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Unformatted text preview: wing formula: Ordinary Coherence Function COH pq = γ pq
2 GXF pq 2 = GFF qq GXX pp (5.42) where: GXX pp = Auto power spectrum of the output p GFF qq = Auto power spectrum of the input q GXF pq = Cross power spectrum between output p and input q • • • Partial coherence is deﬁned as the ordinary coherence between a conditioned output and another conditioned output, between a conditioned input and another conditioned input, or between a conditioned input and a conditioned output. The output and input are conditioned by removing contributions from other input(s). The removal of the effects of the other input(s) is formulated on a linear least squares basis. The order of removal of the inputs during "conditioning" has a deﬁnite effect upon the partial coherence if some of the input(s) are mutually correlated. There will be a partial coherence function for every input/output, input/input and output/output combination for all permutations of conditioning. The usefulness of partial coherence, especially between inputs, for experimental modal analysis is of limited value. Multiple coherence is deﬁned as the correlation coefﬁcient describing the linear relationship between an output and all known inputs. There is a multiple coherence function for every output. Multiple coherence can be used to evaluate the importance of unknown contributions to each output. These unknown contributions can be measurement noise, nonlinearities, or unknown inputs. Particularly, as in the evaluation of ordinary coherence, a low value of multiple (524) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + coherence near a resonance will often mean that the "leakage" error is present in the frequency response function. Unlike the ordinary coherence function, a low value of multiple coherence is not expected at antiresonances. The antiresonances for different response locations occur at the same frequency. Though one response signal may have a poor signaltonoise ratio at its antiresonance, other inputs will not at the same frequency. The formulation of the equations for the multiple coherence functions can be simpliﬁed from the normal computational approach to the following equation. Multiple Coherence Function MCOH p = q=1 t=1 Σ Σ Ni Ni H pq GFF qt H * pt GXX pp (5.43) where: H pq = Frequency response function for output p and input q H pt = Frequency response function for output p and input t GFF qt = Cross power spectrum between output q and output t GXX pp = Auto power spectrum of output p • • • • If the multiple coherence of the p − th output is near unity, then the p − th output is well predicted from the set of inputs using the least squares frequency response functions. Example: H 1 Technique: Two Inputs/One Output Case
To begin to understand the size of the problem involved, start with the two input, one output case. ˆ X p − η p = H p1 F 1 + H p2 F 2
(5.44) (525) +UCSDRLRJA CN20263663/664 Revision: June...
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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
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