# Simulating Slosh Induced Damping, with Application to Aircraft Wing-like Structures
## 4 Results and Discussion
### Impact of Initial Excitation Displacement with One Compartment  
The impact of the initial excitation displacement on the slosh-induced damping ratio with no baffles was considered. The instantaneous non-dimensional tip vertical displacement of the system, as well as the instantaneous non-dimensional vertical fluid force of the water tank, over ten oscillations and different initial excitation amplitudes, are shown in Figure 5a,b. These show the vertical fluid force of the water tank, \(F_{y}\), non-dimensionalised according to the gravitational force of the water tank, \(F_{g}\). The fluid force follows the positive upwards \(y\)-direction. The tip vertical displacement and vertical fluid force of the beam-tank system with frozen water are also shown in Figure 5a,b. Compared with the system with frozen water, the tip vertical displacement of the beam-tank system with sloshing water shows clear damped sinusoidal curves, as in Figure 5a. The vertical fluid force shows similar spike curves but different amplitudes for different \(D_{i}\). In order to cross compare the tip vertical displacement and vertical fluid force among different initial excitation amplitudes, we normalised the non-dimensional tip vertical displacement and the non-dimensional vertical fluid force by the value of the initial excitation amplitude, as shown in Figure 5c,d. There are two key approaches for investigating the beam-tank model. The first approach is to establish a non-inertial frame of reference based on the tank's body-fitted coordinates. Under this approach, the focus is the fluid and, therefore, we can investigate the cause and mechanism of the sloshing patterns under different conditions. The forces acting on the water consist of the gravitational, inertial and forces due to the sloshing motion. As both the inertial force and the sloshing force are functions of time, the total force acting on the water is also a function of time. The second approach is to establish an inertial frame of reference based on global coordinates. The focus then is the whole system instead of the fluid. This second approach is a convenient way to investigate how different sloshing patterns affect the free-decay motion of the system. Using this approach, we are interested in the forces acting on the beam by the sloshing water. There is no inertial force to be considered as the observer is in the inertial frame of reference. The presented study follows this second method in order to investigate the impact on the free decay motion of the system of the sloshing water. The vertical fluid forces shown in Figure 5b,d are those acting on the beam by the fluid. Figure 4: Damping ratio of the free decay beam–tank system with both sloshing water and frozen water as a function of the number of compartments. The vertical motion (along \(y\)-axes direction) of the tip of the beam-tank system can be qualitatively analogized with a vertical 1-DOF mass-spring-damper model with gravity. If there is no tank in the hollow beam, the vertical motion of its tip is similar to a hinged spring as an idealised beam is used in this study. If a tank with frozen water is introduced into the hollow beam then it is equivalent to hanging a mass block on the vertical spring and letting it naturally oscillate. There is no damping involved in the system. The only force acting on the beam (or spring) is the constant gravitational force of the frozen water (or the block). The impact of the gravitational force is to change the equilibrium position of the system, i.e., an offset in both the tip vertical displacement and the vertical fluid force. Finally, if a sloshing tank is introduced, this is then equivalent to a sloshing mass block hanging from the spring, and there will be sloshing forces acting on the beam (or spring) on top of the gravitational force of the water. This study will pay particular attention to the impact of the sloshing forces on the oscillation motion of the system.