4. MODAL CHARACTERIZATION METRICS The fundamental set of matrix equations describing forced response of a linear structural dynamic FEM (with structural damping) are > @^ ` > @^ ` ^ ` ( ) ) 1 i K u F t M u e K . (1) For the case of undamped free vibration, the orthonormal mode transformation and properties with respect to system mass and stiffness are ^ ` > @^ `q u ) , > @ > @> @ > @ ) ) , M T , > @ > @> @ > @O ) ) K T . (2) 4.1 MODAL PARTICIPATION FACTORS AND MODAL EFFECTIVE MASS (CRAIG-BAMPTON MODEL) In the case of a supported (e.g., base-fixed) system, the displacement degrees of freedom are partitioned into interior (or free) and boundary (or fixed) degree of freedom subsets, as shown below: ¿ ¾ ½ ¯ ® ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª ( ) ( ) ) (1 F t F t u u K K K K i u u M M M M b i b i bb bi ib ii b i bb bi ib ii K (3) The Craig-Bampton[3] modal transformation describes the interior degrees of freedom in terms of boundary fixed modes and “constraint modes” associated with unit boundary displacements. ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª , ) < ¿ ¾ ½ ¯ ® » » ¼ º « « ¬ ª , ) ¿ ¾ ½ ¯ ® b bb bq ib iq b bb bq ii ib iq b i u q u K K q u u 0 0 1 (4) It should be noted that when the boundary is statically determinate, the “constraint modes” are rigid body vectors, referenced at the boundary. When the above transformation is applied, the resulting Craig-Bampton component dynamic equations are ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª < , ) ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª, ( ) ( ) 0 0 0 ) (1 2 F t F t u q K i u q P M P b i bb T ib qb T iq b bb bi ib q b bb bq qb qq Z K , (5) The boundary mass and stiffness matrix partitions reduce to (a) the 6X6 rigid body mass matrix and (b) a null 6X6 boundary stiffness matrix, respectively, if the boundary is statically determinate. It is of interest to consider the response of such a system to simple harmonic boundary accelerations. The modal accelerations, in this situation are: > @^ ` ( ) ( ) ( ) q f h f P u f b nb n n , 2 2 ( / ) 1 ( / ) ( ) n n n n f f i f f h f K (6) And the boundary reaction loads, for a statically determinate boundary, are: ^ ` > @ ^ ` ( ) ( ) ( ) 1 P P h f u f F f M b N n bn nb n bb b » ¼ º « ¬ ª ¦ (7) The modal participation products are called modal effective mass matrices (one per mode), which when summed are approximately the total system rigid body mass (equal only if the boundary is massless). Modal effective mass is a modal metric that indicates direction of modal activity as well as degree with which the boundary reacts to modal response. > @ > @> @ bn nb n P P Meff , > @ > @ bb N n n Meff M | ¦ 1 (8) 370
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