32 A.W. Phillips and R.J. Allemang Unlike the historical approach to estimation of the modal vectors, many recent modal parameter estimation algorithms, including the autonomous procedures, are based upon numerical processing methods like singular value decomposition (SVD). The solutions that are identified, based upon the data associated with a cluster of estimates of the same modal vector, have no direct physical or causal constraint. An example of a physical or causal constraint would be the expectation of realvalued, normal modes for systems where no expectation of non-proportional damping is likely. SVD methods will identify the most dominant unitary (orthogonal and unit length) vectors in a cluster, yielding a complex-valued vector in general. Experience has shown that when modes are very close in frequency with minimal spatial resolution, the complex-valued vectors will still show significant independence. However, when these complex-valued vectors are examined closely, the non-dominant portion of the complex-valued vector often correlates very highly with one or more nearby modal vectors. This can be examined by several variants of the MAC calculation and the weighted MAC calculation. This is discussed in the next section and is the subject of a companion paper associated with this work [7]. 4.2.2.1 Special Forms of the Modal Assurance Criterion To understand the nature of the possible modal vector contamination in a complex-valued modal vector, three conventional MAC calculations can be performed (1) between the real parts of the modal vectors and the complex-valued modal vectors (rMAC), (2) between the imaginary parts of the modal vectors and the complex-valued modal vectors (iMAC) and (3) between the real parts of the modal vectors and the imaginary parts of the modal vectors (riMAC). These three MAC calculations and the interpretation of these MAC values will be sensitive to the rotation and normalization of the complexvalued modal vector estimates. The following use and discussion assumes that the complex-valued modal vectors have been rotated so that the central axis of the complex-valued modal vector is centered on the real axis. These three MAC computations identify (1) that the real part of the modal vector is the dominant part of the complex-valued modal vector (rMAC), (2) that the imaginary part of the modal vector is the dominant part of the complex-valued modal vector (iMAC) and (3) that the real and imaginary parts of the modal vector are, or are not, related to one another. All MAC computations in this case are, as always, bounded from zero to one. If near normal modes are expected, (1) the rMAC should be close to one, (2) the iMAC should be close to zero and (3) the riMAC should also be close to zero. Note in the following definitions, complex-valued modal vectors c and d can again be any of the modal vectors that the user wishes to include in the evaluation. rMACcd D Ref cg H f dg f dg H Ref cg Ref cg H .Ref cg/ f dg H n d o (4.7) iMACcd D Imf cg H f dg f dg H Imf cg Imf cg H .Imf cg/ f dg H n d o (4.8) riMACcd D Ref cg H .Im f dg/ .Imf dg/ H Ref cg Ref cg H .Ref cg/ Imf dg H Imf dg (4.9) The above MAC evaluations identify whether, and how, the contamination of a complex-valued modal vector is related to another of the identified modal vectors. However, the MAC computation is normalized to unity vector length, vector by vector, for the vectors used in the calculation. A weighted MAC can be used to determine the degree or scale of the contamination. The following three definitions of the weighting for each of the above MAC calculations limits the associated MAC value to a fraction of the zero to one scale. If near normal modes are expected, (1) the weighting and rwMAC should be close to one, (2) the weighting and iwMAC should be close to zero and (3) the combined weighting and riwMAC should also be close to zero. Note in the following definitions, complex-valued modal vectors c and d can again be any of the modal vectors that the user wishes to include in the evaluation. rwMACcd DrWc rMACcd where rWc D Ref cg H / .Ref cg f cg H f cg (4.10)
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