## 1 Introduction  
The aim of this paper is to investigate the dynamic properties of the sloshing phenomenon inside rectangular tanks characterised by different bottom shapes. The interest is to investigate two particular aspects of the fluid-structure interaction, namely the natural sloshing frequency of the fluid inside the tank and its damping feature. The latter is related, among the other features, to the beating phenomenon [1], characterised by the undesired dynamic forces induced on the substructure due to sloshing motion. The understanding of the aforementioned aspects is important in the framework of the vibration control [2; 3; 4], especially in the design process of containers or passive damping devices exploiting fluid motion, since the motions of the container and the liquid are coupled and the resonance phenomena could occur [5; 6; 7; 8; 9; 10; 11; 12; 13]. The dynamics of the coupled system generally depend on the container geometry and the amount of the inner liquid. At a first glance, such a problem may be analysed as the case of a bifilar pendulum, even if the physics of such a pendulum is quite different from the sloshing motion inside a container which moves along the horizontal direction. In the case of a container suspended as a bifilar pendulum, gravity directly provides the restoring force, working in concert with, or in opposition, to the hydrodynamic forces, so that both in-phase and antiphase oscillations can occur [14; 15; 16]. In the horizontal sloshing case, gravity does not directly affect the motion of the container, being indirectly involved through the liquid motion, whereby the hydrodynamic pressure of the liquid on its walls provides the restoring force. Hence, the sloshing liquid is expected to be out of phase with the oscillating tank, always accumulating in the direction opposite to that of the containers' motion. In some cases, the interaction between the liquid motion and the substructure could produce significant advantages in the overall dynamics of the coupled system. A classical example is represented by the so called Tuned Sloshing Dampers (TSD) [17; 18; 19; 20; 21]. Theirworking mechanism is based on the dynamic response of the liquid motion inside the tank and its interaction with the dynamic input, generally transmitted by the structure on which it is installed [22]. The sloshing forces transferred to the structure are called _base shear forces_. In real life applications, structural systems are typically induced to move due to external actions, e.g., wind loads or earthquakes and the liquid motion tends to come with a phase shift with respect to the structural motion, depending on the physical properties of the liquid itself, the tank geometries and the structural dynamics properties. This phenomenon results in a dissipation energy by virtue of the energy exchange between the structural system in motion and the device itself. During the dynamic motion, an energy amount is temporarily stored and can be effective in vibration damping when released back to the structure. This energy exchange becomes more effective if a tuning process is properly carried out, so that the sloshing forces are able to reduce the structural vibration and mitigate resonance effects [23]. When excited, the water wave motion quickly develops, but on the contrary, the established wave motion does not vanish immediately after the excitation is over. This results in the so-called beating phenomenon, consisting of long lasting undesired oscillating pulses transmitted to the substructure with an amplitude modulation instead of an exponentially decaying trend. This behaviour of continuous wave motion is often regarded as an adverse effect, for instance within the structural control framework. The beating phenomenon is due to a fraction of the energy absorbed by the device being transferred back to the structure rather than being dissipated within the device itself, which is a result of the complex _liquid-tank_ system [24].