# Simulating Slosh Induced Damping, with Application to Aircraft Wing-like Structures
## 1 Introduction  
Liquid sloshing in a partially filled container is of significant interest in multiple disciplines. It is a non-linear and transient phenomenon that is challenging to simulate. The walls of the container will be subject to impact loads when liquid sloshing occurs. These loads can be significant when the natural frequency of the sloshing system and the excitation frequency are close and such large impulsive impact loads can result in damage on the structure of the container walls [1; 2; 3]. Alongside the issue of impact loading, liquid sloshing will, in general, provide added damping to the system. If considered correctly, this has the potential to be exploited in order to reduce the overall vibration of the system. The mechanics of the added damping and the energy dissipation changes, as well as relevant civil applications, are of widespread interest [4; 5]. Studies specifically related to the mechanics of added damping generation and the associated energy dissipation of a sloshing tank have been reported in several articles. A series of experiments was carried out by [PERSON] [6] in which they used rectangular basins to investigate the mechanics of the energy dissipation of standing sloshing waves. Dissipation of the waves was also observed and found to be due to the viscosity caused by surface tension. Another notable example can be found in [PERSON] et al. [7], where standing waves were investigated both experimentally and numerically. They observed dimpled crests when a steep wave appears, while the temporal symmetry of the resultant standing sloshing wave became broken due to energy dissipation. A follow-up study was performed in [PERSON] et al. [8], in which three recurrent modes of breaking waves were observed by increasing the forcing amplitude. The authors also found that energy dissipation was almost double when breaking waves occur. One of the earliest civil applications of slosh-induced damping is the tuned sloshing damper (TSD), which aims to dampen the vibrations of a building. On-site measurements were carried out by [PERSON] et al. [9] to quantify the performance of the tuned sloshing-damper system for four tall buildings. Results showed that the TSD system can help a tall building to reduce half to one-third of the acceleration responses caused by strong winds by carefully selecting the water-mass ratio according to the total mass of the building. [PERSON] and [PERSON] [10] performed both experimental and analytical investigations on the nonlinear vibrations of a structure coupled with a sloshing water tank, targeting applications such as elevated water tanks or storage tanks for liquefied natural gas. The system was subjected to a sinusoidal excitation with a series of amplitudes. They found that the shapes of the resonance curves of the structure change with different water depths of the sloshing tank. The amplitude of the structure is very small when the system is close to the tuning frequency and under a small excitation amplitude (i.e., an amplitude equal to 0.205 N, 0.139 N and 0.152 N, respectively), while a complex oscillation motion occurs when the excitation amplitude is large (i.e., equal to 0.64 N). In similar work, [PERSON] et al. [11] considered a driven pendulum connected to a TSD to investigate its induced wave breaking and the associated energy dissipation generated by the shallow water sloshing of a TSD system. Large roll motions of up to 55\({}^{\circ}\) and subsequent complex flows occurred in this system and so a particle-based Lagrangian smoothed particle hydrodynamics (SPH) solver was employed to simulate the system in terms of its frequency behaviour in order to allow large deformation within the solution. For small excitation, low-amplitude travelling waves were observed. While for large excitation, violent fluid motions and breaking waves were observed. They also noted a significant change in the frequency response of the system with large excitation. As the second part of the same study, [PERSON] et al.