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Unformatted text preview: esponse functions in the presence of multiple inputs, special care must be taken to assure that the input spectrum matrix is not singular. Therefore, techniques have been investigated to evaluate the form of the input spectrum matrix before taking any data. Singular, in this case, implies that:
• Input forces may not be coherent at any frequency.
• Independent, uncorrelated noise sources must be used. (Random, Random Transient, Periodic Random) (5-30) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + • The impedance of the structure at the input locations may tend to correlate the inputs at resonance. • There are no zero’s in the input spectrum matrix. Ordinary and Partial Coherence Functions The historical approach that was used to try to evaluate the correlation between the forces utilized ordinary and partial coherence functions. The ordinary coherence function measures the degree of linear dependence (or correlation) between the spectra of two signals. The partial coherence function measures the degree of linear dependence between the spectra of two signals, after eliminating in a least squares sense, the contribution of some other signals. Both functions can be used in systematic procedure to verify that the forces are not correlated or that the input cross spectra matrix [GFF] is not singular. For cases involving more than two inputs, this approach is very difﬁcult and requires considerable judgement. In reality, only the ordinary coherence function, for the case of two inputs, is still used. COH ik = | GFF ik |2 GFF ii GFF kk (5.53) where:
• • • GFF ik = Cross power spectrum between inputs i and k GFF ii = Auto power spectrum of input i GFF kk = Auto power spectrum of input k Principal/Virtual Input Forces (Virtual Forces) The current approach used to determine correlated inputs involves utilizing principal component analysis to determine the number of contributing forces to the [GFF] matrix. In this approach, the matrix that must be evaluated is: (5-31) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + GFF 11 . [GFF] = . . GFF Ni 1 where:
• • •
* GFF ik = GFF ki (Hermitian Matrix) * GFF ik = Σ F i F k . . . . . . GFF 1Ni . . . GFF Ni Ni (5.54) GFF is the power spectrum of a given input. Principal component analysis involves an eigenvalue decomposition of the [GFF] matrix . Since the eigenvectors of such a decomposition are unitary, the eigenvalues should all be of approximately the same size if each of the inputs is contributing. If one of the eigenvalues is much smaller at a particular frequency, one of the inputs is not present or one of the inputs is correlated with the other input(s). [ GFF ] = [ V ] [ Λ ] [ V ] H
(5.55) Since the eigenvectors of such a decomposition are unitary, the eigenvalues should all be of approximately the same size if each of the inputs is contributing. If one of the eigenvalues is much smaller at a particular frequency, one of the inputs is not present or one of the...
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This note was uploaded on 09/29/2013 for the course MECHANICAL ME taught by Professor Regalla during the Fall '11 term at Birla Institute of Technology & Science, Pilani - Hyderabad.
- Fall '11