M}}}, \tag{5}\] \[\ddot{\theta} =-\frac{d\cos^{2}(\theta)}{m_{\text{P}}}\dot{\theta}-\frac{g}{l} \sin(\theta)+\frac{1}{l}cos(\theta)\dot{x}. \tag{6}\]  
Considering the real system, the actuating force \(F(t)\) is caused due to a direct linear drive, which is driven by a cascade control loop with a very high dynamic stiffness. Furthermore, the drive's mass is essentially greater than the pendulum's mass. These two statements lead to the simplification \(m_{\text{P}}/m_{\text{M}}\approx 0\), from which  
\[F(t)=m_{\text{M}}\cdot\dot{x}(t) \tag{7}\]  
results and, hence, the dynamical behavior of the system can be solely described with Equation (6).