• Based upon the pole surface density threshold, identify all possible pole densities above some minimum value. This will be a function of the number of possible solutions represented by the consistency diagram. • Sort the remaining solutions into frequency order based upon damped natural frequency ( ωr). • Construct the 10th order, pole weighted vector (state vector) for each solution. • Normalize all pole weighted vectors to unity length with dominant real part. • Calculate the Auto-MAC matrix for all pole weighted vectors. • Retain all Auto-MAC values that have a pole weighted MAC value above a threshold, 0.8 works well for most cases. All values below the threshold are set to 0.0. • Identify vector clusters from this pole weighted MAC diagram that represent the same pole weighted vector. This is done by a singular value decomposition (SVD) of the pole weighted MAC matrix. The number of significant singular values for this MAC matrix represents the number of significant pole clusters in the pole weighted vector matrix and the value of each significant singular value represents the size of the cluster since the vectors are unitary. Note that the singular value is nominally the square of the number of vectors in the cluster and will likely be different, mode by mode. • For each significant singular value, the location of the corresponding pole weighted vectors in the pole weighted vector matrix (index) is found from the associated left singular vector. This is accomplished by multiplying the left singular vector by the square root of the singular value and retaining all positions (indexes) above a threshold (typically 0.9). The positions of the non-zero elements in this vector are the indexes into the pole weighted vector matrix for all vectors belonging to a single cluster. • For each identified pole cluster, perform a singular value decomposition (SVD) on the set of pole weighted vectors. The significant left singular vector is the dominant (average) pole weighted vector. Use the zeroth order portion of this dominant vector to identify the modal vector and the relationship between the zeroth order and the first order portions of the dominant vector to identify the modal frequency and modal damping values. • Estimate appropriate statistics for each mode identified based upon the modes that are grouped in each cluster. • For the modal parameters identified, complete the solution for modal scaling using any MIMO process of your choice. • User interaction with the final set of values can exclude poorly identified modes based upon physical or statisical evaluations. Once the final set of modal parameters, along with their associated statistics, is obtained, quality can be assessed by many methods that are currently available. The most common example is to perform comparisons between the original measurements and measurements synthesized from the modal parameters. Another common example is to look at physical characteristics of the identified parameters such as reasonableness of frequency and damping values, normal mode characteristics in the modal vectors, and appropriate magnitude and phasing in the modal scaling. Other evaluations that may be helpful are mean phase correlation (MPC) on the vectors, an Auto-MAC looking for agreement between the modal vectors from conjugate poles or any other method available. 4. Statistical Evaluation Parameters Statistical evaluation parameters can be estimated for each common cluster of pole weighted modal vectors on the basis of the complex modal frequency, the modal vector, and the modal scaling. The number of pole weighted vectors will in general be different in each cluster so the statistics will be based upon the number of estimates available (sample size N). Examples of the statistics currently computed for each modal parameter are described in the following sections. 4.1 ModalFrequencyStatistics The weighted modal frequency for the cluster is found by constructing the pole weighted vector (typically 10th order) for each pole retained in a cluster, then taking the SVD of the group of pole weighted vectors and selecting the singular vector associated with the largest singular value. This chosen singular vector contains both the shape and the modal frequency information. The modal frequency is identified by dividing the first order portion by the zeroth order portion of the vector in a least squares sense. (Note that it is also possible to solve the frequency polynomial which would result from using the complete vector.) Also, for numerical reasons, the pole weighted vector is actually computed in the Z-domain. 387
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