This study investigates the displacement transmissibility of single-degree-of-freedom systems with a Coulomb friction contact between a mass and a fixed or oscillating wall. While forced vibration and base motion problems have been extensively investigated, little work has been conducted on the joined base-wall problem. Based on the work of Den Hartog (Trans Am Soc Mech Eng 53:107–115, 1930), analytical expressions of the displacement transmissibility are derived and validated numerically. The mass absolute motion was analysed in the joined base-wall motion case with a new technique, with results such as: (1) the development of a method for motion regime determination; (2) the existence of an inversion point in transmissibility curves, after which friction damping amplifies the mass response; (3) the gradual disappearing of the resonant peak when the ratio between friction and elastic forces is increased. Moreover, numerical analysis provides further insight into the frequency region where mass sticking occurs in the base motion problem.@en