2. NOMENCLATURE Matrices and Vectors: ^ `F Force Vector (Eq. 1) > @ <T Reduction Transformation (Eq. 16) > @, Identity Matrix (Eq. 2) ^ `U Displacement vector (Eq. 1) > @K Stiffness Matrix (Eq. 1) ^ `q Modal displacement vector (Eq. 7) > @ M Mass Matrix (Eq. 1) > @) Modal matrix (Eq. 7) > @P Modal Participation Matrix (Eq. 10) > @< Load Patch Transformation (Eq. 13) ^ `Q Generalized Force Vector (Eq. 9) > @O Eigenvalue Matrix (Eq. 8) Variables: )E Modal Coherence Metric (Eq. 18) Z frequency (radians/sec) (Eq. 11) Subscripts: A Approximate (Eq. 18) n Normal Mode (Eq. 10) CB Craig-Bampton (Eq. 8) o Omitted (Eq. 1) E Exact (Eq. 18) < Load Patch (Eq. 13) a Analysis (Eq. 1) 3. COVENTIONAL ORDER REDUCTION STRATEGIES While many authors have developed and refined dynamic finite element model reduction procedures since the late 1960s, two related strategies retain the broadest acceptance. They are (a) Guyan Reduction[4] and (b) the Craig-Bampton Method[5]. 3.1 GUYAN REDUCTION The underlying idea that defines Guyan Reduction is static condensation…and Guyan’s monumental formulation was published as a one-half page technical note! The degrees of freedom describing a structural dynamic system are first separated into “analysis” and “omitted” subsets, which lead to the partitioned matrix equations (ignoring damping) ¿ ¾ ½ ¯ ® ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª o a o a oo oa ao aa o a oo oa ao aa F F U U K K K K U U M M M M (1) If only the “analysis” partition of the mass matrix were non-zero, and external forces were only applied to “analysis” degrees of freedom, the relationship between “analysis” and “omitted” degrees of freedom would be ^ `a oo oa aa o a U U K K U » ¼ º « ¬ ª , ¿ ¾ ½ ¯ ® 1 (2) In that situation, the reduction transformation in equation 2 would be exact. However, when the “omit” partitions are nonzero, the reduction transformation is approximate (its columns are Rayleigh-Ritz shape functions). Application of the reduction transformation, in a symmetric manner following the Rayleigh-Ritz method yields, respectively, the (statically exact) reduced stiffness matrix and (dynamically approximate) mass matrix > @ > @ > @ > @ ao oo oa aa oo oa aa oo oa ao aa ao oo aa aa K K K K K K K K K K K K K 1 1 1 » ¼ º « ¬ ª , » ¼ º « ¬ ª , (3) 360
RkJQdWJsaXNoZXIy MTMzNzEzMQ==