With　the　increasing　performance　requirements　of　vibration　suppression　devices　for　precision　instruments　and　structural　safety,　there　is　an　urgent　need　to　explore　efficient　and　stable　vibration　absorbers.　This　paper　aims　to　investigate　the　response　mechanism　of　a　geometrically　nonlinear　coupled　system　under　harmonic　excitation,　taking　into　account　the　damping　of　the　primary　system,　which　is　often　overlooked　in　research.　The　slow　variation　equation　of　the　system　is　derived　by　the　complexification-averaging　method,　from　which　the　bifurcation　analysis　is　conducted,　and　the　influences　of　the　two　types　of　damping　on　the　bifurcation　are　thoroughly　discussed.　Moreover,　two　motion　triggering　conditions　are　analyzed,　namely,　the　threshold　for　the　transition　from　intra-well　periodic　to　inter-well　chaos,　and　the　threshold　for　the　occurrence　of　strongly　modulated　response,　some　typical　parameters　are　selected　for　numerical　verification.　Furthermore,　the　damping　dissipation　ratio　is　defined　to　evaluate　the　damping　efficiency　of　the　system,　and　the　numerical　results　indicate　that　the　optimal　working　efficiency　can　be　achieved　when　the　system　first　performs　steady-state　motion　across　two　potential　energy　wells.　Finally,　the　energy　spectrum　is　calculated　by　the　Romberg　algorithm　to　verify　the　vibration　reduction　performance　of　the　proposed　model.