Civil Engineering Reference
In-Depth Information
There are two independent eigen-value solutions to this problem. First, there is one that only
involves
z
(
x,t
) displacements, defined by
⎡
4
⎤
n
⎛
⎞
L
π
2
EI
m
a
0
⎢
−
ω
⎥
⋅
=
⎜
⎝ ⎠
y
z
⎢
⎥
⎣
⎦
which will render
n
eigen-values and corresponding eigen-vectors
0
⎡
⎢ ⎥
EI
nx
π
2
y
(
)
()
n
x
sin
1
0
ω
=
π
and
φ
=
1
1
⎢
⎢
⎣ ⎦
n
4
n
mL
The second solution involves a combined motion of
y
(
x,t
) and θ(
x,t
) displacements. It is defined by
4
⎡
⎤
n
⎛ ⎞
π
2
2
⎢
EI
−
ω
m
ω
me
⎥
⎜ ⎟
z
L
a
⎡⎤
⎢
⎝ ⎠
⎥
y
=
0
⎢⎥
⎢
⎥
2
22
a
n
⎛
n I
⎞
π
π
⎢⎥
⎛ ⎞
⎣⎦
⎢
⎥
θ
2
w
2
me
GI
m
ω
⎜
+
⎟
−
ω
⎜ ⎟
⎢
t
⎥
⎜
⎟
θ
L
2
L
⎝ ⎠
⎝
⎠
⎣
⎦
and it will render two different eigen-values and corresponding eigen-vectors:
⎡
⎢ ⎥
1/2
1
⎧
⎫
⎡
⎤
2
ˆ
ˆ
K
1
+
ω
1
−
ω
nx
π
⎪
⎛
⎞
⎪
ˆ
⎢
2
⎥
()
θ
eK
ω
=
+
+
φ
x
=
sin
0
⎢
⎢ ⎥
⎢
⎣ ⎦
⎨
⎬
⎜
⎟
2
2
n
2
n
mem
⎢
2
2
⎥
L
−
⎝
⎠
⎪
⎪
θ
a
ˆ
⎩
⎣
⎦
⎭
θ
2
⎡
⎢ ⎥
1/2
1
⎧
⎫
⎡
⎤
2
K
ˆ
ˆ
1
+
ω
1
−
ω
nx
π
⎪
⎛
⎞
⎪
ˆ
⎢
2
⎥
()
θ
eK
ω
=
−
+
φ
x
=
sin
0
⎢
⎢ ⎥
⎢
⎣ ⎦
⎨
⎬
⎜
⎟
3
3
n
2
n
mem
⎢
2
2
⎥
L
−
⎝
⎠
⎪
⎪
θ
⎣
⎦
a
ˆ
⎩
⎭
θ
3
4
2
2
⎡
⎤
n
L
π
n
π
n
π
K
K m
⎛
⎞
=
⎜
⎝ ⎠
⎛ ⎞
⎛ ⎞
,
ˆ
K
EI
K
z
,
ˆ
mK
θ
θ
where:
,
K
=
⎢
GI
+
EI
⎥
=
ω =
⎜ ⎟
⎜ ⎟
z
z
θ
t
w
L
L
K
θ
⎝ ⎠
⎝ ⎠
⎢
⎥
⎣
⎦
⎡
⎤
⎡
⎤
ˆ
ˆ
ˆ
ˆ
1
ω
−
1
ω
−
1
1
ω
−
1
ω
−
1
⎛
⎞
⎛
⎞
ˆ
ˆ
2
2
a
ˆ
e K
a
ˆ
e K
⎝ ⎢
⎣ ⎦
It may be of some interest to develop the modal mass associated with these mode shapes. The
cross sectional mass matrix is given by
=
+
−
⎦
=
−
−
⎢
⎥
⎢
⎥
⎜
⎟
⎜
⎟
θ
θ
2
3
e
2
2
e
2
2
⎢
⎝
⎠
⎥
⎣
[
]
M
=
diagmmm
θ
, and thus
0
L
L
nx
π
T
2
M
∫
φ M φ
dx
m
∫
sin
dx
mL
/ 2
=
=
=
1
1
0
1
n
n
n
0
0
L
L
nx
π
(
)
(
)
T
2
2
2
M
∫
φ M φ
dx
m
a
ˆ
m
∫
sin
dx
m
a
ˆ
m
L
/ 2
=
=
+
=
+
2
2
0
2
θθ
θθ
n
n
n
2
L
2
0
0
L
T
M
=
∫
φ M φ
dx
=
M
3
3
0
3
2
n
n
n
n
0
