Periods and damping times of quasi-mode fundamental kink oscillations have been computed for non-uniform cylindrical models of coronal loops. The radial inhomogeneity length-scale, the density contrast and the internal Alfvén velocity are three equilibrium quantities that determine the theoretical values of the period and damping times in 1-D equilibrium models. From a seismological point of view this means that observed values of period and damping time can be recovered by an infinite number of equilibrium models. In other words, observed values of period and damping time of the fundamental kink oscillation mode do not allow a unique identification of even a 1-D equilibrium model. Only if there is additional information on one of the three equilibrium quantities, can we use the observed values of period and damping time to determine the two remaining equilibrium quantities. However, it is not all bad news. It turns out that, even without additional information, we can determine upper limits to the internal Alfvén velocity. We apply this scheme to the set of 11 loop oscillation events studied in Goossens et al. 2002 and find constraints on the equilibrium parameters for these 11 events.