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Unformatted text preview: 14) 2×2 The solution for H pq is found by the eigenvalue decomposition of the [GFFX] matrix as follows: [GFFX p ] = [V ] Λ [V ] H (5.15) where:
• Λ = diagonal matrix of eigenvalues Solution for the H pq matrix is found from the eigenvector associated with the smallest (minimum) eigenvalue (λ 1 ). The size of the eigenvalue problem is second order resulting in ﬁnding the roots of a quadratic equation. This eigenvalue solution must be repeated for each frequency and that the complete solution process must be repeated for each response point X p . Alternately, the solution for H pq is found by the eigenvalue decomposition of the following matrix of auto and cross power spectrums: GXX pp [GXFF p ] = GFX qp GXF pq H GFF qq (5.16) 2×2 [GXFF p ] = [V ] Λ [V ] H (5.17) where:
• Λ = diagonal matrix of eigenvalues The solution for H pq is again found from the eigenvector associated with the smallest (minimum) (511) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + eigenvalue (λ 1 ). The frequency response function is found from the normalized eigenvector associated with the smallest eigenvalue. If [GFFX p ] is used, the eigenvector associated with the smallest eigenvalue must be normalized as follows: H {V }λ min = pq −1 (5.18) If [GXFF p ] is used, the eigenvector associated with the smallest eigenvalue must be normalized as follows: −1 {V }λ min = H pq (5.19) One important consideration of the three formulations for frequency response function estimation is the behavior of each formulation in the presence of a bias error such as leakage. In all cases, the estimate differs from the expected value particularly in the region of a resonance (magnitude maxima) or antiresonance (magnitude minima). For example, H 1 tends to underestimate the value at resonance while H 2 tends to overestimate the value at resonance. The H v algorithm gives an answer that is always bounded by the H 1 and H 2 values. The different approaches are based upon minimizing the magnitude of the error but have no effect on the phase characteristics. In addition to the attractiveness of H 1 , H 2 and H v in terms of the minimization of the error, the availability of auto and cross power spectra allows the determination of other important
2 functions. The quantity γ pq is called the scalar or ordinary coherence function and is a frequency dependent, real value between zero and one. The ordinary coherence function indicates the degree of causality in a frequency response function. If the coherence is equal to one at any speciﬁc frequency, the system is said to have perfect causality at that frequency. In other words, the measured response power is caused totally by the measured input power (or by sources which are coherent with the measured input power). A coherence value less than unity at any frequency indicates that the measured response power is greater than that due to the measured input. This is due to some extraneous noise also contributing to the output pow...
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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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