A　new　and　efficient　computation　of　the　normal　forms　is　developed　in　this　paper　for　high　dimensional　non-linear　systems,　and　the　computational　method　is　applied　to　non-planar　non-linear　oscillations　of　a　cantilever　beam.　The　method　developed　here　has　the　advantage　of　directly　calculating　the　coefficients　of　the　normal　forms　and　the　associated　near　identity　non-linear　transformations　for　three　different　cases,　that　is,　(1)　the　case　of　two　pairs　of　pure　imaginary　eigenvalues;　(2)　the　case　of　one　non-semisimple　double　zero　and　a　pair　of　pure　imaginary　eigenvalues;　and　(3)　the　case　of　two　non-semisimple　double　zero　eigenvalues.　The　final　partial　differential　equations　of　various　resonant　cases　appear　in　a　canonical　form　whose　solutions　can　be　conveniently　obtained　using　polynomial　equations.　With　the　aid　of　the　Maple　software,　a　symbolic　program　for　computing　the　normal　forms　of　high　dimensional　non-linear　systems　is　given.　Comparing　the　method　developed　here　with　other　methods　of　computing　the　normal　forms,　it　is　understood　that　we　may,　respectively,　obtain　the　normal　forms,　the　coefficients　of　the　normal　forms　and　the　associated　near　identity　non-linear　transformations　for　three　resonant　cases　by　using　a　same　main　Maple　symbolic　program.　Moreover,　the　method　is　easy　to　apply　to　engineering　problems.　The　normal　forms　of　the　averaged　equations　and　their　coefficients　for　non-planar　non-linear　oscillations　of　the　cantilever　beam　under　combined　parametric　and　forcing　excitations　are　calculated　for　two　resonant　cases.　(C)　2003　Elsevier　Ltd.　All　rights　reserved.