490 S.E. Obando and P. Avitabile MAC 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -0.1 -0.05 0 0.05 0.1 0.15 Time (sec) Displacement (in) Reference Solution 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -0.1 -0.05 0 0.05 0.1 0.15 Time (sec) Displacement (in) Predicted Solution TRAC TRAC TRAC MAC (t2) (t1) (t1) (t2) (t2) (t1) Fig. 44.7 Physical interpretation of MAC and TRAC. MAC is a vector to vector correlation taken at an instance in time ti, while TRAC compares the response for its full duration at each DOF in the model. In the figure, the shape of two models (blue andgrey) are correlated at a particular time step and the TRAC is computed for the displacement vector at one DOF TRACji D h ˚ X1j.t/ T fX2i.t/gi 2 h ˚X1j.t/ T ˚ X1j.t/ i hfX2i.t/g T fX2i.t/gi (44.21) where X1 and X2 are time response vectors for a particular DOF. TRAC values close to 1.0 indicate strong similarity between vectors, while values close to 0.0 indicate minimal or no similarity. The MAC is calculated between the shapes of the full space reference solution and estimated solution obtained from the reduced order model at each time step. Similarly, the TRAC is used to compare the time response from the reduced order model to the time response from the full space finite element solution at each degree of freedom. A diagram detailing the two correlation techniques is shown in Fig. 44.7. 44.2.7 Numerical Computations The computation of the time response developed in this work is based on the Equivalent Reduced Model Technique (ERMT), a technique developed by Avitabile and Thibault [1, 35]. This technique uses an exact reduced model representation for the calculation of the system response. Newmark integration technique [36–39] is used to perform the direct integration of the equations of motion for the ERMT solution process due to similarity with the HHT (Hilber–Hughes–Taylor [38, 39]) method commonly used in FEA software. From the known initial conditions for displacement and velocity, the initial acceleration vector is computed using the equation of motion and the applied forces as
RkJQdWJsaXNoZXIy MTMzNzEzMQ==