^ ` > @> @ ^ ` > @^ ` \ \ \ U K K U T U T < < < 1 1 1 . (16) Finally, the load patch based “Guyan Reduction” stiffness and mass matrices are > @ > @ > @> @ > @ 1 1 < < K T K T KT T \ \ \\ , > @ > @ > @> @\ \ \\ M T M T T . (17) It should be noted that, in the limiting case of concentrated, point patch loads, the present formulation reduces to Guyan Reduction. The qualities associated with this alternative reduction strategy are demonstrated with two basic examples in the following sections. 4.2 STRETCHED STRING EXAMPLE Consider a stretched string of length (L=72”) and tension (T=9.85x107xUA), which is modeled by 400 “string” elements. Vibration modes associated with (a) exact system analysis, (b) Guyan Reduction (reduced to four analysis degrees of freedom) and (c) the alternative reduction strategy (reduced using load patches) are summarized in Figures 1 and 2. Guyan Reduction Unit Point Loads New Reduction Unit Load Patches Figure 1: Stretched String Example Load Patterns for Model Order Reduction to Four Degrees of Freedom Exact and Guyan Reduction Modes Exact and New Reduction Modes Figure 2: Comparison of the First Four Stretched String Mode Shapes Computed via Reduction to 4 DOF The limitations associated with Guyan Reduction, for this example, are clearly illustrated by the straight segment mode shapes associated with point unit loads. In contrast, modes associated with distributed unit patch loads more closely follow the exact mode shapes, indicating the advantage of the new reduction strategy. A quantitative comparison of modal frequency and mode shape errors associated with Guyan Reduction and the new reduction strategy is provided in Figure 3. 363
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