# Simulating Slosh Induced Damping, with Application to Aircraft Wing-like Structures
## 4 Results and Discussion
### The Slosh-Induced Damping Ratio of an Oscillated Aircraft-Wing-like Structure  
As per Section 2, an idealised cantilever beam with no structural damping is employed as part of this decoupled analysis in order to highlight only the damping induced by the attached sloshing tank. For a beam-tank system with frozen water, the tip displacement of the system under a free-decay oscillation will maintain the same amplitude from cycle to cycle (with the shape of its time history being a basic sinusoidal wave), as there is no damping source involved in the system. The time history of the tip displacement of the system can be expressed as a damped sinusoidal function as  
\[D=D_{amp}\cdot e^{(-\lambda t)}cos(2\pi\frac{t}{T}-\phi), \tag{7}\]  
where \(D\) is the instantaneous tip displacement of the system along the \(y\)-axis direction, \(e\) is Euler's number, \(\lambda\) is the decay rate, and \(\phi\) is the phase angle at \(t=0\). The damping ratio of the tip displacement is denoted as \(\zeta\) and is defined as  
\[\zeta=\frac{\lambda}{\sqrt{\lambda^{2}+(\frac{2\pi}{T})^{2}}}. \tag{8}\]  
The tip displacement damping ratio is a non-dimensional parameter to quantify the decay rate of the free oscillation of the beam-tank system relative to its natural frequency. We note that \(\zeta=0\) for the frozen water case, as would be theoretical expected, and confirm the levels of numerical dissipation are not adversely affecting our results. Meanwhile, the beam-tank system with sloshing water is expected to have a non-zero value for the damping ratio. In this study, the ratio of a case is obtained by finding the best fit of the time history data of the vertical tip displacement of the system during \(t/T=0\) and \(t/T=10\), according to the damped sinusoidal function.  
The global view of the damping ratio of the free decay beam-tank system with both sloshing water and frozen water as a function of the number of compartments, \(N_{c}\), and within the parameter range of this study, is shown in Figure 4. For compartment number \(N_{c}=1\) (i.e., with no baffles placed inside the tank), the slosh-induced damping of \(D_{i}=1.0\) is close to that of \(D_{i}=0.5\), with a difference of approximately 2%. In contrast, the slosh-induced damping of \(D_{i}=0.25\) is much smaller than the other two cases, at around 48% of that with \(D_{i}=1.0\). By including baffles into the tank and increasing the compartment number from one to two, four and eight, the slosh-induced damping with \(D_{i}=1.0\) does not appear to change significantly. For \(D_{i}=0.5\), the slosh-induced damping with two compartments is quite close to that with one compartment. Continuing to increase the compartment number to four and eight, the slosh-induced damping with \(D_{i}=0.5\) increased by 18% and 24%, respectively. The damping of \(D_{i}=0.25\) with different compartment numbers shows a distinctly non-monotonic function, with significant variation within the parameter range of this study. As the number of compartments is increased from one to four, the slosh-induced damping dropped by approximately 59%. As the number of compartments is increased to eight, the slosh-induced damping also increases by 153% compared to with one compartment.