In　this　paper,　the　nonlinear　free　vibration　of　the　nanotube　with　damping　effects　is　studied.　Based　on　the　nonlocal　elastic　theory　and　Hamilton　principle,　the　governing　equation　of　the　nonlinear　free　vibration　for　the　nanotube　is　obtained.　The　Galerkin　method　is　employed　to　reduce　the　nonlinear　equation　with　the　integral　and　partial　differential　characteristics　into　a　nonlinear　ordinary　differential　equation.　Then　the　relation　is　solved　by　the　multiple　scale　method　and　the　approximate　analytical　solution　is　derived.　The　nonlinear　vibration　behaviors　are　discussed　with　the　effects　of　damping,　elastic　matrix　stiffness,　small　scales　and　initial　displacements.　From　the　results,　it　can　be　observed　that　the　nonlinear　vibration　can　be　reduced　by　the　matrix　damping.　The　elastic　matrix　stiffness　has　significant　influences　on　the　nonlinear　vibration　properties.　The　nonlinear　behaviors　can　be　changed　by　the　small　scale　effects,　especially　for　the　structure　with　large　initial　displacement.　(C)　2014　Elsevier　Ltd.　All　rights　reserved.