2.2. PART 2: GUIDELINES FOR SYSTEMATIC MODEL DEVELOPMENT 15
m
k b u
o
(t)
u(t), F(t)
Figure 2.1: Linear SDOF dynamic system schematic.
2.2.3 LINEAR MULTI-DEGREE-OF-FREEDOM EQUATIONS
Linear structural dynamic systems are generally described in terms of partial differential equa-
tions of mathematical physics or matrix differential equations constructed as assemblies of finite
elements. e typical form for matrix (MDOF) structural dynamic equations is
Œ
M
˚
R
U .t /
C
Œ
B
˚
P
U .t /
C
Œ
K
f
U.t/
g
D
Œ
e
f
F
e
.t/
g
: (2.28)
Symmetric system mass, damping, and stiffness matrices, ŒM ; ŒB, and ŒK, respectively,
are associated with discrete displacement degrees of freedom,
f
U.t/
g
. e external loads,
f
F
e
.t/
g
are allocated to the discrete displacements in accordance with geometric distributions described
by the (linearly independent) columns of the matrix, Œ
e
.
By partitioning the displacements into “interior” and “boundary” subsets,
f
U
g
D
U
i
U
b
; (2.29)
the MDOF equations are expressed as
M
ii
M
ib
M
bi
M
bb
R
U
i
R
U
b
C
B
ii
B
ib
B
bi
B
bb
P
U
i
P
U
b
C
K
ii
K
ib
K
bi
K
bb
U
i
U
b
D
ie
be
f
F
e
g
: (2.30)
e transformation of “interior” displacements to “relative” displacements with respect to
the “boundary” displacements is defined as
U
i
U
b
D
I
ii
K
1
i i
K
ib
0
bi
I
bb
U
0
i
U
b
: (2.31)
Symmetric application of this transformation to Equation (2.30) results in
M
i i
M
0
ib
M
0
bi
M
0
bb
R
U
0
i
R
U
b
C
B
ii
B
0
ib
B
0
bi
B
0
bb
P
U
0
i
P
U
b
C
K
i i
0
0
ib
0
0
bi
K
0
bb
U
0
i
U
b
D
ie
0
be
f
F
e
g
: (2.32)
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