This paper investigates a damping strategy for integrally bladed disks (blisks) based on the use of friction rings. The steady-state forced response of the blisk with friction rings is derived using the so-called dynamic Lagrangian frequency-time method adapted to cyclic structures with rotating excitations. In addition, an original approach for optimal determination of the number of Fourier harmonics is proposed. In numerical applications, a representative compressor blisk featuring several rings is considered. Each substructure is modeled using finite-elements and a reduced-order modeling technique is used for the blisk. The efficiency of this damping technology is investigated, and friction dissipation phenomena are interpreted with respect to frequency responses. It is shown that the friction damping effectiveness depends mainly on the level of dynamic coupling between blades and disk, and on whether the dynamics features significant alternating stick/slip phases. Through parameter studies, design guidelines are also proposed.