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Fergusson and Pilkey (1992 and 1993) review the literature on the use of dynamic
stiffness matrices.
EXAMPLE 10.8
Eigenvalues for a Beam Using a Dynamic Stiffness Matrix
Use the dynamic stiffness matrix to compute the frequencies of the beam of
Examples 10.5 and 10.6. Since an exact stiffness matrix is to be employed, this
can be treated as a single element beam. However, we choose to model the beam
as two elements as shown in Fig. 10.6.
The stiffness relationship for the
i
th element
(
=
i
1or2)is
k
i
dyn
v
i
p
i
=
(1)
where the dynamic stiffness matrix
k
i
dyn
is taken from Table 10.1 with
2
2
λ
=
(
k
z
−
ρω
)/(
EI
)
=−
ρω
/(
EI
)<
0
(2)
2
η
=
(
k
z
−
ρω
)/(
GAk
s
)
=
0 if shear deformation is not included
(
1
/
GAk
s
=
k
∗
+
ρ
r
y
ω
2
0
,k
∗
=
0
)
,
ζ
=
(
P
−
)/(
EI
)
=
0 since
P
=
0 and rotary effects are to
be ignored.
To complete the stiffness matrix of Table 10.1, it is necessary to refer to functions
provided in Chapter 4, Table 4.3. For
λ<
0, Case 1 of Table 4.3 provides
a
2
A
b
2
B
e
1
=
(
+
)/
g
e
2
=
(
aC
+
bD
)/
g
(3)
e
3
=
(
A
−
B
)/
g
e
4
=
(
C
/
a
−
D
/
b
)/
g
in which
EI
ω
a
2
=
(ζ
+
η)
2
/
4
−
λ
−
(ζ
−
η)/
2
=
EI
ω
=
b
2
a
2
2
=
(ζ
+
η)
/
4
−
λ
+
(ζ
−
η)/
2
=
2
EI
ω
a
2
b
2
g
=
+
=
(4)
A
=
cosh
(
a
)
B
=
cos
(
b
)
=
cos
(
a
)
C
=
sinh
(
a
)
D
=
sin
(
b
)
=
sin
(
a
)
Insertion of (4) into (3) gives
EI
ρ
1
2
[cosh
1
2
e
1
=
(
a
)
+
cos
(
a
)
]
e
3
=
[cosh
(
a
)
−
cos
(
a
)
]
ω
EI
ρ
1
/
4
1
2
√
ω
e
2
=
[sinh
(
a
)
+
sin
(
a
)
]
(5)
EI
ρ
3
/
4
1
e
4
=
ω
√
ω
[sinh
(
a
)
−
sin
(
a
)
]
2
and
EI
ρ
[1
1
e
3
−
(
e
3
−
∇=
e
2
−
0
)(
e
4
−
0
)
=
e
2
e
4
=
−
cosh
(
a
)
cos
(
a
)
]
2
ω
2
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