Rocking　structures　are　vulnerable　to　overturn　under　earthquake　shaking.　It　is　necessary　to　reduce　the　overturning　probability　of　rocking　structures.　Recently,　inerters　were　proposed　to　control　the　vibrations　of　rocking　columns,　and　it　was　found　that　the　structure　equipped　with　inerter　has　reduced　probability　of　overturning.　However,　inerter　does　not　have　capacity　to　dissipate　energy,　its　control　performance　is　believed　able　to　be　enhanced　once　being　used　in　conjunction　with　dampers.　Therefore,　this　paper　approaches　rocking　structural　vibration　control　from　two　directions:　by　applying　supplemental　inertia　and　applying　supplemental　damping　to　the　structure.　To　this　end,　inerter,　clutched-inerter　and　viscous　damper　are　selected　as　control　devices,　and　two　combined　scenarios　are　also　considered,　with　configuration　I　consisting　of　an　inerter　and　a　viscous　damper　in　parallel　and　configuration　II　consisting　of　a　clutched-inerter　and　a　viscous　damper　in　parallel.　The　analytical　models　of　a　rocking　frame　with　supplemental　control　devices　in　variable　positions　are　presented　and　the　equations　of　motion　for　each　model　are　derived.　In　addition,　the　maximum　coefficient　of　restitution　of　the　rocking　frame　with　supplemental　control　device　is　derived　based　on　the　angular　momentum-impulse　theorem.　The　effects　and　applicable　conditions　of　the　considered　devices　are　evaluated　by　means　of　assessing　the　overturning　resistances,　maximum　responses　and　fragility　curves　of　rocking　frames　under　different　conditions.　The　results　show　that,　with　appropriate　damping　coefficient　and　apparent　mass　of　the　inerter,　the　combined　configurations　I　and　II　provide　better　control　effectiveness　than　the　cases　with　inerter,　clutched-inerter　or　viscous　damper　applied　individually.