# Simulating Slosh Induced Damping, with Application to Aircraft Wing-like Structures
## 2 Problem Description  
In this study, a simplified aircraft-wing-like structure is used based on a series of experiments that were recently performed at the Airbus Protospace Lab in Fulton (Bristol, UK) to explore the interaction between the sloshing tank and the motion of a beam with moderate initial deflections [18; 19]. The simplified system consists of a hollow cantilevered steel beam with a sloshing water tank attached on top. The length, height and width of the beam are \(L_{\mathrm{s}}=12\) m, \(H=0.3\) m and \(W=0.55\) m along the \(x\)-, \(y\)- and \(z\)-axes, respectively. The thickness of the walls of the beam is \(0.22H\). The dimensions of the beam are determined based on a full-scale single-aisle airliner wing and the first bending frequency of the elastic beam is approximately 2 Hz, with the period \(T=0.5\) s, to match the natural frequency of the wing [19]. In the present simulation, an idealised beam is used, in which the damping matrix of the elastic beam is set to zero and the gravitational force on the beam is ignored. This eliminates any influence from both the mechanical damping and the weight of the beam on the final result. The beam is made of steel, with a density, [PERSON]'s modulus and Poisson's ratio of 8050 kg/m\({}^{3}\), \(2\times 10^{11}\) Pa and 0.3, respectively. The elastic beam is fixed at one end to represent the connection between the aircraft wing and the fuselage and free at the other end. The fuel tank inside the wing of the aircraft, where the sloshing phenomena occur, is also idealised as a hexahedral water tank in this study. The water tank is embedded inside the hollow section of the beam and is mounted close to the free end to represent the fuel tank of the aircraft, as proposed by [PERSON] et al. [18] and [PERSON] et al. [19]. The tank has the same height and width as the beam, while the length is \(L_{f}=3.5\) m. The water tank is filled to 50% throughout this study. Gravitational acceleration is applied to the water tank in the negative \(y\)-direction. The density and dynamic viscosity of the water are 1000 kg/m\({}^{3}\) and 0.001 N \(\cdot\) s/m\({}^{2}\), respectively, and the surface tension of the fluid is set to 0.0728 N/m. The mass ratio between the fully loaded fluid tank and the beam is 0.07, again to match the working conditions of a civil aircraft with reserve fuel [18]. The present study assumes an initial meniscus for the fluid, as in Figure 14 of [PERSON] et al. [5], in order to replicate the wetting effect of water on the tank wall. One objective of this study is to investigate how slosh-induced damping is affected by the number of compartments of the water tank. Four designs are considered in this study, one, two, four and eight compartments. Examples are shown in Figure 1(one compartment) and Figure 1(b) (four compartments). Internal baffles are used to separate the water tank into the separate compartments that block mass and momentum exchange between them. An external load is uniformly applied to the upper \(x\)-\(z\) surface of the beam-tank system along the negative \(y\)-axis direction between \(-0.5\) s and 0 s. The external load is applied in the \(y\)-axis direction because atmospheric turbulence typically excites an aircraft wing along this axis and direction during cruise flight conditions. The \(y\)-axis external load is designed to mimic the effect of the atmospheric turbulence excitation. Timing starts once the external load is removed. After 0 s, the beam-tank system is excited by the external load and subject to free decay oscillation motion. As reported by [PERSON] et al. [5], the largest damping and the highly turbulent fluid motion takes place within the first 1.5 s (around 15 oscillation cycles) of the transition, with this being the most crucial to use for reducing aircraft gust loads. Similarly, a large damping and turbulent fluid motion can be observed in the first 1 s (around 6 oscillation cycles) of the free-decay motion of the sloshing tank, [PERSON] et al. [18].