Graphics Programs Reference
In-Depth Information
The eigenvalues and eigenvectorsof
H
can nowbeobtainedwith the Jacobi method.
Skipping the details, weobtain the following results:
λ
1
=
0
.
14779
λ
2
=
0
.
58235
λ
3
=
1
.
93653
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
0
.
81027
0
.
56274
0
.
16370
z
1
=
z
2
=
z
3
=
0
.
45102
−
0
.
42040
−
0
.
78730
0
.
37423
−
0
.
71176
0
.
59444
(
L
−
1
)
T
z
i
,
The eigenvectors of the original problemare recovered fromEq. (9.22):
y
i
=
which yields
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
.
.
.
0
81027
0
56274
0
16370
u
1
=
0
.
45102
u
2
=
−
0
.
42040
u
3
=
−
0
.
78730
0
.
26462
−
0
.
50329
0
.
42033
These vectors shouldnowbe normalized (each
z
i
was normalized, but the transfor-
mation to
u
i
does n
ot preserve the magnitudes of vectors). The circular frequencies
are
ω
i
=
√
λ
i
/
(
LC
),sothat
.
.
.
0
3844
√
LC
0
7631
√
LC
1
3916
√
LC
ω
1
=
ω
2
=
ω
3
=
EXAMPLE 9.3
n
+1
-1
0
1
2
n
- 1
n
n
+ 2
P
x
L
The propped cantileverbeam carries a compressive axialload
P
. The lateral displace-
ment
u
(
x
) of the beam can be shown to satisfy the differentialequation
P
E I
u
=
u
(4)
+
0
(a)
where
E I
is the bending rigidity. The boundary conditions are
u
(0)
u
(
L
)
u
(0)
=
=
0
u
(
L
)
=
=
0
(b)
(1) Show that displacement analysisofthebeam results in a matrix eigenvalue
problemif the derivatives are approximated by finite differences. (2) Use the
Jacobi method to compute the lowest three buckling loads and the corresponding
eigenvectors.
Solution of Part (1)
We divide the beam into
n
1)
each as shown and enforce the differentialequationat nodes 1 to
n
.Replacing the
derivatives of
u
in Eq. (a) by central finite differences of
+
1 segments of length
L
/
(
n
+
(
h
2
) at the interiornodes
O
Search WWH ::
Custom Search