## 1 Introduction  
The investigation of liquid slosh dates back to the 1960s and was initially motivated by aerospace research [1; 2]. Later on, more practical use cases of this phenomena were investigated [3]. One practical application of the underlying physical problem occurs in packaging machines producing beverages, displayed in Figure 1. Due to high requirements in the production process of daily needs such as food and beverages, packaging machines are intended to produce at maximum productivity. Often, neither the mechanical nor the mechatronical components limit the machine's operating speed and hence the productivity. In fact, the behavior of the materials to be processed limits the maximum achievable operating speed [4]. For instance, residual vibration induced by the applied intermittent motion profile is to be avoided, e.g., due to necessary quality control processes. Hence, the motion optimization of the conveying motion is a powerful method for minimizing the residual vibration. The optimization of the motion profile is a suitable possibility for the given problem, but it is less widespread in industrial practice [5]. Nevertheless, a valid model that predicts the sloshing behavior depending on a motion profile is needed for motion optimization. Model approaches can principally be subdivided into two main classes, namely descriptive models and explanatory models [6]. In the case of modeling liquid slosh, explanatory approaches use the Navier-Stokes equations for describing the liquid-container interaction as a continuum problem. Therefore, material parameters, such as viscosity and density, are considered as distributed parameters. Such a general approach leads to a system of partial differential equations [7; 8; 9]. This system of equations is not solvable analytically, wherefore numerical solution techniques such as the finite difference method or the finite volume method have to be applied [10; 11; 12; 13]. Models with distributed parameters can be used for motion optimization as shown in [14; 15]. Such solutions are associated with enormous calculation times, although the presented solutions already represent an approximation of the partial differential equation system. Another possibility is the modeling with concentrated parameters and physical substitution models such as the pendulum model, also discussed in [7]. This approach leads to an ordinary differential equation of order two, which can be easily considered in common motion optimization algorithms in an efficient way [16; 17]. A new aspect of research in the field of model-based motion optimization is the operating-speed-dependent motion profiles that are discussed in [18; 19]. Applying this approach to the motion optimization for liquid slosh, the previous modeling work has to be re-investigated and discussed regarding their operating-speed-dependent usability. Hence, this paper aims to answer two main questions regarding the modeling of liquid slosh with the purpose for using it in operating-speed-dependent motion optimizations:  
1. Are there limitations of the substitution mechanical slosh models concerning the applicable operating speed? 2. If such limitations can be stated, are the derived models applicable for the whole derived range of operating speeds and at what accuracy? For taking into account the special case of liquid slosh occurring in packaging machines, the following limitations are assumed: only one-dimensional motions and transient excitations are considered due to considering only one machine cycle--in contrast to investigations concerning periodic behavior [20]. Thus, there will be no discussion of resonances and magnification functions, since steady-state behavior must be therefore present. Figure 1: Principle of a filling unit applied in packaging machines. Empty, open containers are conveyed by a linear transport unit that realizes a rise-to-dwell-motion. After filling in the fluid, the filled container is conveyed to a weighing station, where the bottled fluid’s mass is controlled.