It is proposed to investigate in this paper the damped vibrations of an incompressible liquid contained in a deformable tank. A linearized formulation describing the small movements of the system is presented. At first, a diagonal damping is introduced in the reduced equations of the hydroelastic sloshing problem. We obtain a nonclassically damped coupled system with a damping matrix that is not symmetric. Then, by projecting the system onto its complex modes, the frequency and time responses for different type of loads are built. A numerical application is illustrated on a test case.

Internal resonance in the vibration of a floating roof coupled with nonlinear sloshing in a circular cylindrical oil storage tank is investigated. The nonlinear system exhibits internal resonance when nonlinear terms of the governing equation have a dominant frequency close to a certain modal frequency of the system. Numerical results show that when internal resonance occurs, the responses of stresses in a floating roof exhibit a long-duration period of large amplitude despite a short duration of the earthquake excitation applied to the tank. Due to the presence of internal resonance, the underestimation of the stresses associated with the use of the linear theory becomes more marked, and thus the importance of nonlinearity of sloshing in the stress estimation is accentuated. It is illustrated that the magnitudes of the stresses increase with the increase in the liquid-filling level, and that the effect of internal resonance on the stresses noted in the case of sinusoidal excitation appears under real earthquake excitation. A method for reducing the stresses is proposed.

The vibration of a floating roof hydroelastically coupled with nonlinear sloshing is analyzed. Influences of the nonlinearity of sloshing on the magnitude of stresses arising in a floating roof are investigated. Numerical results show that (i) neglecting the nonlinearity of sloshing significantly underestimates the magnitude of the stresses, even when the nonlinear effect is small for the roof displacement; and (ii) the underestimation associated with the use of the linear approximation becomes more marked with the decrease in the liquid depth. The reasons for these results are explained based on the fact that in the nonlinear sloshing, the modal component with circumferential wave number 2 is excited.

A mechanical model for low-gravity sloshing in an axisymmetric tank is developed using a newly developed slosh analysis method. In this method, spherical coordinates, whose origin is at the top of the cone that is tangent to the tank at the contact line of the meniscus with the tank wall, are used to analytically determine the characteristic functions for an arbitrary axisymmetric tank for which it is customary to resort to numerical methods. By this means, fast and cost-efficient computation can be conducted. Parameters of the mechanical model are determined such that the frequency responses of the resultant force and moment to lateral excitation coincide with those of the actual sloshing system. Influences of the Bond number and the liquid-filling level on the parameters of the mechanical model are examined.

A mathematical model is developed for sloshing effects in half-full horizontal cylindrical vessels, under external excitation in the direction of the longitudinal vessel axis. In this geometry the problem is not separable. The velocity potential is expressed in a double series form, and after some mathematical manipulation the problem results in a series of systems of ordinary linear differential equations, one for each longitudinal mode. For the particular case of harmonic motion, a semi-analytical solution is possible. Keeping only the first two terms of the expansion in the transverse direction, an elegant formulation of remarkable accuracy is obtained. Hydrodynamic pressures and forces are calculated for harmonic excitation and for a real seismic motion event. Finally, the equivalence between the half-full horizontal cylinder and an “equivalent” rectangle is demonstrated.