A　vast　amount　of　work　can　be　found　in　the　field　of　beam　vibration　dealing　with　analytical　and　numerical　techniques.　The　solution　procedure　of　a　spreading　residue　harmonic　balance　for　approximating　the　periodic　behavior　of　a　tapered　beam　is　presented　in　this　paper.　In　this　solution　procedure,　no　small　parameter　is　assumed,　all　former　residual　errors　are　introduced　in　the　present　approximation　to　improve　the　accuracy.　The　obtained　results　are　compared　with　the　other　existing　solutions　and　the　exact　ones　which　are　shown　in　graphs.　It　is　evident　that　the　method　considered　here　is　quickly　convergent　and　only　the　first-order　analytical　approximation　leads　to　high　accuracy　of　the　solutions.　In　addition,　the　effect　of　system　parameter　and　vibration　amplitude　on　the　nonlinear　frequency　is　evaluated　and　compared,　and　excellent　agreement　between　approximate　frequencies　and　exact　ones.　It　is　predicted　that　the　solution　method　can　be　found　wide　application　in　the　other　nonlinear　oscillation　problems.　(C)　2016　Elsevier　Inc.　All　rights　reserved.