124 7. RECONCILIATION OF FINITE ELEMENT MODELS AND MODAL TEST DATA
known as Structural Dynamic Modification (SDM)) [4], the Ritz method often produces poor
estimates for the altered system. Augmentation of the truncated baseline system mode shapes
with appropriately defined additional vectors, however, has been found to produce extremely
accurate altered system modal frequencies and mode shapes. Quasi-static residual vectors [5],
appended to a truncated set of mode shapes, were found to produce extremely accurate modes
for offshore oil platform models subjected to localized alterations [6]. Residual Mode Augmen-
tation (RMA), introduced in 2002 [6] and subsequently refined [7], is a procedure that defines
augmented trial vectors, which are appropriate for structures subjected to highly distributed, as
well as localized, alterations.
7.1.2 SENSITIVITY ANALYSIS STRATEGIES
e present discussion focuses on Ritz procedures that address structural sensitivities due to
stiffness and mass alterations described by large (as opposed to small) parametric variations.
erefore, formulations that address computation of eigenvalue and mode shape derivatives are
not considered.
e matrix equations describing exact free vibration of baseline and altered structures,
respectively, are
Œ
K
O
Œ
ˆ
O
Œ
M
O
Œ
ˆ
O
Œ
O
D
Œ
0
; (7.1)
Œ
K
O
C p K
Œ
ˆ
Œ
M
O
C p M
Œ
ˆ
Œ
D
Œ
0
: (7.2)
It is implicitly assumed that the stiffness and mass changes scale linearly with respect to
the parameter, p. erefore, changes in “beam” depth may not be directly applied, since the
axial stiffness (AE) scales linearly with depth and the flexural stiffness (EI) scales as the cube
of depth. e appropriate formulation for Equation (7.2) permits linear sensitivity of “AE” and
“EI” separately.
e relationship between mode shapes of the baseline and altered structures is expressed
as the cross-orthogonality of orthonormal mode shape sets
Œ
COR
D
ˆ
T
O
Œ
M
O
Œ
ˆ
; (7.3)
where the modal self-orthogonality properties are
Œ
OR
O
D
ˆ
T
O
Œ
M
O
Œ
ˆ
O
D
Œ
I
O
;
Œ
OR
D
ˆ
T
Œ
M
O
C pM
Œ
ˆ
D
Œ
I
: (7.4)
e most fundamental Ritz approximation, commonly used in SDM [4] employs a trun-
cated set of low frequency eigenvalues as the reduction transformation described by
Œ
ˆ
D
Œ
ˆ
OL
Œ
'
: (7.5)
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