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　(TMDNS)　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　timeconsuming　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　infinity　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.