A finite element method is developed to treat the steady-state vibration of two axisymmetric structures—a base substructure and an attached dampersubstructure—that are driven by traveling wave excitation and that couplethrough a spatially distributed hysteretic friction interface. The base substructure is representative of a rotating brake rotor or gear, and the damper is a ring affixed to the base under preload and intended to control vibration through friction along the interface. In the axisymmetric approximation, the equation of motion of each substructure is reduced in order to the number of nodal degrees of freedom through the use of a propagation constant phase shift. Despite nonlinearity and with contact occurring at an arbitrarily large number of nodal points, the response duringsticking, or during a combination of sticking and slipping motions, can be determined from a low-order set of computationally tractable nonlinear algebraic equations. The method is applicable to element types for longitudinal and bending vibration, and to an arbitrary number of nodal degrees of freedom in each substructure. In two examples, friction damping of the coupled base and damper is examined in the context of in-plane circumferential vibration (in which case the system is modeled as two unwrapped rods), and of out-of-plane vibration (alternatively, two unwrapped beams). The damper performs most effectively when its natural frequency is well below the base's natural frequency (in the absence of contact), and also when its natural frequency is well separated from the excitationfrequency.