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Unformatted text preview: function(s) satisfy the following single and multiple input relationships: Single Input Relationship X p = H pq F q
(5.1) (54) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + Multiple Input Relationship X1 X 2 . . X p N H 11 H 21 . . H p1 . . . . . . . H 1q . . . H pq N F1 F 2 . . Fq N =
×1 (5.2) . . . . . . . o o × Ni i ×1 An example of a two input, two output case for Equation (5.2) is shown in Equation (5.3) and Figure 51. X1 = X2 H 11 H 21 H 12 F 1 H 22 F 2 (5.3) Figure 51. Two Input, Two Output FRF Concept 5.2.1 Noise/Error Minimization
The most reasonable, and most common, approach to the estimation of frequency response functions is by use of least squares (LS) or total least squares (TLS) techniques [13,67] . This is a standard technique for estimating parameters in the presence of noise. Least squares methods minimize the square of the magnitude error and, thus, compute the best estimate of the (55) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + magnitude of the frequency response function but have little effect on the phase of the frequency response function. The primary difference in the algorithms used to estimate frequency response functions is in the assumption of where the noise enters the measurement problem. The different assumptions of the source of the error is noted graphically in Figure 52. X . . . . . . . . . . . . .
ex ev ef F Figure 52. Least Squares Concept Three algorithms, referred to as the H 1 , H 2 , and H v algorithms, are commonly available for estimating frequency response functions. Table 52 summarizes this characteristic for the three methods that are widely used. (56) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + Frequency Response Function Models Technique Solution Method H1 H2 Hv LS LS TLS Assumed Location of Noise Force Inputs no noise noise noise Response noise no noise noise TABLE 52. Summary of Frequency Response Function Estimation Models Consider the case of N i inputs and N o outputs measured during a modal test. Based upon the assumed location of the noise entering the estimation process, Eqs. (5.4) through (5.6) represent the corresponding model for the H 1 , H 2 , and H v estimation procedures. H 1 Technique [H] N o × Ni {F } Ni × 1 = {X } N o × 1 − {η } N o × 1
(5.4) H 2 Technique [H] N o × Ni { {F} Ni × 1 − {υ } Ni × 1 } = {X } N o × 1
(5.5) H v Technique [H] N o × Ni { {F} Ni × 1 − {υ } Ni × 1 } = {X } N o × 1 − {η } N o × 1
(5.6) Note that in all methods, the inversion of a matrix is involved. Therefore, the inputs (references) that are used must not be fully correlated so that the inverse will exist. Extensive evaluation tools (using eigenvalue decomposition) have been developed in order to detect and avoid this condition [8] . (57) +UCSDRLRJA CN20263663/664 Revision: June 12, 2001 + 5.2.2 Single Input FRF Estimation
Figure 5.3 represents the model of the measurement situation for a single input, single output frequency...
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 Fall '11
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