## 5 Conclusions  
In this paper the results of an experimental study on the characteristics of the free motion of water in prismatic sloshing devices with different bottom shapes have been presented. The performed experimental campaign consisted of a series of tests on a tank filled with water at nine different depths and each case tested for three different bottom geometries, namely flat, sloped and circular. The tank was subject to an impulsive random force and data about the related displacement and the dynamic inertia force have been acquired and processed in order to characterise the oscillating motion of the filling liquid and to focus on the so called \"beating phenomenon\". Since the interest was in the natural sloshing frequency and damping features, including the beating phenomenon, the sole measurement of the base shear force was deemed sufficient for the purpose of the work. Thence, further understanding of the free water surface dynamics considering different bottom geometries is left for future work.  
Mechanical models of tanks filled with water in motion resemble the classical _mass-spring-damper_ system and from this point of view equations to obtain natural frequency values have been formulated. An important objective of this work was to compare the equations outcomes with the experimental results and evaluate the effectiveness of such theoretical approaches to actual laboratory cases, even in the circumstance of aspect ratios (\(h/L\)) higher than those allowed by the linear wave theory and in order to assess the validity of classical methods for practical tuning procedures. Furthermore, damping behaviour has been investigated considering the related aspect of the beating phenomenon.  
Experimental natural frequency values for the box-shaped tank have shown a very good agreement with the corresponding values from [PERSON]'s linear wave theory with a percentage error always under 2% for all the water depth ratios tested. The relative error was higher when compared with [PERSON]'s formulation because of the different theoretical approach which was more affected by the variation of liquid mass.  
For the other two bottom shapes, namely sloped at 45\({}^{\circ}\) and circular, the best analytical approach resulted to be the one proposed by [PERSON] that, starting from the linear wave theory of [PERSON] and simplifying the wave dynamics equation with a small angles hypothesis, retrieved a modified expression for the natural sloshing frequency employing the wet perimeter. A further modification was to adapt the wet perimeter defined solely for wedged corners also for the case of the circular one.  
The sloped shaped bottom showed a steady behaviour throughout all the nine different liquid depths with a slight increase of the oscillating frequency only for water heights over 9 cm. This result is notable as it demonstrates that the sloped shape is the most robust with respect to the frequency \"mistuning\" that may be caused by varying water levels as it may happen in water storage tanks used for passive structural control purposes.  
Conceptualisation, A.A., N.C., C.B. and F.U.; methodology, N.C.; formal analysis, A.A.; investigation, N.C. and A.A.; data curation, A.A.; writing--original draft preparation, A.A.; writing--review and editing, A.A., N.C., C.B. and F.U.; supervision, C.B. and F.U.; project administration, C.B.; funding acquisition, C.B. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the Italian Ministry of Education, University and Research (MIUR) through the funded Project of Relevant National Interest \"Combined numerical and experimental methodology for fluid structure interaction in free surface flows under impulsive loading\" (Prot. 20154 EHYW9).  
Not applicable.  
The authors declare no conflict of interest.