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

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