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Use Eq. (4.84) to replace the constants
C
1
,C
2
,C
3
,
and
C
4
in Eq. (4.83) by the state variables
and set
x
=
,
giving
p
z
(τ )
EI
V
a
3
3!
EI
−
M
a
2
2
EI
+
w
=
w
−
θ
−
d
τ
b
a
a
0
p
z
(τ )
EI
2
2
EI
+
V
a
M
a
EI
θ
b
=
θ
a
+
−
d
τ
0
(4.85)
V
b
=
V
a
−
p
z
(τ )
d
τ
0
M
b
=
V
a
+
M
a
−
p
z
(τ )
d
τ
0
U
i
z
a
z
i
,
where
z
VM
]
T
,
giving
In matrix notation this appears as
z
b
=
+
=
[
wθ
3
6
EI
2
2
EI
−
−
−
1
2
2
EI
EI
01
U
i
U
i
=
()
=
(4.86)
00
1
0
00
1
0
p
z
(τ )
0
b
d
τ
w
EI
−
0
p
z
(τ )
b
V
b
M
b
θ
d
τ
EI
z
i
=
=
(4.87)
−
0
τ
−
0
p
z
(τ )
p
z
(τ )
d
d
τ
First Order Form of the Governing Equations
A typical method of developing transfer matrices, which applies to both simple and difficult
problems, is that of integration of first order equations in the state variables [Wunderlich,
1966 and 1967]. In Chapter 1, Eq. (1.133), first order governing equations for a beam were
derived. Retain the sign convention of Chapter 1, i.e., use Sign Convention 1. If shear
deformation is ignored, then Chapter 1, Eq. (1.135) becomes
w
V
M
0
−
10
0
w
V
M
0
0
−
=
+
d
dx
0
EI
000 0
001 0
001
/
p
z
0
(4.88)
z
=
A
z
+
P
z
=
gives
∗
Integration of these relations
(
Az
+
P
)
e
A
x
x
x
e
A
x
z
a
+
e
−
A
τ
P
z
=
(τ )
d
τ
(4.89)
=
x
a
∗
The solution to the matrix relation in Eq. (4.88) is analogous to the familiar solution of the first order scalar
differential equation
e
Ax
x
0
z
=
z
0
e
Ax
e
−
A
τ
p
Az
+
p
:
z
=
+
(τ )
d
τ
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