Negative　stiffness　element　has　been　applied　to　improve　the　control　performance　of　tuned　mass　dampers　(TMDs)　recently,　and　two　tuned　mass　dampers　enhanced　with　negative　stiffness　(TMD-NS)　element,　namely　KDamper　(Without　loss　of　generality,　it　is　referred　to　as　TMD-NS　I　in　the　present　study)　and　Extended　KDamper　(EKD,　TMD-NS　II),　have　been　developed.　Previous　studies　have　demonstrated　the　control　effectiveness　of　TMD-NS　I　and　II.　However,　there　still　exist　some　issues　to　be　addressed:　(1)　previous　studies　normally　optimize　TMD-NS　via　the　intricate　and　time-consuming　numerical　methods,　the　analytical　solutions　for　the　optimal　design　parameters　of　TMD-NS　II　remain　unknown;　(2)　a　comprehensive　and　exhaustive　evaluation　that　compares　the　control　effectiveness　of　TMD-NS　I　and　II　is　absent　from　existing　literature.　To　fill　these　research　gaps,　this　study　derives　closed-form　optimum　solutions　for　TMD-NS　II　using　the　H∞　approach.　The　control　effectiveness　of　TMD-NS　I　and　II　in　suppressing　the　seismic　responses　of　structures　is　investigated　systematically.　Specifically,　the　analytical　model　of　an　undamped　SDOF　system　equipped　with　TMD-NS　I　or　II　is　first　developed　within　a　unified　framework,　and　corresponding　dynamic　equations　of　motion　are　formulated.　Subsequently,　the　optimal　parameters　of　TMD-NS　I　and　II　are　derived　based　on　the　classical　“fixed-point”　theory.　Based　on　the　derived　optimal　parameters,　the　control　effectiveness　of　TMD-NS　I　and　II　are　examined　by　using　a　damped　SDOF　system　subjected　to　harmonic　excitations　and　real　earthquake　ground　motions.　Finally,　a　5-storey　isolated　benchmark　building　model　is　adopted　to　further　investigate　the　effectiveness　of　TMD-NS　in　the　seismic　protection　of　engineering　structures.　The　results　reveal　that　the　derived　closed-form　solutions　are　accurate　in　capturing　the　optimal　parameters　of　TMD-NS.　In　addition,　both　the　optimized　TMD-NS　I　and　II　outperform　the　conventional　TMD　in　reducing　the　seismic　responses　of　structures.　Furthermore,　TMD-NS　I　proves　more　effective　in　reducing　the　absolute　acceleration　of　the　isolated　building,　whereas　TMD-NS　II　demonstrates　better　performance　in　mitigating　the　isolating　deformation.　In　a　nutshell,　both　the　TMD-NS　I　and　II　are　highly　effective　alternatives　to　conventional　TMDs,　showcasing　superior　performance　in　vibration　reduction　and　robustness.　©　2024　Elsevier　Ltd