Abstract Vibrations of plates, shells and plate–shell systems coupled with sloshing, quiescent and inviscid fluid have been advantageously studied by inserting the sloshing condition into the eigenvalue problem. Here a formulation of this particular eigenvalue problem for symmetric matrices is obtained. In fact, in the previous studies, this technique has given eigenvalue problems for non-symmetric matrices for which the problem of the existence of complex eigenvalues arises. The present analysis deals with compressible and incompressible fluids and the discretization of the system is obtained by using the Rayleigh–Ritz method. The Rayleigh quotient of the system is manipulated to obtain expressions suitable for symmetric formulations of the eigenvalue problem. In particular, the Rayleigh quotient is transformed into a simpler expression where the potential energies of the compressible fluid and free surface waves do not appear. The method is applied to a vertical, simply supported, circular cylindrical shell partially filled by an incompressible sloshing liquid. A case with large interaction between sloshing and bulging modes is considered and interesting phenomena are observed.