Civil Engineering Reference
In-Depth Information
⎡
⎢
ζ
()
ω
V
(
ae
ij
i
*
*
*
*
*
∫
P
H
BA
P
H
=
⋅
φφ
+
φφ
+
φφ
+
φφ
+
φφ
yy
1
zy
5
θ
y
5
yz
5
zz
1
2
⎢
⎣
i
j
i
j
i
j
i
j
i
j
B
m
ω
ρ
i
L
exp
4
i
⎡
⎤
)
(
)
⎤
*
*
*
2
*
2
2
2
∫
+
φφ
BA
+
φφ
BP
+
φφ
BH
+
φφ
B A
dx
φ
+
φ
+
φ
dx
⎢
⎥
z
1
y
2
z
2
2
y
z
θ
θ
θ
θ
θ
⎥
⎦
⎢
θ
i
j
i
j
i
j
i
j
i
i
i
⎥
⎣
⎦
L
(8.9)
i
j
⎫
and
. The problem is further simplified if
=
1, 2
⎬
⎭
T
()
[
]
⎫
x
0
0
φ
≈
φ
⎪
⎬
⎪
1
z
1
(8.10)
T
()
[
]
φ
x
00
≈
φ
2
⎭
θ
2
with corresponding eigen-frequencies
ωω
=
and
ωω
=
, modal eigen-damping
1
z
2
θ
ratios
and
, and with modal mass properties
mm
and
2
mm
θ
. In
ζζ
=
ζζ
=
=
=
1
z
2
1
z
θ
that particular case
⎧
2
⎡
⎤
κ
κ
(
)
⎡
⎤
/
0
10
ωω
⎡ ⎤
⎪
ae
ae
ˆ
zz
z
r
z
(
)
θ
,
V
⎢
⎥
E
ω
=
−
⎢
⎥
−
⎨
⎢ ⎥
η
r r
01
2
κ
κ
⎢
(
)
⎥
⎣ ⎦
⎢
⎥
0
/
⎪
ωω
⎣
ae
ae
⎦
⎣
⎦
⎩
θ
z
θθ
r
θ
(8.11)
ζζ
−
−
ζ
⎫
⎡
⎤
ωω
/
0
⎡
⎤
⎪
z
ae
ae
r
z
zz
z
θ
2
i
+
⋅
⎢
⎥
⎬
⎢
⎥
0
ωω
/
−
ζ
ζ
−
ζ
⎣
⎦
⎢
⎥
⎪
r
θ
⎣
ae
θ
ae
⎦⎭
θ
z
θθ
and
2
∫
dx
∫
dx
φ
φφ
z
z
θ
2
2
()
()
2
3
⎛
ω
V
⎞
⎛
ω
V
⎞
B
B
ρ
ρ
L
L
z
exp
z
exp
*
4
*
3
H
H
κ
=
κ
=
⎜
⎟
⎜
⎟
ae
zz
⎜
⎟
ae
z
⎜
⎟
2
m
2
θ
2
m
2
ω
∫
dx
ω
∫
dx
φ
φ
⎝
⎠
⎝
⎠
z
z
z
z
z
z
L
L
(8.12)
2
∫
dx
∫
dx
φ
φφ
z
θ
θ
2
2
()
()
4
V
3
V
B
⎛
ω
⎞
B
⎛
ω
⎞
ρ
ρ
L
L
exp
exp
θ
*
3
θ
*
4
A
A
κ
=
κ
=
⎜
⎟
⎜
⎟
ae
⎜
⎟
ae
z
⎜
⎟
2
2
θθ
2
m
ω
∫
θ
2
m
ω
∫
φ
dx
φ
dx
⎝
⎠
⎝
⎠
θ
θ
θ
θ
θ
θ
L
L
(8.13)
