4 Normalization of Experimental Modal Vectors to Remove Modal Vector Contamination 31 Mode Frequency, Hz Mode Frequency, Hz Auto−MAC of Reference File 0.00 0.00 0.15 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.06 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.66 0.00 0.04 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.81 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.90 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.92 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.93 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.94 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.15 0.00 0.99 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.05 0.00 0.97 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.06 0.00 1.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.05 0.00 0.02 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.04 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.15 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.06 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.09 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.09 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.03 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.27 0.00 0.05 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.27 1.00 0.00 0.00 0.00 0.15 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 362.56 363.87 557.05 761.23 764.16 1222.98 1224.05 1328.03 1328.82 2019.30 2023.81 2322.02 2322.48 2338.06 −2338.06 −2322.48 −2322.02 −2023.81 −2019.30 −1328.82 −1328.03 −1224.05 −1222.98 −764.16 −761.23 −557.05 −363.87 −362.56 362.56 363.87 557.05 761.23 764.16 1222.98 1224.05 1328.03 1328.82 2019.30 2023.81 2322.02 2322.48 2338.06 MAC Value 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 4.1 MAC of modal vectors and conjugate modal vectors where: Apqr DLpr qr OR Apqr D pr qr MAr (4.5) In the above equations, and particularly in Eq. 4.5, it should be noted that the residue should be purely imaginary for a normal mode case utilizing displacement over force FRF data. For the anticipated normal mode situation, there is no constraint on the numerical characteristics of either the modal participation coefficient or the modal vector coefficient individually as long as the product of these two terms yields the correct residue characteristic. 4.2.2 Modal Assurance Criterion The traditional modal assurance criterion (MAC) computation [2], restated in Eq. 4.1, is widely used in modal parameter estimation and structural dynamics [3–6] to sort the numerous possible solutions of modal vectors from either modeling or experiment. MACcd D ˇ ˇ ˇf cg Hf dg ˇ ˇ ˇ 2 f cg Hf cgf dg Hf dg D f cg Hf dgf dg Hf cg f cg Hf cgf dg Hf dg (4.6) Once modal vectors are estimated in any modal parameter estimation procedure, the MAC computation is often utilized to evaluate the quality of the solutions. This begins with an evaluation of the MAC between all of the modal vectors in the final set to ascertain whether the modal set is an independent set of vectors. This often involves the modal vectors associated with the conjugate poles. Since the conjugate poles and vectors are estimated separately, if non-conjugate relationships exist between the associated modal vector estimates (between the modal vector for a pole and the modal vector of the conjugate pole), the MAC between these two vectors will not be unity as expected. A number of users have noted that this often correlates with modal vectors that are exhibiting some unexpected characteristics or contamination. Figure 4.1 is a graphical representation of this situation. While the MAC values are acceptable, the comparisons between modal vectors and the associated conjugate modal vectors do exhibit slightly lowered consistency or correlation.
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