The　coupling　of　frame　structures　with　external　oscillating　bodies,　such　as　rigid　walls,　dynamic　mass　absorbers,　elastoplastic　dampers,　or　tuned　mass　dampers,　can　be　effective　in　reducing　the　displacements　of　the　structure　and　protecting　it　against　seismic　loads.　This　paper　proposes　connecting　an　external　oscillating　body　to　the　first　story　of　a　frame　structure　and　studying　the　effectiveness　of　the　coupling　by　evaluating　the　reduction　in　the　displacements　of　the　frame　structure.　The　inertial　effects　of　the　oscillating　body　are　increased　by　introducing　a　virtual　mass,　provided　by　an　inerter　device.　The　oscillating　body,　characterized　by　physical　and　virtual　masses,　is　connected　to　the　frame　structure　through　a　hysteretic　device.　The　study　is　performed　on　a　dynamically　equivalent　three-degree-of-freedom　(3-DOF)　model,　whose　equations　are　written　using　a　direct　approach.　To　verify　the　effectiveness　of　the　protection　device,　named　hysteretic　mass　damper　inerter　(HMDI),　in　reducing　the　displacements　of　the　frame　structure,　the　displacement　of　the　first　story　and　the　drifts　of　the　upper　stories　are　compared　to　those　of　the　frame　structure　not　connected　to　the　external　oscillating　body.　An　initial　spectral　analysis,　performed　on　the　linearized　mechanical　system,　clarifies　the　role　of　the　parameters　of　the　external　device　in　the　dynamic　behavior　of　the　coupled　system.　An　additional　seismic　analysis　is　performed　by　using　three　single　earthquake　records　first,　and　then　a　set　of　seven　additional　earthquake　records　selected　to　be　compatible　with　the　design　spectrum　of　Los　Angeles.　Specific　spectral　gain　maps　are　used　to　organize　the　results　of　an　extensive　parametric　analysis.　They　show　that　the　HMDI　reduces　both　displacements　and　drifts　of　the　structure　in　large　ranges　of　the　parameters　that　characterize　the　HMDI.　It　is　found　that　the　proposed　protection　method　is　particularly　effective　for　low-　and　medium-rise　frame　structures.