The　synchronous　in-unison　motions　in　vibrational　mechanics　and　the　non-synchronous　out-of-unison　motions　are　the　most　frequently　found　periodic　motions　in　every　fields　of　science　and　everywhere　in　the　universe.　In　contrast　to　the　in-unison　normal　modes,　the　out-of-unison　complex　modes　feature　a　RV　phase　difference.　By　the　complex　mode　analysis　we　classify　the　out-of-unison　planar　motion　into　two　types,　gyroscopic　motions　and　elliptic　motions.　It　is　found　that　the　gyroscopic　and　elliptic　motions　have　different　characteristics　for　a　two　degree-of-freedom　(2DOF)　system.　The　gyroscopic　motion　involves　two　distinct　frequencies　with,　respectively,　two　corresponding　complex　modes.　However,　the　elliptic　motion　the　nonlinear　non-gyroscopic　2DOF　system　with　repeated　frequencies　involves　only　single　frequency　with　corresponding　two　complex　modes.　The　study　of　the　differences　and　similarities　of　the　gyroscopic　and　elliptic　modes　sheds　new　light　on　the　in-depth　mechanism　of　the　planar　motions　in　the　universe　and　the　man-made　engineering　systems.　(C)　2016　Elsevier　Ltd.　All　rights　reserved.