Civil Engineering Reference
In-Depth Information
⎡
⎤
Φφφ φ φ
=
⎣
""
(9.138)
1
2
i
N
⎦
mod
and where
…
) contains the mode shape values, each associated with
the corresponding global degree of freedom number
φ
(
i
=
1,2,
,
N
mod
k
=
1,2,
,
N
…
, i.e.
r
T
⎡
⎤
φ
=
⎣
φφ
""
φ
φ
(9.139)
i
12
k
N
r
⎦
It should be noted that these eigen-modes are identical to those in Chapters 4, 6 and 7,
but their mathematical description differs in the way their components are organised.
While the eigen-modes in Chapters 4, 6 and 7 are based on a description of components
associated with motion in horizontal, vertical and torsion directions, the eigen-modes in
Eq. 9.137 contain displacement components associated with the global degrees of
freedom in the finite element model of the system.
It should also be noted that because
a
K
depends of the mean wind velocity, so will
the total stiffness of the system, and thus, the resonance frequencies and the associated
mode shapes will change with increasing mean wind velocities. These changes are not
negligible with respect to the overall behaviour of the system in the close vicinity of an
instability limit, where it will usually be necessary to update system quantities due to
these effects (resonance frequencies in particular, but it can not be ruled out that in some
cases it may also be necessary to update mode-shapes). Sufficiently far from an
instability limit these effects are small such that the modal solution strategy of most wind
engineering problems may be based on the eigen-frequencies
ω
and corresponding
i
mode shapes
φ
as determined from the eigen-value problem in still air, i.e. from
2
i
⎡
⎤
KMφ 0
(9.140)
−
ω
=
⎣
⎦
i
Introducing Eq. 9.136 into Eq. 9.135, and pre-multiplying the entire equation by
Φ
()
(
)
()
(
)
T
T
T
T
T
()
T
()
t
t
t
t
Φ M Φη
+
Φ CΦΦC Φη ΦKΦΦK Φη ΦR
−
+
−
=
(9.141)
ae
ae
and defining the modal quantities
T
T
⎫
⎫
M Φ MΦ
C Φ CΦ
K Φ MΦ
=
C Φ C Φ
K Φ K Φ
R Φ R
=
ae
ae
⎪
⎪
⎪
⎪
T
T
=
and
=
(9.142)
⎬
⎪
⎬
⎪
ae
ae
T
T
=
=
⎪
⎭
⎪
⎭
