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Unformatted text preview: Modal expansion (decomposition) [ ] [ ] x e e e x e I x x x x x x x x x x n i n n n i n i i n n n i = = = = + + + = = = = 1 1 1 1 2 1 1 1 1 1 1 1 1 , , 1 : basis a form , vectors unit the D, In = = = = = = = j i j i x x x x ij i n j n j j ij j j T i T i i i if , if : ity orthogonal use we formally, of t coefficien the find To 1 1 1 e e x e e could x u u u x u x u M m m m M M U T r n s n s r r r r s rs r s s T r T r r r = = = = = = 1 1 1 so : ity orthogonal mass of property s ' use can we , in of t coefficien the find To ) i.e., , and , " normalized mass " is (If or , : orthogonal mass are of Columns . 1 . = = = = = = r rs s T r T rs r s T r T m M I MU U U m M diagonal M MU U U u u u u ) so becomes this , normalized mass is (If x u u u x u M M M U T r n s n s r r s rs s s T r T r = = = = = = 1 1 . 1 = = = n r r r U U u x x as d represente be can vector any so D, n in basis a form columns its so r, nonsingula is matrix modal the system, DOF n an For [ ] [ ] x x M U M M MU U M U n r T T T r 1 , , 1 = = = = : once at , ts coefficien modal the of all find To [ ] ) : simpler this makes ion normalizat (Mass x x M U I MU U M U T T T = = = [ ] [ ] ) if ( so , : Velocities I M M U M U M M MU U M U t U t T T T T = = = = = = x x x x 1 ) ( ) ( x u x u x x M m M m U U U U T r r r T r r r r = = = = 1 , 1 1 1 : s ' want you which for columns the just of all know to have even t don' invert to have t don' : , over Advantages Modal response to ICs (revisited) + = + = = = = sin cos ) ( ) ( ) ( ) ( 1 r r r r r r r r r n r r r t b t a t b a t t t U t if if where response, free In u x = = = = = = = = = =...
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This note was uploaded on 09/18/2011 for the course ASE 365 taught by Professor Staff during the Spring '10 term at University of Texas at Austin.
 Spring '10
 Staff

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