Civil Engineering Reference
In-Depth Information
The content of the normalised modal load matrix
S
S
⎡
⎤
ˆˆ
ˆˆ
QQ
QQ
11
1 2
()
⎢
⎥
S
ω
=
⎢
ˆ
Q
S
S
⎥
ˆˆ
ˆˆ
⎣
QQ
QQ
⎦
21
2 2
is given in Eq. 6.59:
2
2
3
3
⎛ ⎞
ρρ
BB V
⎛ ⎞
V
ˆ
()
2
S
J
ω
=
⋅
⋅
⋅
⋅
⎜ ⎟
⎜ ⎟
⎝ ⎠
⎝ ⎠
⎜ ⎟
ˆˆ
ij
Q
ij
22
mmB
ω
B
ω
i
j
i
j
where the reduced joint acceptance function
ˆ
i
J
is given in Eq. 6.60. An expanded version of the
joint acceptance function itself is given in in Eq. 6.61. Under the present circumstances it
simplifies into
(
)
S
ω
,
Δ
x
2
2
11
()
()(
)
ww
J
∫∫
x
x
C I
′
dx dx
=
φ
⋅
φ
⋅
⋅
z
1
z
2
L
w
1
2
1
1
2
σ
w
L
exp
(
)
S
ω
,
Δ
x
2
()
()
2
ww
2
2
J
∫∫
x
x
C BC
′
′
I
dx dx
J
J
=
φ
⋅
φ
⋅
⋅
,
=
12
z
1
2
L
M
w
1
2
θ
21
12
1
2
2
σ
w
L
exp
(
)
S
ω
,
Δ
x
2
2
22
()
()(
)
ww
J
∫∫
x
x
BC
′
I
dx dx
=
φ
⋅
φ
⋅
⋅
1
2
Mw
1
2
θ
θ
2
2
2
σ
w
L
exp
ˆ
(
)
(
)
(
)
S
,
xS
o
,
x
I
V
Introducing
ω
Δ=
ω
⋅
ω
Δ and
=
σ
, then the content of the
ww
w
ww
ww
normalised modal load matrix is given by
2
2
⎛
⎞
⎛
⎞
VB
C BC
′
′
ρ
VBC
′
ρ
ˆ
ˆ
()
()
()
()
LM
()
()
L
S
ω
=
⋅
J
ω
S
ω
,
S
ω
=
⎜
⋅
J
ω
⎟
S
ω
,
⎜
⎟
ˆˆ
ˆˆ
⎜
11
⎟
w
21
w
QQ
2
QQ
⎜
⎟
2
m
2
mm
11
ω
12
ωω
⎝
⎠
⎝
⎠
11
12
1 2
2
2
⎛
⎞
ρ
VB C
′
ˆ
()
()
()
()
()
M
S
ω
=
S
ω
and
S
ω
=
⋅
J
ω
S
ω
⎜
⎟
ˆˆ
ˆˆ
ˆˆ
⎜
22
⎟
w
QQ
QQ
QQ
2
2
m
21
12
22
ω
⎝
⎠
22
where:
2
⎛
⎞
ˆ
ˆ
2
()
()
(
)
2
J
∫∫
x
x
Co
,
x dx dx
∫
dx
=
φ
⋅
φ
⋅
ω
Δ
⎜
φ
⎟
11
z
1
z
2
ww
1
2
z
⎜
⎟
1
1
1
⎝
⎠
L
L
exp
⎛
⎞
ˆ
ˆ
2
()
()
(
)
2
2
J
∫∫
x
x
Co
,
x dx dx
∫ ∫
dx
dx
=
φ
⋅
φ
⋅
ω
Δ
⎜
φ
⋅
φ
⎟
21
z
1
2
ww
1
2
z
θ
⎜
θ
⎟
1
2
1
2
⎝
⎠
L
L
L
exp
2
⎛
⎞
ˆ
ˆ
2
()
()
(
)
2
J
∫∫
x
x
Co
,
x dx dx
∫
dx
=
φ
⋅
φ
⋅
ω
Δ
⎜
φ
⎟
22
1
2
ww
1
2
θ
θ
⎜
θ
⎟
2
2
2
⎝
⎠
L
L
exp
ˆ
(
)
(
)
sin
x L
Co
,
x
exp
C
x V
Since
φφ
==
π
, and
ω
Δ=
− ⋅
ω
⋅ Δ
the present situation is
z
ww
wy
θ
1
2
equivalent to that which was encountered in Example 6.2, and thus,
