Sloshing-height estimation for liquid-filled containers under four-dimensional motions including spatial translation and rotation about a fixed direction: modelling and experimental validation

Sloshing is the nonlinear dynamical behaviour of a liquid undergoing free-surface oscillations within an accelerated, partially filled vessel. In industrial scenarios, when liquid-filled containers are transported between stations in processing or packaging lines, liquid sloshing often needs to be limited to prevent undesirable effects, such as spillage or disruption of operations. In this case, reliable and near real-time models are needed to optimize the liquid behavior under specific container motions. Since typical pick-and-place operations involve 3D translational paths combined with a rotation about a fixed-orientation axis, this paper extends the sloshing-height-estimation models previously developed by the authors for translational motions and cylindrical containers, to take into account an additional rotation either about a vertical axis (known as SCARA motion) or around a horizontal axis with a fixed direction (denoted as Tilting motion). The presented approach, based on two equivalent discrete mechanical models, i.e. the mass-spring-damper and the pendulum, exhibits meaningful merits: it is computationally cheap, it requires no experimental assessment of the model parameters, and it needs no external sensor readings. Several sloshing-height formulations are proposed for the two models and an extensive experimental campaign is conducted to assess the effectiveness and the limitations of the most promising formulations for both models. Experiments cover configurations in which the ratio of the static liquid height to the container radius ranges from 0.6 to 1.6, and including dynamic motions with container accelerations up to 7.2m/s2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<^>2$$\end{document} and 11.0rad/s2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<^>2$$\end{document}. All experimental data sets are distributed on a public repository for future use by the community.