380 S. Richardson et al. 35.5 Shape Difference Indicator The Shape Difference Indicator is defined with the formula, SDI D 1 f ug fvg2 fugh fugCfvgh fvg! 2 (35.2) or SDI D0 @ 2 real fug h fvg fugh fugCfvgh fvg 1 A 2 (35.3) real fugh fvg Dthe real part of the vector dot product fugDcomplex comparison shape (m-vector) fvgDcomplex baseline shape (m-vector) mDnumber of matching DOFs between the shapes h–denotes the transposed conjugate vector SDI values also range between 0 and 1. If two shapes have identical shape components, SDID1. If two shapes have different shape components, SDI<1. Several examples illustrate typical SDI values. • If fvgDfug, SDI D1 • If fvgD0 or fugD0, SDID0 • If fvgD2fug, SDID0.64 • If fvgD10fug, SDID0.04 35.6 Applying MAC and SDI to Order-Based ODS’S The following Figs. 35.4, 35.5, 35.6, 35.7, 35.8 and 35.9 are bar charts of the MAC and SDI values between order-based ODS pairs for the baseline and seven unbalance cases. Each bar chart depicts an 8 by 8 matrix of MAC or SDI values. Each diagonal bar of value 1 is the MAC or SDI value of each ODS correlated with itself. Each bar chart also shows the worst case of a MAC or SDI value in the box at the top. Shape #1 is the baseline (balanced) case. Shapes #2 through #8 are the seven unbalance cases in order. A MAC or SDI value less than1 indicates that a pair of ODS’s are different from one another. A bar value close to zero is a clear indication that two ODS’s are different. Note If all of the off-diagonal bars in these bar charts are close to zero, the metric can be used to uniquely identify each unbalance condition. Figure 35.4 shows the MAC values for the first order ODS data. Many of the MAC values for the off-diagonal pairs are close to1. This bar chart shows that MAC values cannot be used on first order ODS data to uniquely identify each unbalance case. Figure 35.5 shows the SDI values for the first-order ODS data. Overall, these values are closer to zero than the MAC values, but the ODS’s for the worst case pair (cases 2 and 7) are still closely correlated (SDID0.93). Hence, SDI values cannot be used on first order ODS data to uniquely identify each unbalance case. Figure 35.6 is the MAC values for the second order ODS’s. This bar chart still shows that the MAC values cannot be used on second order ODS data to uniquely identify each unbalance case. Figure 35.7 is the SDI values for the second order
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