In order to shift the frequencies of a liquid with a free surface away from dangerous system frequencies, the surface of the liquid may be covered with a flexible membrane or a thin elastic plate. The magnitude of such a frequency shift depends on the parameters of the liquid and those of the structural system and should be known for the design of an aerospace vehicle. For that reason, we treat a rigid circular cylindrical large aspect ratio container filled with incompressible and viscous liquid, being covered by an elastic structure. The coupled frequencies and the decay magnitudes are determined for the lower vibration modes. The fact that we deal with a large aspect ratio tank, where the liquid close to the bottom of the container exhibits only small velocities, suggests that the adhesive bottom conditions may be deleted, while those at the side wall of the container are mainly contributing to the damping and are for that reason considered in the analysis. We determine the complex frequencies of the hydroelastic system. The results show that the coupled frequencies increase with increasing liquid height ratio and reach soon a nearly constant value, which is being reached much earlier for higher modes. The decay magnitude also reaches quickly a constant value. Higher modes are damped out fast. In addition, it was found that the coupled oscillation frequencies are always larger than those of the frictionless liquid, and that the difference between them decrease considerably with increasing mode numbers.