6.2. PART 2: CORRELATION OF TEST AND FEM MODES 113
f
g
D
ˆ
f
q
g
: (6.10)
e “left-hand” generalized eigenvectors are defined as
ˆ
L
D
ˆ
1
: (6.11)
Expansion of the generalized eigenvectors using the real trial vectors, ŒV , defines the test
eigenvectors (complex modes) in terms of the physical DOF, specifically,
Œ
ˆ
TEST
D
Œ
V
ˆ
;
Œ
ˆ
L
TEST
D
ˆ
L
V
T
: (6.12)
e state-space eigenvectors resulting from experimental modal analysis using SFD are
exactly orthogonal to one another, in particular,
Œ
OR
TEST
D
Œ
ˆ
L
TEST
Œ
ˆ
TEST
Œ
I
; (6.13)
eliminating the need for satisfaction of NASA and/or USAF test mode weighted orthogonality
criteria! Moreover, orthogonality of state-space eigenvectors is not dependent on the TAM mass
matrix, ŒM
TAM
.
6.2.3 THEORETICAL SYSTEM MODES IN STATE-SPACE FORM
Modes associated with an undamped or damped theoretical model may be formulated in a man-
ner similar to the considerations employed in the previous discussion. If one substitutes the
Guyan [4] (or other suitable) reduction transformation for SVD trial vectors,
f
U
FEM
g
D
Œ
‰
TAM
f
U
TAM
g
; (6.14)
the reduced-order TAM modal dynamic equations become
Œ
M
TAM
˚
R
U
TAM
C
Œ
B
TAM
˚
P
U
TAM
C
Œ
K
TAM
f
U
TAM
g
D
f
0
g
: (6.15)
Converting the above system to a state-space description, the following state-space eigen-
value problem is posed:
Œ
A
TAM
Œ
ˆ
TAM
D
Œ
ˆ
TAM
Œ
TAM
; (6.16)
Œ
A
TAM
D
M
1
TAM
B
TAM
M
1
TAM
K
TAM
I 0
: (6.17)
e (generally complex) right- and left-eigenvectors for the state-space TAM automati-
cally satisfy perfect orthogonality in the same manner as the test eigenvectors, specifically,
Œ
ˆ
L
TAM
D
Œ
ˆ
1
TAM
;
Œ
ˆ
L
TAM
Œ
ˆ
TAM
D
Œ
I
: (6.18)
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