Tuned mass dampers are useful devices for limiting vibrations in various machines. Their main advantage is that they can be used as add-on elements and do not require integration into the main structure. However, conventional tuned mass dampers that use viscous or material damping are not attractive from the energetic point of view because they also dissipate energy in the case of unobjectionable small vibrations. Dry friction used as a lock-up element for the tuned mass damper could aid in overcoming this disadvantage. The dynamics of such a system are investigated in this paper. A special analytic technique based on variables’ rescaling is applied to obtain approximate analytical predictions of the steady-state amplitudes for both the main system and the mass damper. The results show that the lock-up mass damper can efficiently limit the amplitudes at resonance for a wide range of parameters but cannot supply perfect vibration suppression at any tuned frequency. Additionally, the efficiency of the system is limited to a certain excitation range. Excitation that increases over a certain threshold leads to notably high vibration amplitudes that cannot be handled by the described simple approach. All analytical results are confirmed by numerical simulations of the full system.