The　study　on　the　optimal　design　theory　for　vibration　reduction　structures　with　complex　stiffness　and　damping　is　necessary.　The　dynamic　equations　for　structures　including　tuned　mass　dampers　(TMDs)　with　complex　stiffness　and　damping　were　established.　The　method　for　determining　their　optimal　frequency　ratio　and　optimal　damping　ratio　was　presented　based　on　the　fixed　point　theory.　The　steady　state　responses　of　the　such　kind　structures　to　harmonic　excitation　and　stationary　random　excitation　were　analyzed,　respectively.　The　theoretical　formulas　for　the　optimal　damping　and　frequency　were　derived　when　the　displacement　amplitude　minimization　of　the　main　structure　was　set　as　the　optimal　objective.　The　results　showed　that　the　overall　vibration　reduction　effect　of　this　kind　of　structures　were　close　to　that　of　structures　with　viscous　damping,　and　the　former　is　better　in　a　certain　frequency　range　with　the　same　mass　ratio.　The　random　responses　of　this　kind　of　structures　under　Davenport　wind　speed　spectrum　were　analyzed.　The　results　showed　that　the　vibration　reduction　effect　of　TMDs　with　complex　stiffness　and　damping　is　less　than　that　under　white　noise　excitation,　the　vibration　reduction　effect　of　the　such　kind　structures　is　less　than　that　of　structures　with　viscous　damping,　but　the　difference　is　slight;　so,　if　TMDs　are　applied　in　vibration　control,　adopting　complex　stiffness　and　damping　is　a　profitable　option.　©,　2015,　Chinese　Vibration　Engineering　Society.　All　right　reserved.