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The global stiffness matrix is assembled as
12
−
6
−
12
−
6
0
0
2
2
=
−
6
4
6
2
0
0
k
aa
k
ab
0
−
12
6
24
0
−
12
−
6
EI
k
ba
k
bb
+
k
bb
k
bc
K
=
(3)
2
2
2
3
−
6
2
08
6
2
k
cb
k
cc
0
0
0
−
12
6
12
6
2
2
0
0
−
6
2
6
4
w
a
θ
a
w
b
θ
b
w
c
θ
c
]
T
The corresponding global displacement vector is
V
=
[
or, for harmonic
motion,
V
=
φ
. Similarly, the global mass matrix is
156
−
22
54
13
0
0
=
ρ
420
−
22
4
2
−
13
−
3
2
0
0
m
aa
m
ab
0
54
−
13
312
0
54
13
m
ba
m
bb
+
m
bb
m
bc
M
=
(4)
13
−
3
2
0
8
2
−
13
−
3
2
m
cc
0
cb
0
0
54
−
13
156
22
2
2
0
0
13
−
3
22
4
The frequencies can be determined by solving the generalized linear eigenvalue prob-
lem of Eq. (10.34). Equation (10.34) is obtained by applying the displacement boundary
conditions to
φ
and ignoring the rows in
(
K
−
λ
M
)
φ
=
0 corresponding to the unknown
reactions.
Thus, with
w
a
θ
a
w
b
θ
b
w
c
θ
c
]
T
θ
a
w
b
θ
b
θ
c
]
T
w
a
=
w
c
=
0
,
φ
=
[
reduces to
φ
y
=
[
and the
eigenvalue problem becomes
2
2
2
2
4
6
2
0
4
−
13
−
3
0
θ
a
w
−
λ
=
EI
6
24
0
−
6
−
13
312
0
13
ρ
420
b
0
(5)
3
2
2
2
2
2
2
2
08
2
−
3
0
8
−
3
θ
b
2
2
2
2
0
−
6
2
4
0
13
−
3
4
θ
c
Use of a standard eigenvalue solution procedure will lead to the desired frequencies.
An alternative technique for finding the frequencies is to establish the characteristic
equation from the determinant of the coefficients of
0
,
where
K
dyn
is referred to as the
dynamic stiffness matrix
. The boundary conditions must be taken into
account in establishing the determinant. Let
(
K
−
λ
M
)
φ
=
K
dyn
φ
=
2
M
K
−
ω
=
K
dyn
=
[
D
ij
]
i, j
=
1
,
2
,
...
,
6
,
(6)
and apply the displacement boundary conditions
(w
a
=
w
c
=
0
).
This leads to
D
11
D
12
D
13
D
14
D
15
D
16
w
=
0
a
D
21
D
22
D
23
D
24
D
25
D
26
θ
a
w
D
31
D
32
D
33
D
34
D
35
D
36
b
=
0
(7)
D
41
D
42
D
43
D
44
D
45
D
46
θ
b
w
=
D
51
D
52
D
53
D
54
D
55
D
56
0
c
D
61
D
62
D
63
D
64
D
65
D
66
θ
c
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