• 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 choosing. • 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. ApplicationExamples Several case histories involving data from two applications are discussed in the following sections. These two applications were chosen as extremes of data cases, representing a real but very easy modal parameter identification situation (circular plate) and a real but very difficult modal parameter identification situation (civil infrastructure bridge test). The circular plate is lightly damped with many repeated roots but can be readily handled by almost any algorithm within the UMPA framework. The bridge is more moderately damped with significant noise on the data (traffic was maintained on part of the bridge while testing proceeded) and cannot be handled by any of the algorithms within the UMPA framework without significant effort and user interaction. The case histories include cases where the base vector of the algorithm is a function of long and short dimension to demonstrate the sensitivity of the solution to having an adequate spatial basis for determining the consistent vector characteristics. For details concerning the meaning and values of the threshold parameters or a more complete explanation of the CSSAMI procedure, please see the companion papers [25-26] . 4.1 Application Example: C-Plate Laboratory Test Data The first example of the CSSAMI procedure is performed on a laboratory test object consisting of a circular plate. This test object is very lightly damped and nearly every peak in the data is associated with a repeated root caused by the symmetry of the test object. This test object has been tested many times and the autonomous UMPA method estimated modal parameters consistent with past analysis. The FRF data in this case has 7 responses and 36 inputs (taken with an impact test method). In this case, a Polyreference Time Domain (PTD) method is used but, for every possible pole estimated over a model order range from 2 to 20, a complete long dimension modal vector was estimated so that sufficient spatial information (base vector 405
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