1 Integrating Multiple Algorithms in Autonomous Modal Parameter Estimation 3 Therefore, the nomenclature of the number of outputs (No) and number of inputs (Ni) has been replaced by the length of the long dimension of the data matrix (NL) and the length of the short dimension (NS) regardless of which dimension refers to the physical output or input. This means that the above reciprocity relationship can be restated as: ŒH.!i/ NL NS DŒH.!i/ T NS NL (1.2) Note that the reciprocity relationships embedded in Eqs. 1.1 and 1.2 are a function of the common degrees of freedom (DOFs) in the short and long dimensions. If there are no common DOFs, there are no reciprocity relationships and the data requirement for modern modal parameter estimation algorithms (multiple references) will not be met. Nevertheless, the importance of Eqs. 1.1 and 1.2 is that the dimensions of the FRF matrix can be transposed as needed to fit the requirement of specific modal parameter estimation algorithms. This impacts the size of the square matrix coefficients in the matrix coefficient, polynomal equation and the length of the associated modal (base) vector. 1.2.2 Autonomous Modal Parameter Estimation The interest in automatic modal parameter estimation methods has been documented in the literature since at least the mid 1960s when the primary modal method was the analog, force appropriation method [1–3]. Following that early work, there has been a continuing interest in autonomous methods that, in most cases, have been procedures that are formulated based upon a specific modal parameter estimation algorithm like the Eigensystem Realization Algorithm (ERA), the Polyreference Time Domain (PTD) algorithm or more recently the Polyreference Least Squares Complex Frequency (PLSCF) algorithm (which thebasis of the commercial version of the PLSCF, the PolyMAX ® method and the rational fraction polynomial algorithm with Z-domain generalized frequency (RFP-z)) [8]. A relatively complete list of autonomous and semi-autonomous procedures that have been reported can be found in a recent paper [4]. Each of these past procedures have shown some promise but have not yet been widely adopted. In many cases, the procedure focused on a single modal parameter estimation algorithm and did not develop a general procedure. Most of the past procedural methods focused on modal frequency (pole) density but depended on limited modal vector data to identify correlated solutions. Currently, due to increased computational speed and availability of memory, procedural methods can be developed that were beyond the computational scope of available hardware only a few years ago. These methods do not require any initial thresholding of the solution sets and rely upon correlation of the vector space of hundreds or thousands of potential solutions as the primary identification tool. The discussion in the following sections of the use of multiple modal parameter estimation algorithms in autonomous modal parameter estimation is based upon recent implementation and experience with an autonomous modal parameter estimation procedure referred to as the Common Statistical Subspace Autonomous Mode Identification (CSSAMI) method. The strategy of the CSSAMI autonomous method is to use a default set of parameters and thresholds to allow for all possible solutions from a given data set. This strategy allows for some poor estimates to be identified as well as the good estimates. The philosophy of this approach is that it is easier for the user to evaluate and eliminate poor estimates compared to trying to find additional solutions. The reader is directed to a series of previous papers in order to get an overview of the methodology and to view application results for several cases [4–7]. Note that much of the background of the CSSAMI method is based upon the Unified Matrix Polynomial Algorithm (UMPA) [8]. This means that this method can be applied to both low and high order methods with short or long dimension modal (base) vectors. This also means that most commercial algorithms could take advantage of this procedure. Note that high order, matrix coefficient polynomials normally have coefficient matrices of a dimension that is based upon the short dimension of the data matrix, NS. In these cases, it may be useful to solve for the complete, unscaled or scaled, modal vector of the large dimension, NL. This will extend the temporal-spatial information in the modal (base) vector so that the vector will be more sensitive to change. This characteristic is what gives the CSSAMI autonomous method a robust ability to distinguish between computational and structural modal parameters. 1.2.3 Pole Weighted Modal Vectors The key to estimating the modal parameters utilizing the CSSAMI autonomous procedure is formulating clusters of pole weighted modal vectors, or state vectors, from the estimates of modal parameters that are represented in a consistency diagram. These state vectors are formed from the modal vector estimates found as the consistency diagram is developed.
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