> @ > @ » ¼ º « ¬ ª , » ¼ º « ¬ ª , oo oa aa oo oa ao aa ao oo aa aa M M K K M M K K M 1 1 . (4) The reduced (approximate) applied force vector is ^ ` ^ ` ^ `o ao oo a aF F K K F ] [ 1 (5) It should be noted that the reduction transformation matrix columns (in equation 2) are physically consistent with normalized deflection shapes associated with application of individual unit “analysis” set loads. This central idea in Guyan Reduction is the primary motivating principal employed in development of the “new” method to be discussed later in this paper. 3.2 CRAIG-BAMPTON MODELS The Craig-Bampton method, while one of the earliest systematic approaches for component mode synthesis (CMS), continues to be the most widely applied CMS strategy. The matrix equations for of a single Craig-Bampton component are a logical extension to Guyan Reduction. The degrees of freedom describing a component dynamic system are first separated into “boundary-analysis” and “interior-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 . (6) By augmenting the Guyan Reduction transformation (equation 2) with a truncated set of “boundary-fixed” modes, the CraigBampton reduction transformation is defined as, ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª ) , ¿ ¾ ½ ¯ ® n a on oo oa an aa o a q U U K K U 1 0 . (7) Application of the reduction transformation, in a symmetric manner following the Rayleigh-Ritz method yields, respectively, the Craig-Bampton component reduced stiffness and mass matrices, > @ » ¼ º « ¬ ª nn na an aa CB K K O 0 0 , > @ » ¼ º « ¬ ª , nn na an aa CB P M P M , (8) and the reduced (approximate) applied force vector, ^ ` ^ ` > @^ ` ¿ ¾ ½ ¯ ® ) ¿ ¾ ½ ¯ ® o T on o ao oo a n a F F K K F Q F ] [ 1 . (9) 3.3 REDUCED MODEL “ENHANCEMENTS” One may attempt to define an enhancement of the Guyan Reduction method based on a frequency domain solution of the Craig-Bampton component equations, ¿ ¾ ½ ¯ ® ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª , n a n a nn na an aa n a nn na an aa Q F q U K q U P M P O 0 0 , (10) 361
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