132 E. J. Perez et. al inverse pole matrix. The results of our analysis built from Equation 13 are shown in figure 11 as our final conclusion for the predictability of these transfer functions. [H] = [Φ][M][Φ]T = Φ11 Φ12 · · · Φ1m Φ21 Φ22 · · · Φ2m .. . .. . . . . .. . Φn1 Φn2 · · · Φnm M1 0 · · · 0 0 M2 · · · 0 .. . .. . . . . .. . 0 0 · · · Mx Φ11 Φ21 · · · Φn1 Φ12 Φ22 · · · Φn2 .. . .. . . . . .. . Φ1m Φ2m · · · Φnm (13) Mx = −ω2 ω2 nx −ω2 +2iζ xωnxω (14) Conclusion One of the key purposes of this paper was to investigate the differences between the transfer paths and transmissibility matrices of MIMO and SIMO systems. Towards this goal there were two main findings. The first was that as one of the forces applied in a MIMO system became an order of magnitude larger than any of the other forces, the transfer path between any two points converged to the path that would be seen in a SIMO case where just the larger force was present. The second key finding was that the MIMO transfer paths could not be constructed as a linear combination of SIMO cases. In the effort to predict FRFs at locations that were never forced, the first method used measured transfer paths and FRFs in a calculation like Equation 12. This method did predict some peaks but was often vastly inaccurate. The error in this method was found to be in the assumption that the transfer path would be the same between the measured and predicted cases. This method necessitates a transfer path based on the frequency information from the unknown forcing case, however obtaining this data would be contrary to the original goal of not forcing at the location. Therefore, this method was unsuitable for experimental only prediction without the integration of some numerical simulation data. The failure of the first method due to the inability to gather the needed transfer paths inspired an attempt to integrate numerical simulation predictions for transfer path into the calculation. The results of this method were seen in Figure 10. This method produced predictions that were improved from the prior method. However it was still erroneous, likely due to FE model inaccuracies used for numerical transfer path prediction or boundary condition differences between the model and the experiment. Another shortcoming of this method is that its requirement for highly accurate modeling makes solving for FRFs using simulation more reasonable than the method proposed. The third attempt at FRF prediction followed roughly the method outlined in Equation 13. This method produced many of the best results as can be seen in Figure 11. Fig. 11 Prediction of FRF from shaker 2 to accelerometer 7 from shaker 1 SIMO data.
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