128 7. RECONCILIATION OF FINITE ELEMENT MODELS AND MODAL TEST DATA
"
k
O
C
N
X
iD1
p
i
Œ
k
i
#
Œ
'
"
m
O
C
N
X
iD1
p
i
Œ
m
i
#
Œ
'
Œ
D
Œ
0
; (7.29)
Œ
k
O
D
N
ˆ
T
OL
K
O
N
ˆ
OL
;
Œ
m
O
D
N
ˆ
T
OL
M
O
N
ˆ
OL
;
Œ
k
i
D
N
ˆ
T
OL
K
i
N
ˆ
OL
;
Œ
m
i
D
N
ˆ
T
OL
M
i
N
ˆ
OL
: (7.30)
Recovery of mode shapes in terms of physical DOF is accomplished with
Œ
ˆ
D
N
ˆ
OL
Œ
'
: (7.31)
7.1.5 RMA SOLUTION QUALITIES
Since its introduction in 2001, RMA has exhibited the capability to accurately follow modal
sensitivity trends over an extremely wide range of parametric variation. e simple cantilevered
(planar) beam example, provided in Figure 7.1, demonstrates typical RMA performance (“100%”
is baseline). Actual cross-orthogonality checks are also excellent.
Support
Illustrative Cantilevered Beam Modal Frequency Sensitivity
Percent of Baseline “EI2”
Modal Frequency (Hz)
EI1 EI2 EI3
Baseline: EO is Uniform
10
4
10
3
10
2
10
1
10
0
10
-1
10
0
10
1
10
2
Exact Reference Perturbed Modes
Exact Extreme Perturbed Modes
Augmented Mode-Based Sensitivity
Figure 7.1: RMA sensitivity performance for a cantilevered beam example.
7.1. PART 1: FINITE ELEMENT MODEL MODAL SENSITIVITY 129
7.1.6 ILLUSTRATIVE EXAMPLE: ISPE MODAL TEST
In 2017, an early finite element model of the ISPE test article, provided by Dr. Eric Stewart of
NASA/MSFC, was employed for an RMA convergence study. Details of the ISPE model with
parametric sensitivity regions are summarized in Figure 7.2.
Negligible
Sensitivity
Figure 7.2: ISPE finite element model and parametric sensitivity regions.
In response to concerns brought up by Dr. Eric Stewart of NASA/MSFC regarding RMA
solution convergence, an investigation of the matter was conducted. Specifically, the role of
the SVD tolerance parameter .tol/, defined in Equation (7.26), was evaluated. An objective
convergence criterion was developed based on comparison of parametric alterations resulting
from the solution of the exact modal equation,
"
K
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C
X
i
p
i
K
i
#
Œ
ˆ
e
D
"
M
O
C
X
i
p
i
M
i
#
Œ
ˆ
e
Œ
e
; (7.32)
and the approximate modal equations (see Equations (7.28)–(7.30), developed for a specific
value of “tol”),
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