Vibration　energy　harvester　(VEH)　has　proven　to　be　a　favorable　potential　device　to　supply　continuous　energy　for　electric　components.　Researchers　have　done　a　lot　of　investigations　on　VEH　in　recent　years.　However,　there　are　few　studies　on　delayed　VEH　with　viscoelastic　property　under　random　excitation.　A　new　viscoelastic　energy　harvester　with　time　delay　subjected　to　narrow-band　random　excitation　is　investigated　in　this　paper.　Firstly,　we　utilize　the　multiple　scales　method　to　obtain　the　analytical　approximations　of　the　steady-state　amplitude　close　to　the　primary　resonance.　Secondly,　Monte　Carlo　simulations　are　applied　to　verify　the　correctness　of　the　proposed　method.　Results　demonstrate　the　satisfactory　accuracy　of　the　method　presented　in　this　paper.　Then,　the　phenomena　of　stochastic　jump,　stochastic　bifurcation,　and　the　extension　of　the　effective　bandwidth　of　VEH　can　be　found.　Finally,　the　influence　of　system　parameters　on　the　mean　square　voltage　and　mean　output　power　are　discussed.　It　is　shown　that　a　negative　feedback　gain　is　profitable　to　the　enhancement　of　the　performance　of　VEH,　and　we　find　that　the　piezoelectric　coupling　term　has　a　positive　effect　on　the　mean　square　voltage　and　mean　output　power.　The　mean　square　voltage　and　mean　output　power　can　reach　the　maximum　at　an　appropriate　time　delay　and　feedback　gain.　These　results　indicate　that　the　narrow-band　random　excitation　has　an　important　effect　on　the　dynamics　of　the　delayed　vibration　energy　harvester　with　viscoelastic　property,　namely　a　large　dose　of　noise　intensity　can　decline　the　responses　in　the　energy　harvester.　©　2022　Elsevier　Ltd