6.2. PART 2: CORRELATION OF TEST AND FEM MODES 115
1. Complex mode orthogonality is perfectly satisfied by left-hand eigenvector counterparts
to state-space complex eigenvectors for test and TAM eigenvector sets (Equations (5.19)
and (6.13), respectively).
2. Complex mode cross-orthogonality (correlation) and complex modal coherence are de-
fined based on the complex least-squares process (Equations (6.22) and (6.24), respec-
tively).
Due to (a) the presence of many closely spaced shell breathing modes of the test article
and (b) non-proportional damping (which is a basic reality of all built-up structural dynamic
systems), the test article’s modes are generally complex (and are not readily approximated as
“real” modes).
6.2.5 ILLUSTRATIVE EXAMPLE: ISPE MODAL TEST
e ISPE TAM model is described 75 real, undamped modes in the 0–65 Hz frequency band
computed from the reduced TAM model (expressed in state-space form according to Equa-
tions (6.15)–(6.18)). ISPE modal test data consists of 63 complex modes in the 0–65 Hz fre-
quency band, estimated by the SFD-2018 algorithm. Following the “recipe” based on Equa-
tions (6.13), (6.22), and (6.24), complex mode orthogonality, cross-orthogonality, and modal
coherence matrices are summarized in Figure 6.2.
e TAM and test mode orthogonality matrices are perfect identity matrices. e cross-
orthogonality matrix appears similar in form to its state-space counterpart (see Figure 6.1), and
the modal coherence matrix indicates that the majority of lower frequency (
50%) of test modes
are strong linear combinations of the TAM modes.
Recalling the fact, introduced in Chapter 3, that the orthogonality matrix can be “un-
packed” to describe a subject system’s modal kinetic energy distribution (Equation (3.3)), a corre-
sponding “unpacking” of the complex state-space mode orthogonality matrix (Equation (6.13))
similarly describes the kinetic energy distribution of the complex modes.
In particular, the modal orthogonality and kinetic energy distribution relationship pair for
a complex state-space modal set is defined, in terms of left- and right-hand eigenvectors as,
Œ
OR
D
Œ
ˆ
L
Œ
ˆ
D
Œ
ˆ
V;L
Œ
ˆ
V
C
Œ
ˆ
U;L
Œ
ˆ
U
;
Œ
KE
ˆ
D conj
ˆ
L
˝
Œ
ˆ
D conj
ˆ
V;L
˝
Œ
ˆ
V
C conj
ˆ
U;L
˝
Œ
ˆ
U
:
(6.25)
Note that Œˆ
in the present context corresponds to the complex conjugate transpose
of the matrix, Œˆ; the same operator designation applies to the left-hand eigenvector matrix.
Partitioning of the complex state-space left- and right-hand eigenvectors into “velocity” and
“displacement” partitions (designated by the subscripts “V ” and “U ”, respectively), in order to
describe the modal kinetic energy distributions in terms of the measured DOFs, the above defi-
nitions of complex state-space eigenvector orthogonality and modal kinetic energy distributions
are independent of the TAM mass matrix.
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