Civil Engineering Reference
In-Depth Information
()
()
()
x
⎡
φ
φ
⎤
y
N
mod
⎢
⎥
(
)
∑
()
( ) ()
r
x t
,
x
t
Φη
(4.7)
x
t
=
⋅
η
=
⋅
⎢
⎥
z
i
⎢
⎥
i
1
=
x
φ
⎣
θ
⎦
i
where
N
mod
is the number of modes that has been deemed necessary for a sufficiently
accurate or representative solution.
Fig. 4.3
Mode shapes
()
()
Φ
and the vector
η
that contains the so-called gener-
The mode shape matrix
alised coordinates are defined by
()
()
()
()
⎡
⎤
⎫
x
x
...
x
...
x
Φφ
=
⎣
φ
φ
⎦
⎪
⎬
1
i
N
mod
(4.8)
T
()
()
()
()
⎡
⎤
⎪
η
t
t
.....
t
.....
t
=
⎣
η
η
η
⎦
⎭
1
i
N
mod
The introduction of Eq. 4.7 into the equilibrium equations, followed by consecutive
weighing with each (orthogonal) mode shape and span-wise integration will then render
N
equivalent modal equilibrium conditions
mod
M η C η K η Q
⋅+
⋅+
⋅=
(4.9)
0
0
0
tot
