Civil Engineering Reference
In-Depth Information
Example 4.2
The modal quantities given in Eq. 4.61 may be obtained from a fully expanded vector format given
by:
T
T
T
T
T
T
M
(
m
m
m
m
m
m
=
φ
φ
+
φ
φ
+
φ
φ
+
φ
φ
+
φ
φ
+
φ
φ
ae
y
yyy
z
zyy
θθ
yy
y
yzz
z
zzz
θθ
zz
ij
i
j
i
j
i
j
i
j
i
j
i
j
T
T
T
+
φ
m
φ
+
φ
m
φ
+
φ
m
φ
)
⋅ Δ
x
yy
zz
θθ
θθ
θ
θθθ
i
j
i
j
i
j
T
T
T
T
T
T
C
=
(
φ φφφφ φφ φφφφφ
c
+
c
+
c
+
c
+
c
+
c
ae
y yyy
z zyy
yy
y yzz
z zzz
zz
θθ
θθ
ij
i
j
i
j
i
j
i
j
i
j
i
j
T
T
T
+
φφφφφφ
c
+
c
+
c
)
⋅ Δ
x
yy
zz
θθ
θθ
θ θθθ
i
j
i
j
i
j
T
T
T
T
T
T
K
=
(
φφφφφφφφφφφφ
k
+
k
+
k
+
k
+
k
+
k
ae
y yyy
z zyy
yy
y yzz
z zzz
zz
θθ
θθ
ij
i
j
i
j
i
j
i
j
i
j
i
j
T
T
T
+
φφφφφφ
k
+
k
+
k
)
⋅ Δ
x
yy
zz
θθ
θθ
θ θθθ
i
j
i
j
i
j
where
is the spanwise mesh separation (above assumed constant). If the coefficients vary
along the span, their numerical values need to be given on the diagonal of an
N
by
N
matrix, where
N
is the number of nodes.
Δ
Thus, for a full description of the motion induced load effects altogether twenty-
seven motion dependent load coefficients are required. First Eqs. 4.59 is introduced into
4.57 and all terms associated with structural motion are gathered on the left hand side
(
)
(
)
(
)
⎡
2
⎤
()
()
i
−−
MM CC KK a a
ω
+−
ω
+−
⋅
ω
=
ω
(4.63)
0
ae
0
ae
0
ae
⎣
⎦
η
Q
K
−
1
and then the result is pre-multiplied with
, recalling that
⎡
2
⎤
⎫
diag
M
K
=
ω
⎪
0
⎣
i
i
⎦
(4.64)
⎬
diag
⎡
2
M
⎤
C
=
ωζ
⎪
⎣
⎦
⎭
0
iii
It is convenient to introduce a reduced modal load vector
T
T
i
()
(
)
⎡
⎤
∫
φ
x
a
x
,
dx
⋅
ω
q
⎢
⎥
L
()
1
()
exp
−
⎢
⎥
a
ω
=⋅
K
a
ω
⎢
"
"
(4.65)
ˆ
0
Q
Q
2
M
⎥
ω
i
i
⎢
⎥
⎣
⎦
T
(
)
⎡
⎤
x
,
a
a
a
where
a
ω
=
⎣
. The following is then obtained
q
q
q
q
⎦
y
z
θ
