We develop a variational method for solving the spectral problem of free vibrations of a shell of revolution nonclosed in the meridional direction. This method is equally efficient for both medium and small values of the relative thickness of the shell. Coordinate systems of functions are constructed with regard for the structure of formal asymptotic expansions of a fundamental system of solutions of the initial equations. As an example, we calculate frequencies and forms of vibrations of a circular cylindrical shell and show that the algorithm proposed for solving the considered problem guarantees the uniform convergence of solutions and their first three derivatives in the entire region of integration of the equations.