This study investigates the stability of liquid movement in tanks subjected to combined horizontal and vertical periodic loads, with particular emphasis on damping effects. The liquid is modeled as incompressible, inviscid, and irrotational, reducing the governing equations to the Laplace formulation with kinematic and dynamic free-surface conditions. By exploiting rotational symmetry, the problem is simplified to one-dimensional singular equations, which are solved using a boundary element method combined with the normal mode approach. Spectral analysis yields natural frequencies and vibration modes of the liquid in a rigid shell, forming the basis for equations that describe free-surface dynamics. Since in-flight loads cannot be precisely defined, fuzzy logic is employed to capture uncertainty, and stability regimes are determined under vertical and combined excitations. The modeling approach neglects the influence of internal baffles, compressibility of the gas above the liquid, and capillary effects, which are left for future research. Rayleigh damping is incorporated, demonstrating that even minor damping coefficients significantly reduce surface oscillations. The analysis underlines that damping plays a decisive role in suppressing sloshing instabilities and ensuring the structural reliability of launch vehicle fuel tanks.