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Unformatted text preview: 12, 2001 + F1 F2 H Xp ηp + ˆ Xp Figure 5-7. Two Input, One Output Model If more than one output is measured, the equations become: F1 * * * * {X p } [F 1 F 2 ] = [H p1 H p2 ] [F 1 F 2 ] F2 (5.45) Therefore, for input locations 1 and 2, each output is used with the two inputs to compute two frequency response functions. Therefore, there will be 2 × N o frequency response functions to be computed. H 11 H 21 H 31 . . H No1 H 12 GXF 11 H 22 GXF 21 H 32 GXF 31 = . . . . H N o 2 GXF N o 1 GXF 12 GXF 22 GXF 32 GFF 11 GFF 21 . . GXF N o 2 GFF 12 GFF 22 −1 (5.46) For each output location, one formulation of the equations to be solved can be developed by replacing the inverse of the [GFF] matrix with the equivalent adjoint of the [GFF] matrix divided by the determinant of the [GFF] matrix. In this way, it is clear that the frequency response functions can be found as long as the determinant of the [GFF] matrix is not zero. (5-26) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + H p1 = GXF p1 GFF 22 − GXF p2 GFF 21 det[GFF] (5.47) H p2 = GXF p2 GFF 11 − GXF p1 GFF 12 det[GFF] (5.48) where: • • det[GFF] = Determinant of [GFF] matrix det[GFF] = GFF 11 GFF 22 − GFF 21 GFF 12 For the two input, one output case several possible coherence functions can be formulated. While the ordinary coherence between the output and each input can be formulated, these coherence functions may not provide useful information due to the possible interaction between the two forces. Ordinary Coherence (Output p and Input 1) COH p1 = |GXF p1 |2 GFF 11 GXX pp (5.49) Ordinary Coherence (Output p and Input 2) COH p2 = |GXF p2 |2 GFF 22 GXX pp (5.50) The ordinary coherence between the two inputs is a useful function since this is a measure of whether the forces are correlated. If the forces are perfectly correlated at a frequency, the inverse of the [GFF] matrix will not exist and the frequency response functions cannot be estimated at that frequency. In this case, the ordinary coherence between the two forces cannot be unity, although values from 0.0 to 0.99 are theoretically acceptable. The limit is determined by the accuracy of the measured data and the numerical precision of the computation. Ordinary Coherence (Input 1 and Input 2) (5-27) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + COH 12 |GFF 12 |2 = GFF 11 GFF 22 (5.51) Multiple coherence is always a good measure of whether the output response is caused by the combination of the measured inputs. Multiple coherence is is used in multiple input situations in the same way that ordinary coherence is used in the single input situations. Multiple Coherence MCOH p = q=1 t=1 Σ Σ 2 2 H pq GFF qt H * pt GXX pp (5.52) (5-28) +UC-SDRL-RJA CN-20-263-663/664 Revision: June 12, 2001 + Summary of Methods H 1 Technique: Underestimates amplitude at resonances. Causes damping to be overestimated. Underestimates amplitude at anti-resonances. • • H 2 Technique: Overestimates amplitude at resonances. Causes dampi...
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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.

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