4.2. PART2: FREQUENCYRESPONSE FUNCTION ESTIMATES FROM MEASUREDDATA 79
Coherent output autospectra and cumulative coherence function pairs are therefore defined
as follows:
.
G
YY
/
1
D
ˇ
ˇ
H
YZ
1
ˇ
ˇ
2
G
Z
1
Z
1
;
2
Y 1
D
ˇ
ˇ
H
YZ
1
ˇ
ˇ
2
G
Z
1
Z
1
=G
YY
for input x
1
.t/; (4.41)
.
G
YY
/
2
D
ˇ
ˇ
H
YZ
1
ˇ
ˇ
2
G
Z
1
Z
1
C
ˇ
ˇ
H
YZ
2
ˇ
ˇ
2
G
Z
2
Z
2
;
2
Y 2
D
ˇ
ˇ
H
YZ
1
ˇ
ˇ
2
G
Z
1
Z
1
C
ˇ
ˇ
H
YZ
2
ˇ
ˇ
2
G
Z
2
Z
2
=G
YY
for inputs x
1
.t/ C x
2
.t/; (4.42)
.
G
YY
/
K
D
ˇ
ˇ
H
YZ
1
ˇ
ˇ
2
G
Z
1
Z
1
C
ˇ
ˇ
H
YZ
2
ˇ
ˇ
2
G
Z
2
Z
2
C C
ˇ
ˇ
H
YZ
N
ˇ
ˇ
2
G
Z
K
Z
K
;
2
YK
D
ˇ
ˇ
H
YZ
1
ˇ
ˇ
2
G
Z
1
Z
1
C
ˇ
ˇ
H
YZ
2
ˇ
ˇ
2
G
Z
2
Z
2
C C
ˇ
ˇ
H
YZ
N
ˇ
ˇ
2
G
Z
K
Z
K
=G
YY
;
for inputs; x
1
.t/ C C x
k
.t/: (4.43)
e cumulative coherence function family has the property
0
2
Y 1
2
Y 2
: : :
2
YK
1: (4.44)
When graphically displayed as a function of frequency, the coherence function family ap-
pears as a waterfall plot series that clearly indicates the relative contributions of the accumulated
excitation sources. e benefits of such a display will be demonstrated in a series of illustrative
examples.
Finally, it is noted that MI/MO analysis represents a simple extension to MI/SO analysis.
4.2.4 ILLUSTRATIVE EXAMPLE: ISPE MODAL TEST
e ISPE test article was excited with broad band random excitation forces at four separate
locations, and MI/MO correlation and spectral analysis was preformed employing 8192 length
windows with 50% overlap processing. Cumulative coherences for all 265 TAM response chan-
nels plus 4 drive point response channels were computed. MI/MO plots associated with all 4
excitation forces and 4 drive point responses are detailed in Figure 4.8.
e plots in Figure 4.8 are arranged in a 4 4 matrix “plot map.” For example, the plots
in the (2,3) position correspond to excitation 2, drive point response 3. Each plot “i; j ” entry
includes the cumulative coherence (top), FRF phase angle (middle), and FRF magnitude (bot-
tom). An important feature of the plot matrix is FRF “reciprocity” (e.g., the “i; j ” component
is consistent with the “j; i” component, with regard to both magnitude and phase of the FRF).
Greater detail can be discerned from the plot group for the “1, 1” component shown in Fig-
ure 4.9. Of particular interest is the (top) cumulative coherence plot, which (a) indicates the
successive contributions of the first three excitations (to drive point 1 response autospectum)
that are substantially below unity and (b) indicates the significant role of the fourth excitation
producing near unit cumulative coherence.
80 4. MEASURED DATA ANALYSIS
Figure 4.8: Representative ISPE MI/MO FRF and cumulative coherence plots.
File: ISPETESTDFRFs…MI/MO: Navg=16, df=0.025, Input(1)=2302104X-, Output(1)=2302104X+ 21-Sep-2019 3:54 PM
1
0.5
0
200
0
-200
CoherencePhase
10
-1
10
-2
10
-3
10
-4
Magnitude
15 20 25 30 35 40 45 50
15 20 25 30 35 40 45 50
Hz
Hz
15 20 25 30 35 40 45 50
Hz
Figure 4.9: ISPE MI/MO FRF and cumulative coherence plots for excitation 1, drive point
response 1.
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