MODEL REDUCTION Model reduction is necessary in order to develop expansion approaches for modal data for the unmeasured translational DOF as well as for rotational DOF. For this work, the expansion is needed for augmenting the limited set of real-time operating data to provide a full field displacement solution. The reduction techniques are the basis of the expansion discussed in this work. These techniques have been presented in earlier work cited in the references; only summarizing equations are presented below. Several model reduction methods have commonly been used for expansion of measured data. Four common methods are Guyan [22], Dynamic Condensation [23], SEREP [24], and a Hybrid method [25]. In these methods, the relationship between the full set of degrees of freedom and a reduced set of degrees of freedom can be written as n {X }=[T]{X }a @ @ @ (1) All of these methods require the formation of a transformation matrix that can project the full mass and stiffness matrices to a smaller size. The reduced matrices can be formulated as > @ > @ > @> @ M T M T n T a (2) > @ > @ > @> K T K T n T a (3) For the specific work in this paper, only the SEREP method has been used for the expansion of mode shapes. The System Equivalent Reduction Expansion Process (SEREP) produces reduced matrices for mass and stiffness that yield the exact frequencies and mode shapes as those obtained from the eigensolution of the full size matrix. The SEREP transformation is formed as > @ > @> @g a n UT U U (4) The SEREP transformation is developed with analytical mode shapes of the structure. Equation 1 is used for expansion of real-time operating data and is written as > @ > @> a n T RTO ERTO (5) CORRELATION TOOLS The correlation tools utilized to compare the results of the expanded RTO [ERTOn] and "reference time solution" [REFn] are briefly discussed here in order to help clarify the differences between the techniques. Correlation tools such as, the Modal Assurance Criterion (MAC) [26] and the Time Response Assurance Criterion (TRAC) [27] will be used to verify the accuracy of the expanded RTO in each case. The MAC as used for this work will identify the correlation of the expanded real time operating displacement solution obtained with the reference solution. The MAC can be computed at each time step t to compare the transient displacement solution with time. The MAC is written as ^ ` ^ ` > @ ^ ` ^ ` > @^ ` ^ ` > n T n n T n 2 n T n ERTO REF REF ERTO REF ERTO MAC_RTO (6) 204
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