The　exact　solutions　of　the　nonlinear　vibration　systems　are　extremely　complicated　to　be　received,　so　it　is　crucial　to　analyze　their　approximate　solutions.　This　paper　employs　the　spreading　residue　harmonic　balance　method　(　SRHBM)　to　derive　analytical　approximate　solutions　for　the　fifth-order　nonlinear　problem,　which　corresponds　to　the　strongly　nonlinear　vibration　of　an　elastically　restrained　beam　with　a　lumped　mass.　When　the　SRHBM　is　used,　the　residual　terms　are　added　to　improve　the　accuracy　of　approximate　solutions.　Illustrative　examples　are　provided　along　with　verifying　the　accuracy　of　the　present　method　and　are　compared　with　the　HAM　solutions,　the　EBM　solutions,　and　exact　solutions　in　tables.　At　the　same　time,　the　phase　diagrams　and　time　history　curves　are　drawn　by　the　mathematical　software.　Through　analysis　and　discussion,　the　results　obtained　here　demonstrate　that　the　SRHBM　is　an　effective　and　robust　technique　for　nonlinear　dynamical　systems.　In　addition,　the　SRHBM　can　be　widely　applied　to　a　variety　of　nonlinear　dynamic　systems.