Based　on　the　harmonic　balance　method,　a　solution　procedure　of　residue　harmonic　was　developed　for　studying　the　system　in　which　the　motion　of　a　particle　is　a　rotating　parabola.　Firstly,　the　higher-order　analytical　vibration　frequency　and　steady　state　response　were　obtained.　Secondly,　we　study　the　change　trend　of　vibration　frequency　as　nonlinear　coefficient,　initial　amplitude,　linear　stiffness　coefficient　of　the　system　in　steady　state,　the　effects　of　initial　amplitude,　nonlinear　coefficient　on　vibration　frequency　response　were　presented.　The　results　show　that　the　presented　second-order　residue　harmonic　balance　approximations　to　vibration　frequency　and　steady　response　are　more　accurate　than　some　existing　results,　and　that　the　relative　error　of　the　solution　is　greatly　reduced,　which　are　in　good　agreement　with　the　exact　ones.　The　vibration　frequency　is　decreased　with　the　increase　of　the　nonlinear　coefficient　and　the　initial　amplitude,　but　it　increased　with　the　increase　of　the　linear　stiffness　coefficient.　©　2018,　Editorial　Office　of　Journal　of　Vibration　and　Shock.　All　right　reserved.