Environmental Engineering Reference
In-Depth Information
masses, provided these are arranged at the associated centres of gravity, suffices to
determine the first eigenmode with sufficient accuracy. For details see also [8].
4.3.2 The energy method
Only the first eigenmode (fundamental vibration) is significant for the gust response of
a tower. The higher eigenmodes can certainly be very relevant for analysing vortex-
induced transverse vibrations. However, owing to the (generally) low critical wind
speeds v
crit
, these do not play a role in heavyweight towers made from reinforced or
prestressed concrete.
The eigenmode and natural frequency of the fundamental vibration can be determined
for any distribution of mass on the basis of the conservation of energy principle, which
states that the sum of potential energy U and kinetic energyWmust be constant at every
point in time, that is
E
0
¼
U
þ
W
¼
const
:
Replacing the unknown eigenmode y (x) by the deflection curve that would result from
the horizontal effect of the distributed dead load
(x) enables us to determine the kinetic
energy and potential energy approximately by evaluating the following integrals:
g
Z
x
¼
L
1
2
g
U
¼
ðÞ
y
ðÞ
dx
x
¼
0
Z
x
¼
L
g
ðÞ
2
g
2
W
¼
½
y
ðÞ
_
dx
x
¼
0
where g
¼
acceleration due to gravity
¼
9.81m/s
2
These integrals are evaluated with the formula for the fundamental mode
y
ðÞ¼
^
v ð Þ
At rest, only the kinetic energy is available at time t
¼
0:
y
ðÞ
sin
Z
x
¼
L
g
ðÞ
2
g
v ^
2
E
0
¼
½
y
ðÞ
dx
x
¼
0
At maximum deflection, only the potential energy is available at time t
¼p
/(2
v
):
Z
x
¼
L
1
2
g
E
0
¼
ðÞ^
y
ðÞ
dx
x
¼
0
Equating the two expressions gives us the natural angular frequency of the fundamental
mode:
R
x
¼
L
x
¼
0
g
ðÞ^
y
ðÞ
dx
2
v
¼
R
x
¼
L
g
ðÞ
g
2
^
y
ðÞ
½
dx
