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Insert the series expansions of Eqs. (13.61) and (13.62) in the governing equations of
Eq. (13.58) to obtain
d
w
m
dx
=−
θ
m
m
2
θ
L
y
d
m
m
K
m
dx
=−
ν
w
+
m
(13.64a)
m
4
m
2
L
y
L
y
dV
m
dx
=
2
(
−
ν
)
w
+
ν
−
(
)
K
1
m
m
p
m
x
m
m
2
dm
m
dx
L
y
=
2
K
(
1
−
ν)
θ
m
+
V
m
or
0
−
1
0
0
w
w
0
m
−
ν
m
L
y
2
m
/
0
0
1
K
θ
θ
0
d
dx
m
V
m
m
m
m
V
m
m
m
)
m
L
y
4
ν
m
L
y
2
=
+
2
K
(
1
−
ν
0
0
−
p
m
0
−
ν)
m
L
y
2
0
2
K
(
1
1
0
d
z
dx
=
A
z
+
P
(13.64b)
This system of ordinary differential equations can now be solved, using the techniques of
Chapter 4, to obtain the transfer and the stiffness matrices.
For the case of other boundary conditions at
y
=
0 and
y
=
L
y
,
a series expansion of the
state variables and loadings can be assumed as
w(
x, y
)
w
(
x
)
m
=
φ
m
(
∞
θ(
x, y
)
θ
(
x
)
m
z
=
y
)
(13.65)
V
(
x, y
)
V
m
(
x
)
m
=
1
m
x
(
x, y
)
m
m
(
x
)
where
φ
m
(
y
)
is selected to satisfy the boundary conditions at
y
=
0 and
y
=
L
y
.
A possible
choice for
φ
m
is to use mode shapes for a beam with the appropriate boundary conditions.
These are
Boundary Conditions
Case
y
=
0
y
=
L
y
φ
m
β
m
m
L
y
1
Simply
Simply
sin
β
m
y
Supported
Supported
(
4
m
+
1
)π
2
Fixed
Simply
cosh
β
m
y
−
cos
β
m
y
+
E
m
(
sinh
β
m
y
−
sin
β
m
y
)
4
L
y
Supported
(
2
m
+
1
)π
β
m
y
−
β
m
y
−
E
m
(
β
m
y
−
β
m
y
)
3
Fixed
Fixed
cosh
cos
sinh
sin
2
L
y
where
E
m
The loadings should also
be expanded in the same series. Substitution of these relations into Eqs. (13.58) or (13.60)
leads to governing differential equations in terms of
=
(
cosh
η
−
cos
η
)/(
sinh
η
−
sin
η
)
,
and
η
=
β
m
L
y
.
m
m
m
m
m
w
θ
m
,
m
,
V
m
, and
m
m
. For example,
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