30 A.W. Phillips and R.J. Allemang 4.1 Introduction As modal parameter estimation has evolved over the last 40 years or so, the way that modal vectors are estimated from experimental data has changed, as well. Although the technological advances from single measurement modal parameter estimation to autonomous multiple input, multiple output (MIMO) modal parameter estimation has resulted in improved estimates, the recent developments in autonomous parameter estimation methods has revealed that, while the random errors are reduced, these estimates may still contain small amounts of bias contamination from nearby modes. The results of this contamination are slightly complex estimates of the modal vectors when normal modal vectors are anticipated. The challenge for the analyst is how to deal with this contamination. In testing situations where modal vectors show some contamination, the problem is often ignored or eliminated through a real normalization procedure of the final modal vectors. Frequently, this process is justified because the contamination appears to be dominantly random. However, when the contamination is biased, this justification becomes uncertain. Even with the most sophisticated modal parameter estimation algorithms and numerical procedures, the form of the contamination will often be biased to look like a nearby mode. This indicates that, while the estimated modal vectors may satisfy whatever algorithm and numerical procedures are being utilized, the estimated modal vectors still contain characteristics that can be perceived as non-physical. 4.2 Background The estimation of modal vectors in modern modal parameter estimation algorithms normally involves a two-step process. In the first step, the modal participation vectors, fLgr, are estimated from the eigenvectors of a companion matrix, formulated on the basis of either the short dimension, NS, or the long dimension, NL, of the measured FRF data matrix. Then in a second step, the corresponding modal vectors, f§gr, are found from a weighted least squares set of linear equations involving the selected modal frequencies and modal participations from the eigenvalues and eigenvectors of the companion matrix. In modern algorithms, due to the available speed and memory of modern computers used in testing and data analysis, these two steps are often combined, in what appears to the user, as a single step. The following sections review the relevant theoretical concepts and equations required for discussing the estimation of final, scaled modal vectors. The final scaled modal vectors are often presented as the residues of the partial fraction model of the MIMO FRF data matrix [1]. Alternatively, the final, scaled modal vectors can be presented as a vector proportional to the residue vector with associated modal scaling, such as Modal A.MAr /. 4.2.1 Modal Vectors from Weighted Estimation of Residues The equations that relate the complex modal frequencies, complex modal vectors, complex-valued modal participation vectors and residue vectors to the FRF data are well-known and are restated in the following equations for discussion purposes [1]: Single Reference: Hpq .!/ D N X rD1 Apqr j! r C A pqr j! r (4.1) Multiple Reference: ŒH.!/ D N X rD1 ŒAr j! r C A r j! r D 2N X rD1 ŒAr j! r (4.2) ŒH.!/ NL NS DŒL NL 2N 1 j! r 2N 2N Œ T 2N NS (4.3) ŒH.!/ NS NL DŒL NS 2N 1 j! r 2N 2N Œ T 2N NL (4.4)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==