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Unformatted text preview: response function measurement. υ ˆ F + F H System X
η + ˆ X Figure 53. System Model: Single Input With reference to Figure 5.3 for a case involving only one input and one output (input location q and response location p), the equation that is used to represent the inputoutput relationship is: ˆ ˆ X p − η p = H pq (F q − υ q ) where:
• • (5.7) ˆ F = F − υ = Actual input ˆ X = X − η = Actual output (58) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + • • • ˆ X = Spectrum of the p − th output, measured ˆ F = Spectrum of the q − th input, measured H = Frequency response function = Spectrum of the noise part of the input = Spectrum of the noise part of the output • υ • η • • X = Spectrum of the p − th output, theoretical F = Spectrum of the q − th input, theoretical If υ = η = 0, the theoretical (expected) frequency response function of the system is estimated. If η ≠ 0 and/or υ ≠ 0, a least squares method is used to estimate a best frequency response function, in the presence of noise. In order to develop an estimation of the frequency response function, a number of averages N avg is used to minimize the random errors (variance). This can be easily accomplished through use of intermediate measurement of the auto and cross power spectrums. The estimate of the auto and cross power spectrums for the model in Figure 5.3 can be deﬁned as follows. Note that each function is a function of frequency. Cross Power Spectra GXF pq =
N avg Σ 1 * X p Fq (5.8) GFX qp = N avg Σ 1 Fq X * p (5.9) (59) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + Auto Power Spectra GFF qq =
N avg Σ 1 * Fq Fq (5.10) GXX pp = N avg Σ 1 X p X* p (5.11) where:
• • F * = Complex conjugate of F(ω ) X * = Complex conjugate of X(ω ) H 1 Algorithm: Minimize Noise on Output (η ) The most common formulation of the frequency response function, often referred to as the H 1 algorithm, tends to minimize the noise on the output. This formulation is shown in Eq. (5.12). H pq = GXF pq GFF qq
(5.12) H 2 Algorithm: Minimize Noise on Input (υ ) Another formulation of the frequency response function, often referred to as the H 2 algorithm, tends to minimize the noise on the input. This formulation is shown in Eq. (5.13). H pq = GXX pp GFX qp
(5.13) In the H 2 formulation, an auto power spectrum is divided by a cross power spectrum. This can be a problem since the cross power spectrum can theoretically be zero at one or more frequencies. In both formulations, the phase information is preserved in the crosspower spectrum term. H v Algorithm: Minimize Noise on Input and Output (η and υ ) (510) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + The solution for H pq using the H v algorithm is found by the eigenvalue decomposition of a matrix of power spectrums. For the single input case, the following matrix involving the auto and cross power spectrums can be deﬁned: GFF qq [GFFX p ] = GXF pq GFX qp GXX pp (5....
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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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