4.3 COMPONENT MODE SYNTHESIS ENHANCEMENT It should be noted that the above developed formulation may be adapted for component mode synthesis applications, wherein free vibration of a Craig-Bampton[8] component described by the “boundary” and “interior” displacement partitions is ¿ 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 U U K K K K U U M M M M 0 0 (31) And the modal reduction transformation is ¿ ¾ ½ ¯ ® » ¼ º « ¬ ª ) , ¿ ¾ ½ ¯ ® n a on oo oa an aa o a q U U K K U 1 0 (32) The resulting reduced order Craig-Bampton, free-vibration equation set is of the form ¿ 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 U K q U P M P 0 0 0 0 O . (33) By including all physical degrees of freedom associated with structural alteration in the “boundary” set, multi-parameter sensitivity matrices associated with the “boundary” set may be defined following the formulation provided in sections 4.1 and 4.2, where > @ > @ > @ ¦ ' N i aa i i aa O aa p K K K 1 , , , > @ > @ > @ ¦ ' N i aa i i aaO aa M M p M 1 , , (34) 5. ILLUSTRATIVE EXAMPLE AND RECENT APPLICATIONS A pinned-free Euler-Bernoulli beam finite element model serves as an example to demonstrate the accuracies of the various Rayleigh-Ritz approximations for parametric sensitivity analysis. In addition, two experiences in which the new method was successfully applied are summarized. 5.1 PINNED-FREE BEAM The pinned-free Euler-Bernoulli beam, illustrated in Figure 1 is characterized by uniform flexural stiffness (EI), uniform mass per unit length (UA), length (L), and angular stiffness, (KR) at the pinned end. A finite element model composed of 100 elements is defined, and the baseline structure (with KR = 0) is described by the 10 lowest eigenvalues and mode shapes. Figure 1: Pinned-Free Euler-Bernoulli Beam Results of exact and approximate sensitivity analyses, summarized in Table 1 indicate that eigenvalue and mode shape accuracies for both the classical residual and new method (which generated two residual vectors) are superior to results of the truncated (10 mode) Rayleigh-Ritz method. 380
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