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ω
y
(
x
)sin
t
the problembecomes
2
ω
γ
E I
y
(4)
y
(0)
y
(
L
)
y
(
L
)
=
y
y
(0)
=
=
=
=
0
The corresponding finite difference equations are
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
7
−
4
1
00
···
0
y
1
y
2
y
3
.
y
n
−
2
y
n
−
1
y
n
y
1
y
2
y
3
.
y
n
−
2
y
n
−
1
y
n
/
−
4
6
−
4
1
0
···
0
−
−
···
1
4
6
4
1
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
=
λ
0
···
1
−
4
6
−
4
1
0
···
0
1
−
4
5
−
2
0
···
001
−
21
2
where
L
n
4
2
ω
γ
E I
λ
=
(a) Write down the matrix
H
of the standard form
Hz
=
λ
z
and the transformation
matrix
P
as in
y
Pz
. (b) Write aprogram thatcomputes the lowest twocircular
frequencies of the beamand the correspondingmodeshapes (eigenvectors) using
the Jacobi method. Run the programwith
n
=
=
10.
Note
: th
e analytical solution for
ω
1
=
3
L
2
√
E I
the lowest circular frequencyis
.
515
/
/γ
.
17.
L
/2
L
/4
L
/4
P
P
EI
0
EI
0
2
EI
0
(a)
L
/4
L
/4
0
12
345678910
(b)
The simply supported column in Fig. (a)consists of three segments with the
bending rigidities shown. If only the first buckling mode isofinterest, it is
sufficienttomodel half of the beamas shown in Fig. (b). The differentialequation
for the lateral displacement
u
(
x
) is
P
E I
u
u
=−
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