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Substitution of (5) into Table 10.1 provides
k
11
=
(
EI
/
∇
)(
e
1
e
2
+
λ
e
3
e
4
)
k
i
12
=
(
e
2
)
EI
/
∇
)(
e
1
e
3
−
k
i
13
=−
(
EI
/
∇
)
e
2
k
i
14
=−
(
EI
/
∇
)
e
3
k
i
21
=
k
i
12
k
i
22
=
(
EI
/
∇
)(
e
3
e
2
−
e
1
e
4
)
k
i
23
=−
k
i
14
(6)
k
i
24
=
(
/
∇
)
EI
e
4
k
i
31
=
k
i
13
,
i
32
k
i
23
=
k
i
33
=
k
11
k
i
34
=
(
e
4
)
EI
/
∇
)(
e
1
e
3
+
λ
k
i
41
=
k
i
14
,
i
42
k
i
24
,
i
43
k
i
34
=
=
k
i
22
where the subscript dyn in Table 10.1 has been dropped.
The assembled dynamic stiffness matrix appears as
k
i
44
=
k
11
k
12
k
13
k
14
00
k
21
k
22
k
23
k
24
00
k
31
k
32
k
33
+
k
11
k
34
+
k
12
k
13
k
14
K
dyn
=
(7)
k
41
k
42
k
43
+
k
21
k
44
+
k
22
k
23
k
24
k
31
k
32
k
33
k
34
00
k
41
k
42
k
43
k
44
00
and the corresponding global displacement vector is
c
]
T
=
w
θ
w
θ
w
θ
V
[
(8)
a
a
b
b
c
Introduce the displacement boundary conditions
0
,
and delete the
rows for the reactions corresponding to these prescribed displacements. The
characteristic equation (Eq. 10.42) then becomes
w
=
w
=
a
c
k
22
k
23
k
24
0
k
32
k
33
+
k
11
k
34
+
k
12
k
14
=
0
(9)
k
42
k
43
+
k
21
k
44
+
k
22
k
24
k
41
k
42
k
44
0
A frequency search applied to this relationship leads to the natural frequencies:
ω
=
180
.
74 rad
/
sec
1
ω
=
722
.
96 rad
/
sec
2
ω
=
1626
.
66 rad
/
sec
3
ω
4
=
.
/
2891
84 rad
sec
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