By　studying　the　transverse　dynamics　of　the　axially　moving　beam,　the　linear　and　nonlinear　complex　modes　of　gyroscopic　continua　are　investigated　based　on　the　invariant　manifold　method.　The　nonlinear　partial　differential　equations　are　truncated　into　a　set　of　ordinary　differential　equations　by　the　Galerkin　method.　The　invariant　manifold　method　is　then　used　to　obtain　the　linear　and　nonlinear　complex　modes.　The　gyroscopic　effect　due　to　the　axially　moving　speed　is　discussed　by　comparing　the　proportions　of　the　trial　functions　in　the　complex　mode　during　the　modal　motions.　The　＇traveling　wave＇　phenomena　are　found　for　such　a　gyroscopic　continuum.　The　power　of　the　invariant　manifold　method　in　the　application　to　gyroscopic　systems　is　demonstrated　and　discussed　by　taking　the　axially　moving　continuum　as　an　example.