44 Developments in the Prediction of Full Field Dynamics in the Nonlinear Forced Response of Reduced Order System Models 489 a Expand dynamic response ] T[ 2 ] T[ 1 Reduced Model Full Model b Extract dynamic response at all DOF Full Model 0 0.01 0.02 0.03 0.04 0.05 -0.06 -0.04 -0.02 0 0.02 0.04 Time (s) Displacement (in) Fig. 44.6 Prediction of subcomponent dynamics from expansion of reduced order model time response using uncoupled subsystem component transformation matrices The full field dynamics at all DOF in the model are obtained from the expansion process and the response at the subcomponent level obtained (as in Fig. 44.6b) despite the fact that no ADOF where placed there during the reduction process. 44.2.6 Timer Response Correlation Tools In order to quantitatively compare two different time solutions, two correlation tools were employed: The Modal Assurance Criterion (MAC) and the Time Response Assurance Criterion (TRAC). 44.2.6.1 Modal Assurance Criteria (MAC) The Modal Assurance Criterion (MAC) [33] is widely used as a vector correlation tool. In this work, the MAC was used to correlate all DOF at a single instance in time. The MAC is written as MACij D hfX1ig T ˚X2j i 2 hfX1ig T fX1igi h ˚X2j T ˚ X2j i (44.20) where X1 and X2 are displacement vectors. MAC values close to 1.0 indicate strong similarity between vectors, whereas values close to 0.0 indicate minimal or no similarity. 44.2.6.2 Time Response Assurance Criteria (TRAC) The Time Response Assurance Criterion (TRAC) [34] quantifies the similarity between a single DOF across all instances in time. The TRAC is written as
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