The SEREP transformation is developed with analytical mode shapes for the structure (but can also be evaluated using measured modal vectors as well as done in Chipman’s work [2]). 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) [9] and the Time Response Assurance Criterion (TRAC) [10] will be used to verify the accuracy of the expanded RTO in each case. These functions are summarized here with details found in [2]. 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) Similar to the MAC, the TRAC is a tool used to determine the degree of correlation between two time traces. For the cases presented here, the TRAC is the correlation for one DOF over all time for the expanded time data [ERTOn] compared to the actual measured data [REFn]. The TRAC is written as ^ ` ^ ` > @ ^ ` ^ ` > @^ ` ^ ` > @ (t) REF (t) REF (t) ERTO (t) ERTO (t) REF (t) ERTO TRAC_RTO n T n n T n 2 n T n (7) The values produced by both the MAC and TRAC will range from 0 to 1; values approaching 1 indicate good correlation. A schematic is shown in Figure 3 showing the MAC and the TRAC correlations. Typically, the TRAC is comparison of the displacement - time response solutions (predicted and reference solutions) while the MAC is comparison between the deformed shape of the structure at each time instant as shown in Figure 3. 190
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