16 2. DEFINITION OF TEST ARTICLE FINITE ELEMENT MODELS
e interior partition of the transformed structural dynamics equations is
Œ
M
ii
˚
R
U
0
i
C
Œ
B
i i
˚
P
U
0
i
C
Œ
K
i i
˚
U
0
i
D
Œ
ie
f
F
e
g
Œ
M
ib
˚
R
U
b
Œ
B
ib
˚
P
U
b
: (2.33)
Note the similarity of form of SDOF (Equation (
2.25)) and MDOF (Equation (2.33))
“relative” displacement relationships (with the exception of the MDOF damping term, ŒB
ib
).
It should be noted that for a rigid foundation, ŒB
ib
vanishes.
2.2.4 NORMAL MODES OF UNDAMPED MDOF SYSTEMS
Free vibration of an undamped, linear structural dynamic system is described by solutions of the
real eigenvalue problem (retaining the notation of Equation (2.33)),
Œ
K
i i
f
ˆ
i n
g
Œ
M
i i
f
ˆ
i n
g
n
D
f
0
g
;
n
D !
2
n
; (2.34)
where fˆ
in
g are distinct, orthogonal individual eigenvectors (or mode shapes), and
n
D !
2
n
,
are the corresponding eigenvalues (note: !
n
are circular natural frequencies). e collection of
all (or a truncated set of ) eigenvectors, Œˆ
i
, defines the real mode displacement transformation,
f
U
i
.t/
g
D
Œ
ˆ
i
f
q.t /
g
; (2.35)
which has the following decoupling mathematical properties (for unit mass normalized modes):
Œ
ˆ
i
T
Œ
M
ii
Œ
ˆ
i
D
Œ
I
;
Œ
ˆ
i
T
Œ
K
i i
Œ
ˆ
i
D
Œ
: (2.36)
Note that the terms of the diagonal matrix, Œ, are the real eigenvalues,
n
D !
2
n
.
Application of the real mode displacement transformation to Equation (
2.34), results in
the modal equations
f
R
q
g
C
ˆ
T
i
B
i i
ˆ
i
f
P
q
g
C
Œ
f
q
g
D
ˆ
T
i
ie
f
F
e
g
ˆ
T
i
M
ib
˚
R
U
b
ˆ
T
i
B
ib
˚
P
U
b
: (2.37)
A common approximation for the “modal” damping matrix assumes it is “uncoupled” (or
diagonal). Moreover, the right side damping term,
ˆ
T
i
B
ib
is often assumed negligible, result-
ing in the uncoupled modal equations, which are compared directly to SDOF Equation (2.25)
as follows:
Rq
n
.t/ C 2
n
!
n
Pq
n
.t/ C !
2
n
q
n
.t/ D
ˆ
T
i n
ie
f
F
e
.t/
g
ˆ
T
in
M
ib
˚
R
U
b
.t/
;
# # # # .
Ru
R
.t/ C 2
n
!
n
Pu
R
.t/ C !
2
n
u
R
.t/ D F .t/=m Ru
0
.t/:
(2.38)
While many more consequences of the modal transformation will be discussed throughout
this book, the clear relationship between linear SDOF system dynamics and uncoupled modal
dynamics (Equation (2.37)) will be exploited in the following sections for systematic definition
of MDOF dynamic system modeling requirements.
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