# Simulating Slosh Induced Damping, with Application to Aircraft Wing-like Structures
## 4 Results and Discussion
### Impact of Compartment Numbers  
The impact of compartment numbers on the slosh-induced damping ratio with three initial excitation displacements was also investigated. The instantaneous non-dimensional tip vertical displacement of the system over ten oscillation cycles for \(D_{i}=0.25\) is shown in Figure 13. From the fifth cycle onwards, there is significant decay in the tip displacement with \(N_{c}=8\), compared with other compartment numbers. In contrast, the tip displacement with \(N_{c}=4\) decays less than other compartment numbers from the sixth cycle onwards. Snapshots of the sloshing tank with \(D_{i}=0.25\) with different compartment numbers during the sixth cycle (from \(t/T=5.0\) to \(t/T=6.0\)) are shown in Figures 14-16. Figure 12: Normalised force–velocity diagram for the eighth and ninth oscillation cycles with \(N_{c}=1\). Figure 13: Time history of non-dimensional tip vertical displacement of the system normalised by the initial excitation amplitude over ten cycles for \(D_{i}=0.25\). By placing a baffle in the middle of the tank (i.e., \(N_{c}=2\)), the formation of slosh wave is blocked. Due to the blockage of the mass and momentum exchange between the front and rear half of the tank, the amplitude of the wave crests with \(N_{c}=2\) are smaller than that of the \(N_{c}=1\). There is only one wave crest impact with the ceiling of the tank, but the impact is much more gentle than that of \(N_{c}=1\) at \(t/T=5.0\). A similar shape of sloshing waveform occurs for \(N_{c}=2\) at \(t/T=5.5\) and \(t/T=6.0\) compared with that of \(N_{c}=1\), but the waves formed in the rear half of the tank do not have contact with the tank ceiling at \(t/T=6.0\). By placing three baffles (i.e., \(N_{c}=4\)) in the water tank for \(D_{i}=0.25\), the impact between the sloshing wave and the tank wall has been reduced. More specifically, at \(t/T=5.0\), the tank of \(D_{i}=0.25\) with \(N_{c}=1\) and \(N_{c}=2\) forms falling waves, but for the tank with \(N_{c}=4\), as can be seen from Figure 14c, water has, and will, continue to fall to the front of its respective compartment along the slope. At \(t/T=5.5\), falling waves are seen in all compartments with \(N_{c}=4\), with gentle impact between the sloshing wave and the front top corner of each compartment for the front three. At \(t/T=6.0\), when the tank is back to its lowest position, the free surfaces in all four compartments are generally horizontal. As the number of compartments is increased to eight, we observe one sloshing wave cycle repeated over two cycles, with water slamming to the front and rear baffles and impacting the ceiling of the tank. Different sloshing behaviour to that that has been observed under different compartment numbers is almost certainly due to changes in the natural frequency of the sloshing waves created by the different tank structures. According to [PERSON] [52], the first sloshing frequency with rectangular tanks can be estimated by empirical equation as:  
\[f_{slosh}=\frac{1}{2\pi}\sqrt{\frac{3.16g\;tanh(3.16\frac{h}{L})}{L}}. \tag{9}\]  
where \(f_{slosh}\) is the first frequency, \(g\) is the gravitational acceleration, \(h\) is the height of the water tank and \(L\) is the length of the rectangular tank along direction of excitation. In this study, the water inside the tank is excited along both the longitudinal and the transverse directions due to the free decay motion of the cantilever beam. An increase in the number of compartments reduces the length of each in the longitudinal direction, hence, this increases the sloshing frequency. The slosh phenomenon we observed at \(N_{c}=8\) is due to the sloshing frequency being approximately half of the beam's oscillation frequency. While the beam oscillation frequency is not a multiple of the sloshing frequency when \(N_{c}=4\); thus, a reduced impact between the sloshing wave and the tank wall is observed. An in-depth analysis of this behaviour is expected in future work.