Civil Engineering Reference
In-Depth Information
⎫
[
]
I
=
diag I
I
v
u
w
⎪
⎪
⎬
⎪
(6.40)
ˆ
(
)
2
2
⎡
⎤
S
x
,
diag S
/
S
/
Δω
=
σ
σ
⎪
v
⎣
uu
u
ww
w
⎦ ⎭
and defining
2
K
M
Introducing the modal stiffness
=
ω
i
i
i
(
)
T
mM
/
∫
φφ
x
=
⋅
(6.41)
i
i
i
i
L
then from Eqs. 4.47 and 4.48 the following standard deviations of displacement
responses at
x
are obtained
()
()
()
x
σ
⎡
φ
⎤
⎡⎤
1/2
y
y
r
2
3
⎡
∞
⎤
⎢⎥
=
⎢
⎥
BV
⎛ ⎞
ρ
2
ˆ
ˆ
()
2
()
x
∫
H
J
d
σ
φ
⋅
⋅
ω
⋅
ω
ω
(6.42)
⎢
⎥
⎢
⎥
⎜ ⎟
⎢⎥
z
z
r
i
i
2
mB
ω
⎢
⎥
⎝ ⎠
⎢
⎣⎦
⎣
⎢
⎥
⎣
⎦
i
i
x
0
σ
φ
θ
θ
r
⎦
i
i
where
J
ˆ
i
J
=
(6.43)
i
T
i
∫
φφ
dx
⋅
i
L
Again, neglecting any aerodynamic mass and introducing the notation given in Eqs.
4.25, 4.40 and 5.25, then the frequency response function is given by
−
1
2
⎡
⎤
⎛⎞
ω
ω
(
)
ˆ
()
⎢
⎥
H
1
2
i
ω
=−
κ
−
+
ζ
−
ζ
⋅
(6.44)
⎜⎟
i
ae
i
ae
i
i
ω
ω
⎢
⎥
⎝⎠
i
i
⎣
⎦
where
(
)
ˆ
T
i
∫
dx
φ K φ
ei
2
B
m
ρ
L
exp
κ
=
⋅
(6.45)
(
)
ae
i
2
T
∫
dx
φφ
i
i
i
L
(
)
ˆ
T
i
∫
dx
φ C φ
ei
2
B
m
ρ
L
exp
ζ
=
⋅
(6.46)
(
)
ae
i
4
T
∫
dx
φφ
i
i
i
L
