In　this　paper,　we　construct　a　residue　harmonic　balance　solution　procedure　to　improve　the　accuracy　of　approximate　vibration　period　and　steady　response　for　a　strongly　nonlinear　oscillator　from　the　motion　of　rigid　rod　rocking　back　and　forth　on　a　circular　surface　without　slipping.　In　this　new　solution　procedure,　all　the　unbalanced　residuals　due　to　Fourier　truncation　are　considered　into　next　order　approximation　by　iteration　to　improve　the　accuracy.　The　presented　approximate　analytical　results　are　compared　with　the　other　existing　solutions　such　as　Newton　harmonic　balance　method,　variational　approach　method,　amplitude-frequency　formulation,　He＇s　energy　balance　method　and　the　exact　ones　which　are　shown　in　graphs.　It　is　evident　that　the　method　considered　here　is　quickly　convergent　and　only　the　second　order　approximation　leads　to　high　accuracy　of　the　steady　state　solutions.　In　addition,　four　numerical　examples　are　presented　to　highlight　the　effects　of　system　parameters　on　the　nonlinear　vibration　period,　and　excellent　agreement　between　approximate　periods　and　exact　ones　in　the　steady　state.　It　is　predicted　that　the　solution　method　can　be　found　wide　application　in　the　other　nonlinear　oscillation　problems.　(C)　2017　Faculty　of　Engineering,　Alexandria　University.　Production　and　hosting　by　Elsevier　B.V.