5.2. PART 2: THE SIMULTANEOUS FREQUENCY DOMAIN METHOD 93
left-hand eigenvectors are calculated exactly from the inverse of the complete, raw right-hand
eigenvector set.
5.2.2 EFFECTIVE DYNAMIC SYSTEMS
e SFD method [5], introduced in 1981, has undergone substantial revision and refinement
since that time [6, 7], primarily by this writer and principals at e Aerospace Corporation.
SFD implicitly assumes that FRFs associated with a series of “N ” excitations may be described
in terms of a transformation described by
R
U
1
.f /
R
U
2
.f / : : :
R
U
N
.f /
D
Œ
V
h
R
1
.f /
R
2
.f / : : :
R
N
.f /
i
:
(5.13)
By performing singular value decomposition (SVD) analysis [9] of the FRF collec-
tion, a dominant set of real generalized trial vectors, ŒV , and complex generalized FRFs,
h
R
1
.f /
R
2
.f / : : :
R
N
.f /
i
, is obtained. Unit normalization of the SVD trial vectors is
expressed as
Œ
V
T
Œ
V
D
Œ
I
: (5.14)
eoretically, the generalized FRF array describes the following dynamic system equations
associated with the individual applied forces
h
R
.f /
i
C
Q
B
h
P
.f /
i
C
Q
K
Œ
.f /
D
Q
Œ
F .f /
;
where
Q
B
D
M
1
Œ
B
;
Q
K
D
M
1
Œ
K
;
and
Q
D
M
1
Œ
: (5.15)
e real, effective dynamic system matrices,
Q
B
,
Q
K
, and
Q
, are estimated by linear
least-squares analysis [10].
Estimation of experimental modal parameters is performed by complex eigenvalue anal-
ysis of the state variable form of the effective dynamic system,
(
R
P
)
D
Q
B
Q
K
I 0
P
C
Q
0
f
F
g
; (5.16)
which is of the general type
f
P
g
D
A
f
g
C
f
F
g
: (5.17)
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